diff --git a/-tFIT4oBgHgl3EQf9Cv1/content/tmp_files/2301.11405v1.pdf.txt b/-tFIT4oBgHgl3EQf9Cv1/content/tmp_files/2301.11405v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac5398e179660e764e12ae0198d9022ec35c32d8 --- /dev/null +++ b/-tFIT4oBgHgl3EQf9Cv1/content/tmp_files/2301.11405v1.pdf.txt @@ -0,0 +1,1366 @@ +Revisiting Discriminative Entropy Clustering and its relation to K-means +Zhongwen Zhang Yuri Boykov +University of Waterloo +{z889zhan, yboykov}@uwaterloo.ca +Abstract +Maximization of mutual information between the +model’s input and output is formally related to +“decisiveness” and “fairness” of the softmax pre- +dictions (Bridle et al., 1991), motivating such un- +supervised entropy-based losses for discrimina- +tive neural networks. Recent self-labeling meth- +ods based on such losses represent the state of +the art in deep clustering. However, some impor- +tant properties of entropy clustering are not well- +known or even misunderstood. For example, we +provide a counterexample to prior claims about +equivalence to variance clustering (K-means) and +point out technical mistakes in such theories. +We discuss the fundamental differences between +these discriminative and generative clustering ap- +proaches. Moreover, we show the susceptibility of +standard entropy clustering to narrow margins and +motivate an explicit margin maximization term. +We also propose an improved self-labeling loss; +it is robust to pseudo-labeling errors and enforces +stronger fairness. We develop an EM algorithm +for our loss that is significantly faster than the +standard alternatives. Our results improve the +state-of-the-art on standard benchmarks. +1. Background and motivation +Entropy-based loss functions, e.g. decisiveness and fairness, +were proposed for network training (Bridle et al., 1991; +Krause et al., 2010) and regularization (Grandvalet & Ben- +gio, 2004) and are commonly used for unsupervised and +weakly-supervised classification problems (Ghasedi Dizaji +et al., 2017; Hu et al., 2017; Ji et al., 2019; Asano et al., +2020; Jabi et al., 2021). In particular, the state-of-the-art in +unsupervised classification (Asano et al., 2020; Jabi et al., +2021) is achieved by self-labeling methods using extensions +of decisiveness and fairness. +The community pursues challenging applications of unsu- +pervised classification using deep neural networks, but as +we show in this paper, some important basic properties of +entropy-based clustering are not well-understood or even +examples of linear decision functions over X ∈ R2 +kµ(X) = arg mink ∥X − µk∥ +σv(X) = soft-max(v⊤X) +(a) variance clustering +(b) entropy clustering +Figure 1. Variance vs entropy clustering - binary example (K = 2) +for 2D data {Xi} ⊂ RN (N = 2) comparing linear methods of +similar parametric complexity: (a) K-means [µk ∈ RN] and (b) +entropy clustering based on a linear classifier using K-columns lin- +ear discriminator matrix v = [vk ∈ RN] and soft-max predictions. +Red and green colors in (a) and (b) illustrate optimal linear decision +regions over X ∈ R2 produced by the decision functions kµ(X), +σv(X) for parameters µ and v minimizing two losses: (a) com- +pactness/variance of clusters � +i ∥Xi−µki∥2 where ki = kµ(Xi) +and (b) decisiveness and fairness of predictions � +i H(σi)−H(¯σ) +where H(·) is entropy function, σi = σv(Xi) and ¯σ = avg{σi}. +The decisions kµ(X) in (a) are hard and σv(X) in (b) are soft +(distributions). The softness is visualized by transparency. The +optimal results in (a) and (b) are analyzed in Sec.1.1. The result in +(b) may require margin maximization term, see Fig.3 in Sec.2.1. +understood wrongly. Lapses of clarity call for simple illus- +trative tests, but we did not find any basic low-level exam- +ples of entropy clustering in prior work. We observe that +decisiveness and fairness are general criteria applicable to +any soft-max model, not necessarily deep. Thus, it should +be possible to use them for unsupervised clustering even +with a basic linear classifier using soft-max output. Our +Fig.1(b) shows decision regions for an optimal linear classi- +fier trained for 2D data without any supervision using only +the standard decisiveness & fairness loss. It is natural to +juxtapose such entropy-based linear clustering with the most +popular linear clustering method, K-means, see Fig.1(a). +arXiv:2301.11405v1 [cs.LG] 26 Jan 2023 + +kμ(X)= 0 +kμ(X)=1 +compactness +of clustersv（X）～ onehoto +0v(X) +～ onehot1 +decisiveness & fairness +of predictionsRevisiting Discriminative Entropy Clustering and its relation to K-means +0.0 +0.2 +0.4 +0.6 +0.8 +corruption level +15% +25% +35% +45% +55% +65% +accuracy +forward CE: H(y, +) +forward CE: H(y, +) +reverse CE: H( , y) +Figure 2. Robustness to noisy labels: +reverse cross-entropy +H(σ, y) vs standard cross-entropy H(y, σ). These losses are used +to train VGG-4 network on fully-supervised STL10 data with cor- +rupted labels. The horizontal axis shows the level of corruption, +i.e. percentage η of training images where the correct ground +truth labels were replaced by a random label. We use soft target +distributions ˜y = η ∗ u + (1 − η) ∗ y representing the mixture +of one-hot distribution y for the observed corrupt label and the +uniform distribution u, as recommended in (M¨uller et al., 2019). +The vertical axis shows the test accuracy. Training with reverse +cross-entropy is robust to high levels of labeling errors. +Our paper provides both conceptual and algorithmic contri- +butions briefly summarized below. First, our simple illus- +trative example in Fig.1 works as a counterexample for the +main theoretical claim of a recent TPAMI paper (Jabi et al., +2021) wrongly stating the equivalence between the loss +functions for discriminative entropy clustering and variance +clustering, a.k.a. K-means. We point out specific technical +errors in their proof later in Section 1.2. Our paper also dis- +cusses the susceptibility of standard formulations of entropy +clustering to narrow decision margins and how to avoid +them. We also propose a new formulation of entropy-based +self-labeling loss for clustering. All standard self-labeling +methods replace the entropy H(σ) of soft-max predictions σ +(decisiveness) by cross-entropy H(y, σ) with pseudo-labels +y representing extra hidden variables. In contrast, we pro- +pose reverse cross-entropy H(σ, y) arguably demonstrating +improved robustness to labeling errors, e.g. Fig.2, which +are expected in estimated soft pseudo-labels y. Note that +the second position inside the cross-entropy is natural for +estimated distributions. At the same time, the place of the +first argument is natural for the network predictions σ since +cross-entropy H(σ, y) is an upper-bound approximation for +decisiveness H(σ). We also propose a stronger formula- +tion of the fairness constraint. Our new self-labeling loss +addresses limitations of the standard formulations and it is +amenable to an efficient EM solver derived in our paper. +The rest of this introductory section is organized as fol- +lows. First, Section 1.1 reviews the background and no- +tation for entropy-based clustering with soft-max models. +Then, Section 1.2 reviews the most closely related work +using self-labeling loss formulations of entropy clustering. +We conclude the introduction by summarizing our main +contributions and outlining the structure of the whole paper. +1.1. Background on discriminative entropy clustering +The work of Bridle, Heading, and MacKay from 1991 (Bri- +dle et al., 1991) formulated mutual information (MI) loss +for unsupervised discriminative training of neural networks +using probability-type outputs, e.g. softmax σ : RK → ∆K +mapping K logits lk ∈ R to a point in the probability +simplex ∆K. Such output σ = (σ1, . . . , σK) is often in- +terpreted as a pseudo posterior1 over K classes, where +σk = +exp lk +� +i exp li is a scalar prediction for each class k. +The unsupervised loss proposed in (Bridle et al., 1991) trains +the model predictions to keep as much information about +the input as possible. They derived an estimate of MI as the +difference between the average entropy of the output and +the entropy of the average output +Lmi +:= +−MI(c, X) +≈ +H(σ) − H(σ) +(1) +where c is a random variable representing class prediction, +X represents the input, and the averaging is done over all +input samples {Xi}M +i=1, i.e. over M training examples. +The derivation in (Bridle et al., 1991) assumes that soft- +max represents the distribution Pr(c|X). However, since +softmax is not a true posterior, the right-hand side in (1) +can be seen only as a pseudo MI loss. In any case, (1) +has a clear discriminative interpretation that stands on its +own: H(σ) encourages “fair” predictions with a balanced +support of all categories across the whole training dataset, +while H(σ) encourages confident or “decisive” prediction +at each data point implying that decision boundaries are +away from the training examples (Grandvalet & Bengio, +2004), see Fig.1(b). Our paper refers to unsupervised train- +ing of discriminative soft-max models using predictions’ +entropies, e.g. see (1), as discriminative entropy clustering. +This should not be confused with generative entropy clus- +tering methods where the entropy is used as a measure of +compactness for clusters’ density functions2. +As mentioned earlier, discriminative clustering loss (1) can +be applied to deep or shallow models. For clarity, this paper +distinguishes parameters w of the representation layers of +the network computing features fw(X) ∈ RN for any input +X and the linear classifier parameters v of the output layer +computing K-logit vector v⊤f for any feature f ∈ RN. +The overall network model is defined as +σ(v⊤fw(X)). +(2) +1The term pseudo emphasizes that discriminative training does +not lead to the true Bayesian posteriors, in general. +2E.g., K-means minimizes clusters’ variances. Their logarithms +equal the cluster’s density entropies, assuming Gaussianity. + +Revisiting Discriminative Entropy Clustering and its relation to K-means +A special “shallow” case of the model in (2) is a basic linear +discriminator +σ(v⊤X) +(3) +directly operating on low-level input features f = X. Op- +timization of the loss (1) for the shallow model (3) is done +only over linear classifier parameters v, but the deeper net- +work model (2) is optimized over all network parameters +[v, w]. Typically, this is done via gradient descent or back- +propagation (Rumelhart et al., 1986; Bridle et al., 1991). +Our simple 2D example in Fig.1(b) illustrates “decisive- +ness” and “fairness” losses (1) in the context of a linear +classifier (3) and compares with the standard “compactness” +criterion optimized by K-means, see Fig.1(a). In this “shal- +low” setting both clustering methods are linear and have +similar parametric complexities, about K × N parameters. +K-means (a) finds balanced compact clusters of the least +squared deviations or variance. This can also be interpreted +“generatively”, see (Kearns et al., 1997), as MLE-based fit- +ting of two (isotropic) Gaussian densities, explaining the +failure for non-isotropic clusters in (a). To fix (a) “gener- +atively”, one should use non-isotropic Gaussian densities, +e.g. 2-mode GMM would produce soft clusters similar to +(b). However, this has costly parametric complexity - two +extra covariance matrices to estimate and quadratic decision +boundaries. In contrast, there is no estimation of complex +data density models in (b). Entropy-based loss (1) discrim- +inatively trains a simple linear classifier (3) to produce a +balanced (“fair”) decision boundary away from the data +points (“decisiveness”). Later, we show that the “decisive- +ness” may not be sufficient to avoid narrow decision margins +without an extra margin maximization term, see Fig.3. +In the context of deep models (2), the decision boundaries +between the clusters of training data points {Xi} can be ar- +bitrarily complex since the network learns high-dimensional +non-linear representation map or embedding fw(X). In this +case, loss (1) is optimized with respect to both represen- +tation w and classification v parameters. To avoid overly +complex clustering of the training data and to improve gen- +erality, it is common to use self-augmentation techniques +(Hu et al., 2017). For example, (Ji et al., 2019) maximize +the mutual information between class predictions for input +X and its augmentation counterpart X′ encouraging deep +features invariant to augmentation. +To reduce the complexity of the model, (Krause et al., 2010) +proposed to combine entropy-based loss (1) with regular- +ization of all network parameters interpreted as an isotropic +Gaussian prior on these weights +Lmi+decay += +H(σ) − +H(σ) ++ ∥[v, w]∥2 +c= +H(σ) + KL(σ ∥ u) + ∥[v, w]∥2 +(4) +where +c= represents equality up to an additive constant and +u is a uniform distribution over K classes. Of course, mini- +mizing the norm of network weights as above corresponds +to the weight decay - a common default for network training. +The second formulation of the loss (4) uses KL divergence +motivated in (Krause et al., 2010) by the possibility to gener- +alize “fairness” to balancing with respect to any given target +distribution different from the uniform u. +1.2. Review of entropy-based self-labeling +Optimization of losses (1) or (4) during network training +is mostly done with standard gradient descent or backprop- +agation (Bridle et al., 1991; Krause et al., 2010; Hu et al., +2017). However, the difference between the two entropy +terms implies non-convexity, which makes such losses chal- +lenging for gradient descent. This motivates alternative +formulations and optimization approaches. For example, +it is common to extend the loss by incorporating auxiliary +or hidden variables y representing pseudo-labels for unla- +beled data points X, which are to be estimated jointly with +optimization of the network parameters (Ghasedi Dizaji +et al., 2017; Asano et al., 2020; Jabi et al., 2021). Typically, +such self-labeling approaches to unsupervised network train- +ing iterate optimization of the loss over pseudo-labels and +network parameters, similarly to Lloyd’s algorithm for K- +means or EM algorithm for Gaussian mixtures (Bishop, +2006). While the network parameters are still optimized +via gradient descent, the pseudo-labels can be optimized via +more powerful algorithms. +For example, (Asano et al., 2020) formulate self-labeling +using the following constrained optimization problem with +discrete pseudo-labels y tied to predictions by cross entropy +function H(y, σ) +Lce += +H(y, σ) +s.t. +y ∈ ∆K +0,1 +and +¯y = u (5) +where ∆K +0,1 are one-hot distributions, i.e. corners of the +probability simplex ∆K. Training of the network is done by +minimizing cross entropy H(y, σ), which is convex w.r.t. σ, +assuming fixed pseudo-labels y. Then, model predictions +get fixed and cross-entropy is minimized w.r.t variables y. +Note that cross-entropy H(y, σ) is linear with respect to y, +and its minimum over simplex ∆K is achieved by one-hot +distribution for a class label corresponding to arg max(σ) +at each training example. However, the balancing constraint +¯y = u converts minimization of cross-entropy over all data +points into a non-trivial integer programming problem that +can be approximately solved via optimal transport (Cuturi, +2013). The cross-entropy in (5) encourages the network +predictions σ to approximate the estimated one-hot target +distributions y, which implies the decisiveness. +Self-labeling methods for unsupervised clustering can also +use soft pseudo-labels y ∈ ∆K as target distributions inside + +Revisiting Discriminative Entropy Clustering and its relation to K-means +H(y, σ). In general, soft targets y are commonly used with +cross-entropy functions H(y, σ), e.g. in the context of noisy +labels (Tanaka et al., 2018; Song et al., 2022). Softened +targets y can also assist network calibration (Guo et al., +2017; M¨uller et al., 2019) and improve generalization by +reducing over-confidence (Pereyra et al., 2017). In the con- +text of unsupervised clustering, cross entropy H(y, σ) with +soft pseudo-labels y approximates the decisiveness since +it encourages σ ≈ y implying H(y, σ) ≈ H(y) ≈ H(σ) +where the latter is the decisiveness term in (1). Inspired +by (4), instead of the hard constraint ¯y = u used in (5), +self-labeling losses can represent the fairness using KL di- +vergence KL(¯y ∥ u), as in (Ghasedi Dizaji et al., 2017; Jabi +et al., 2021). In particular, (Jabi et al., 2021) formulates the +following entropy-based self-labeling loss +Lce+kl += +H(y, σ) ++ KL(¯y ∥ u) +(6) +encouraging decisiveness and fairness, as discussed. Simi- +larly to (5), the network parameters in loss (6) are trained +by the standard cross-entropy term. But, optimization over +relaxed pseudo-labels y ∈ ∆K is relatively easy due to the +convexity of KL divergence and linearity of cross-entropy +w.r.t. y. While there is no closed-form solution, the authors +offer an efficient approximate solver for y. Iterating steps +that estimate pseudo-labels y and optimize the model pa- +rameters resembles Lloyd’s algorithm for K-means. The +results in (Jabi et al., 2021) also establish a formal relation +between the loss (6) and the K-means objective. +Our work is closely related to self-labeling loss (6) and +the corresponding ADM algorithm proposed in (Jabi et al., +2021). Their inspiring approach is a good reference point +for our self-labeling loss proposal (10). It also helps to illu- +minate some problems with standard entropy-based losses +and their limited understanding. +In particular, we disagree with the main theoretical claim in +(Jabi et al., 2021) establishing a formal equivalence between +K-means and “regularized” entropy-based clustering with +soft-max models. In fact, our Figure 1 works as a simple +2D counterexample to their claim3. Also, they extend the +entropy-based loss with the classifier regularization ∥v∥2, +but this extra quadratic term is mainly used as a technical +tool in their proof of algebraic similarity between their loss +and the standard K-means loss4. In contrast to related prior +work, we demonstrate that ∥v∥2 is needed in discriminative +entropy clustering for margin maximization. +3The proof of Proposition 2 has a critical technical error - it +ignores normalization for soft-max prediction in their equation (5), +which is hidden via ∝ symbol. Such normalization is critical for +pseudo-posterior models. +4Since they ignore normalization in the softmax prediction, +then ln σ in the cross-entropy H(y, σ) turns into a linear term w.r.t. +logits v⊤x. Adding regularization ∥v∥2 to such loss allows them +to create a quadratic form with respect to v that resembles squared +errors loss in K-means, which is quadratic w.r.t means µk). +1.3. Summary of contributions +Our paper provides conceptual and algorithmic contribu- +tions. First of all, our paper disproves the main theoretical +claim (in the title) of a recent TPAMI paper (Jabi et al., +2021) wrongly stating the equivalence between the stan- +dard K-means loss and entropy-based clustering losses. Our +Figure 1 provides a simple counterexample to the claim, +but we also show specific technical errors in their proof. +Figure 1 helps to motivate entropy clustering with discrimi- +native soft-max models. This general methodology is unde- +servedly little-known to the broader ML community for two +reasons: (A) it was previously presented only in the context +of complex (non-linear, deep) softmax models obfuscating +the basics and (B) because there is confusion even among +the researchers who know about it. Besides clarifying ear- +lier claims about the relation to K-means, we also show that +entropy-based losses may lead to narrow decision margins, +which may contradict one common motivation for decisive- +ness (Grandvalet & Bengio, 2004). Unlike prior entropy +clustering work, we motivate classifier norm regularization +by demonstrating its importance for margin maximization. +We also discuss the limitations of the existing self-labeling +formulations of entropy clustering and propose a new loss, +as well as an efficient pseudo-labeling algorithm. In par- +ticular, we replace standard forward cross-entropy H(y, σ), +where y are soft pseudo-labels, by the reverse cross-entropy +H(σ, y) that is significantly more robust to errors in esti- +mated soft pseudo-labels, see Figures 2 and 4(b). Our for- +mulation of fairness is motivated by a zero-avoiding version +of KL divergence enforcing stronger fairness, see Figure +4(a). We design a new EM algorithm with closed-form EM +steps. In part, our self-labeling formulation of entropy clus- +tering is motivated by its amenability to an efficient EM +solver. Our empirical results improve the state-of-the-art on +many standard benchmarks for deep clustering. +The rest of our paper is organized as follows. Section 2 +motivates our new self-labeling loss for entropy clustering +and derives our EM algorithm. Section 3 compares our ap- +proach with the state-of-the-art entropy clustering methods. +Conclusions are provided in Section 4. +2. Our entropy clustering approach +We are focused on entropy-based losses for clustering with +softmax models that typically enforce “decisiveness” and +“fairness”. First, In Section 2.1 we argue that common for- +mulations of such losses, e.g. (1) or (5), may produce narrow +classification margins. We show that some explicit margin +maximization constraints should be added, which motivates +the classifier norm regularization ∥v∥2 similarly to SVM +methods (Xu et al., 2004). Section 2.2 introduces our new +entropy-based self-labeling loss incorporating strong fair- + +Revisiting Discriminative Entropy Clustering and its relation to K-means +(a) γ = 0 +(b) γ = 0.001 +(c) γ = 0.01 +Figure 3. Margin maximization term γ ∥v∥2 in our loss (7): low- +level clustering results for the softmax linear classifier model (3) +with N = 2 and different weights γ. The dots represent data points. +The optimal softmax clustering of the data and the decision regions +over the whole space are visualized by σ-weighted color trans- +parency, as in Fig.1(b). The “margin” is a weak-confidence “soft” +region around the linear decision boundary lacking color-saturation. +For small γ the classifier can “squeeze” a narrow-margin linear +decision boundary just between the data points, while maintaining +arbitrarily hard “decisiveness” on the data points themselves. +ness and reverse cross-entropy. Section 2.3 derives an effi- +cient EM algorithm for an important sub-problem - estima- +tion of pseudo-labels y. +2.1. Margin maximization via norm regularization +The average entropy term in (1), a.k.a. “decisiveness”, is +recommended in (Grandvalet & Bengio, 2004) as a general +regularization term for semi-supervised problems. They +argue that it produces decision boundaries away from all +training examples, labeled or not. This seems to suggest +larger classification margins, which are good for general- +ization. However, the decisiveness may not automatically +imply large margins if the norm of classifier v in pseudo +posterior models (2, 3) is unrestricted, see Figure 3(a). Tech- +nically, this follows from the same arguments as in (Xu +et al., 2004) where regularization of the classifier norm is +formally related to the margin maximization in the context +of their SVM approach to clustering. +Interestingly, regularization of the norm for all network pa- +rameters [v, w] is motivated in (4) differently (Krause et al., +2010). But, since the classifier parameters v are included, +coincidentally, it also leads to margin maximization. On the +other hand, many MI-based methods (Bridle et al., 1991; +Ghasedi Dizaji et al., 2017; Asano et al., 2020) do not have +regularizer ∥v∥2 in their clustering loss, e.g. see (5). One +may argue that practical implementations of these meth- +ods implicitly benefit from the weight decay, which is om- +nipresent in network training. It is also possible that gradient +descent may implicitly restrict the classifier norm (Soudry +et al., 2018). In any case, since margin maximization is +important for clustering, ideally, it should not be left to +chance. Thus, the norm regularization term ∥v∥2 should be +explicitly present in any clustering loss for pseudo-posterior +models. +We extend MI loss (1) by combining it with the regulariza- +tion of the classifier norm ∥v∥ encouraging margin maxi- +mization, as shown in Figure 3 +Lmi+mm +:= +H(σ) − +H(σ) ++ γ ∥v∥2 +c= +H(σ) + +KL(σ ∥ u) + γ ∥v∥2. +(7) +We note that (Jabi et al., 2021) also extend their entropy- +based loss (6) with the classifier regularization ∥v∥2, but +this extra term is used mainly as a technical tool in relating +their loss (6) to K-means, as detailed in Section 1.3. They +do not discuss its relation to margin maximization. +2.2. Our self-labeling loss function +Below we motivate and put forward some new ideas for +entropy-based clustering losses. First, we observe that the +entropy H(¯σ) in (1) is a weak formulation of the fairness +constraint. Indeed, as clear from an equivalent formulation +in (7), it is enforced by the reverse KL divergence for the +average predictions ¯σ. It assigns a bounded penalty even +for highly unbalanced solutions where ¯σk = 0 for some +k, see the dashed red curve in Fig.4(a). Compare this with +the forward KL divergence KL(u ∥ σ), see the solid red +curve. We propose such zero-avoiding forward version of +KL divergence as a strong fairness loss +Lmi++ +:= +H(σ) + λ KL(u ∥ σ) + γ ∥v∥2. (8) +We will derive our self-labeling loss directly from (8) using +standard splitting technique (Boyd & Vandenberghe, 2004) +to divide optimization of (8) into simpler sub-problems sep- +arating the “decisiveness” and “fairness” terms, as follows. +Introducing auxiliary splitting variables y ∈ ∆K, one for +each training example X, optimization of the loss (8) can +be equivalently written as +min +v,w +H(σ) + γ ∥v∥2 +(decisiveness sub-problem) +min +y +KL(u ∥ y) +(fairness sub-problem) +s.t. +y = σ +(consistency constraint). +This constrained optimization problem can be formulated us- +ing a Lagrangian function enforcing the equality constraint +y = σ via the forward KL divergence for y (motivated +below) +Lour +:= +H(σ) + β KL(σ ∥ y) + λ KL(u ∥ ¯y) + γ ∥v∥2. +(9) +The Lagrangian is optimized with respect to both the net- +work parameters and latent variables y, but we treat the +Lagrange multiplier β as a fixed hyper-parameter. Thus, +the constraint y = σ may not be satisfied exactly and the +Lagrangian (9) works only an approximation of the loss (8). + +Revisiting Discriminative Entropy Clustering and its relation to K-means +(a) strong fairness KL(u∥¯σ) +(b) reverse cross-entropy H(σ, y) +Figure 4. “Forward” vs “reverse”: (a) KL-divergence and (b) cross-entropy. Assuming binary classification K = 2, we can represent all +possible probability distributions as points on the interval [0,1]. The solid curves in (a) illustrate our “strong” fairness constraint, i.e. +the forward KL-divergence KL(u∥¯σ) for the average prediction ¯σ. We show two examples of volumetric prior u1 = (0.9, 0.1) (blue +curve) and u2 = (0.5, 0.5) (red curve). For comparison, the dashed curves represent reverse KL-divergence KL(¯σ∥u) commonly used +for fairness in the prior art. The solid curves in (b) show our reverse cross-entropy H(σ, y) w.r.t the network prediction σ. The dashed +curves show the forward cross-entropy H(y, σ), which is standard in the prior art. The plots in (b) show examples for two fixed estimates +of pseudo-labels y1 = (0.9, 0.1) (blue curves) and y2 = (0.5, 0.5) (red curves). The boundedness of H(σ, y) represents robustness to +errors in y. For example, our loss H(σ, y) turns off the training (sets zero-gradients) when the estimated confidence is highly uncertain, +see y2 = (0.5, 0.5) (solid red). In contrast, the standard loss H(y, σ) trains the network to copy this uncertainty, e.g observe the optimum +σ for the dashed curves. +Also, one can justify hyper-parameter β = 1 empirically, +see Appendix I. Since H(σ) + KL(σ ∥ y) = H(σ ∥ y), we +get the following self-labeling loss formulation +Lour +β = 1 += +H(σ, y) ++ λ KL(u ∥ ¯y) + γ ∥v∥2 +(10) +where the reverse cross entropy H(σ, y) enforces both the +decisiveness and consistency y ≈ σ. +There are some notable differences between our loss (10) +and existing self-labeling losses. For example, consider the +loss (6) proposed in (Jabi et al., 2021). Our loss reverses the +order of both the KL divergence and the cross-entropy terms. +As explained earlier, our version of the KL divergence en- +forces stronger fairness, see Fig.4(a). The reversal of the +cross-entropy is motivated in two ways. First, it makes the +training of network predictions σ robust to errors in noisy +estimates y, see Figure 4(b), as the pseudo-labels y are not +guaranteed to be accurate. On the other hand, compared to +the standard cross-entropy, it enforces stronger consistency +of y with the predictions σ, which work as target distri- +butions for y. Thus, w.r.t. pseudo-labels y, our loss (10) +enforces stronger fairness and stronger consistency y ≈ σ. +The corresponding well-constrained optimization problem +for y allows an efficient EM solver derived in Section 2.3. +2.3. Our EM algorithm for estimating pseudo-labels +To optimize (10) with respect to y, basic Newton’s meth- +ods (Kelley, 1995) can be applied. Although the overall +convergence rate of such second-order methods is fast, the +calculation or approximation of the inverse Hessian is com- +putationally costly as shown in Table 1. This motivates us +to derive the more efficient expectation-maximization (EM) +algorithm (Bishop, 2006) for optimizing y as below. +Here we present a new efficient algorithm for optimizing +our discriminative entropy-based loss (10) with respect to +the pseudo-labels y when the model predictions are fixed, +i.e. σ and v. Using the variational inference (Bishop, +2006), we derive a new EM algorithm introducing a dif- +ferent type of latent variables, K distributions Sk ∈ ∆M +representing normalized support for each cluster over M +data points. We refer to each vector Sk as a normalized +cluster k. Note the difference with distributions represented +by pseudo-posteriors y ∈ ∆K showing support for each +class at a given data point. Since we explicitly use indi- +vidual data points below, we will start to carefully index +them by i ∈ {1, . . . , M}. Thus, we will use yi ∈ ∆K and +σi ∈ ∆K. Individual components of distribution Sk ∈ ∆M +corresponding to data point i will be denoted by scalar Sk +i . +First, we expand our loss (10) introducing the latent vari- +ables Sk ∈ ∆M +Lour +c= +H(σ, y) + λ H(u, ¯y) + γ ∥v∥2 +(11) += +H(σ, y) − λ +� +k +uk ln +� +i +Sk +i +yk +i +Sk +i M + γ ∥v∥2 +≤ +H(σ, y) − λ +� +k +� +i +ukSk +i ln +yk +i +Sk +i M + γ ∥v∥2 +(12) +Due to the convexity of negative log, we apply Jensen’s +inequality to derive an upper bound, i.e. (12), to Lour. Such + +Revisiting Discriminative Entropy Clustering and its relation to K-means +a bound becomes tight when: +Estep : +Sk +i = +yk +i +� +j yk +j +(13) +Then, we fix Sk +i as (13) and solve the Lagrangian of (12) +with simplex constraint to update y as: +Mstep : +yk +i = +σk +i + λMukSk +i +1 + λM � +c ucSc +i +(14) +We run these two steps until convergence with respect to +some predefined tolerance. Note that the minimum y is +guaranteed to be globally optimal since (11) is convex w.r.t. +y (Appendix. A). The empirical convergence rate is within +15 steps on MNIST. The comparison of computation speed +on synthetic data is shown in Table 1. While the number +of iterations to convergence is roughly the same as New- +ton’s methods, our EM algorithm is much faster in terms +of running time and is extremely easy to implement using +the highly optimized built-in functions from the standard +PyTorch library that supports GPU. +number of iterations +running time in sec. +(to convergence) +(to convergence) +K2 +K20 +K200 +K2 +K20 +K200 +Newton +3 +3 +4 +2.8e−2 +3.3e−2 +1.7e−1 +EM +2 +2 +2 +9.9e−4 +2.0e−3 +4.0e−3 +Table 1. Comparison of our EM algorithm to Newton’s methods +(Kelley, 1995). K2, K20 and K200 stand for the number of classes. +Inspired by (Springenberg, 2015; Hu et al., 2017), we also +adapted our EM algorithm to allow for updating y within +each batch. In fact, the mini-batch approximation of (11) is +an upper bound. Considering the first two terms of (11), we +can use Jensen’s inequality to get: +H(σ, y) + λ H(u, ¯y) +≤ +EB[HB(σ, y) + λ H(u, ¯yB)] +(15) +where B is the batch randomly sampled from the whole +dataset. Now, we can apply our EM algorithm to update +y in each batch, which is even more efficient. Compared +to other methods (Ghasedi Dizaji et al., 2017; Asano et al., +2020; Jabi et al., 2021) which also use the auxiliary vari- +able y, we can efficiently update y on the fly while they +only update once or just a few times per epoch due to the +inefficiency to update y for the whole dataset per iteration. +Interestingly, we found that it is actually important to update +y on the fly, which makes convergence faster and improves +the performance significantly (Appendix. C). We use this +“batch version” EM throughout all the experiments. Our full +algorithm for the loss (10) is summarized in Appendix. B. +3. Experimental results +Our experiments start from pure clustering on fixed features +to joint clustering with feature learning. We have also com- +pared different losses on weakly-supervised classification. +Note that our goal is comparing different losses together +with their own optimization algorithms, thus we keeping +our experimental setup as simple as possible to reduce the +distraction factors for analysis. +Dataset +For the clustering problem, we use four standard +benchmarks: MNIST (Lecun et al., 1998), CIFAR10/100 +(Torralba et al., 2008) and STL10 (Coates et al., 2011). The +training and test data are the same. As for the weakly- +supervised setting, we conduct experiments on CIFAR10 +and STL10. We split the data into training and test sets as +suggested by the instructions for the datasets. +Evaluation +As for the evaluation on clustering, we set the +number of clusters to the number of ground-truth categories +and we adopt the standard method (Kuhn, 1955) by finding +the best one-to-one mapping between clusters and labels. +We use the accuracy as the measure for both unsupervised +and weakly-supervised settings while the latter calculates +the accuracy on the test set. +3.1. Clustering with fixed features +In this section, we test our loss (10) with a simple linear +classifier on MNIST (Lecun et al., 1998) by using the (fixed) +original features of the images. We compare it to K-means +and (4). The detailed experimental settings can be found +in Appendix. F. In Table 2, we report the mean accuracy +K-means +MI (Bridle et al., 1991; Krause et al., 2010) +Our +accuracy +53.2% (Hu et al., 2017) +60.2%(3.7) +60.8%(1.1) +Table 2. Comparison of different losses on MNIST without learn- +ing features. +and standard deviation. Note that discriminative clustering +methods perform consistently much better than K-means +(≥ 7%) while our approach achieves a bit higher accuracy +but is more robust. Also, a low-level ablation study can be +found in Appendix. F. +STL10 +CIFAR10 +CIFAR100-20 +MNIST +Kmeans +85.20%(5.9) +67.78%(4.6) +42.99%(1.3) +47.62%(2.1) +MI-GD (Bridle et al., 1991; Krause et al., 2010) +89.56%(6.4) +72.32%(5.8) +43.59%(1.1) +52.92%(3.0) +MI-ADM (Jabi et al., 2021) +81.28%(7.2) +56.07%(5.5) +36.70%(1.1) +47.15%(3.7) +SeLa (Asano et al., 2020) +90.33%(4.8) +63.31%(3.7) +40.74%(1.1) +52.38%(5.2) +Our +92.2%(6.2) +73.48%(6.2) +43.8%(1.1) +58.2%(3.1) +Table 3. Comparison of different methods on clustering with fixed +features extracted from Resnet-50. The numbers are the average +accuracy and the standard deviation over trials. +Besides using low-level features, we also compare our + +Revisiting Discriminative Entropy Clustering and its relation to K-means +method against the state-of-the-art methods using fixed deep +features generated by large models such as Resnet-50 (He +et al., 2016). We still use a one-layer linear classifier for +all loss functions except for Kmeans. The coefficients γ for +the margin maximization terms are set to 0.001, 0.02, 0.009, +and 0.02 for MNIST, CIFAR10, CIFAR100 and STL10 re- +spectively. As illustrated in Figure 3, γ is important for +the optimal decision boundary, especially when features are +fixed. If we jointly learn the representation and cluster the +data, we observed that the results are less sensitive to γ. +Note that this backbone network could be trained together +with the linear classifier even from the scratch. However, we +found that the clustering loss itself is not enough to generate +reasonable features for the backbone network. Thus, we +keep the backbone network fixed and only train the linear +classifier using different clustering loss functions. +3.2. Joint clustering and representation learning +In this section, we train a deep network to jointly learn the +features and cluster the data on the four standard benchmark +datasets: STL10 (Coates et al., 2011), CIFAR10/CIFAR100 +(Torralba et al., 2008) and MNIST (Lecun et al., 1998). +The only extra standard technique we add here is the self- +augmentation, following (Hu et al., 2017; Ji et al., 2019; +Asano et al., 2020). This technique is important for en- +forcing neural networks to learn augmentation-invariant fea- +tures, which are often semantically meaningful. While (Ji +et al., 2019) designed their loss directly based on such tech- +nique, our loss and (Krause et al., 2010; Asano et al., 2020; +Jabi et al., 2021) are more general for clustering without +any guarantee to generate semantic clusters. Thus, for fair +comparison and more reasonable results, we combine this +augmentation technique into network training. The exper- +imental settings and more detailed discussion are given in +Appendix. G. From Table 4, it can be seen that our approach +consistently achieves the best or the most competitive results +in terms of accuracy. +STL10 +CIFAR10 +CIFAR100-20 +MNIST +MI-D⋆ (Hu et al., 2017) +25.28%(0.5) +21.4%(0.5) +14.39%(0.7) +92.90%(6.3) +IIC⋆ (Ji et al., 2019) +24.12%(1.7) +21.3%(1.4) +12.58%(0.6) +82.51%(2.3) +SeLa§ (Asano et al., 2020) +23.99%(0.9) +24.16%(1.5) +15.34%(0.3) +52.86%(1.9) +MI-ADM§ (Jabi et al., 2021) +17.37%(0.9) +17.27%(0.6) +11.02%(0.5) +17.75%(1.3) +Our⋆,§ +25.33%(1.4) +24.16%(0.8) +15.09%(0.5) +93.58%(4.8) +Table 4. Quantitative results of accuracy for unsupervised cluster- +ing methods. We only use the 20 coarse categories for CIFAR100. +We reuse the code published by (Ji et al., 2019; Asano et al., 2020; +Hu et al., 2017) and implemented the optimization for loss of (Jabi +et al., 2021) according to the paper. ⋆: all variables are updated for +each batch. §: loss formula has pseudo-label. +Note that we only use a very small network architecture +(VGG4) here since we observed that more complex archi- +tectures require more additional techniques to obtain rea- +sonable results. For example, Ji et.al. (Ji et al., 2019) also +use auxiliary over-clustering, multiple heads, and more data +to obtain high numbers on STL10 with ResNet structure. +To emphasize on the effects of different loss functions, we +keep the experimental settings as simple as possible. +3.3. Weakly-supervised classification +We also test different methods over different levels of (very) +weak supervision on STL10. In Table 5, we can see that +our approach still shows very competitive results, especially +with weaker supervision. More details are given in Ap- +pendix. H including another test on CIFAR 10. +0.1 +0.05 +0.01 +Only seeds +40.27% +36.26% +26.1% ++ MI-D (Hu et al., 2017) +47.39% +40.73% +26.54% ++ IIC (Ji et al., 2019) +44.73% +33.6% +26.17% ++ SeLa (Asano et al., 2020) +44.84% +36.4% +25.08% ++ MI-ADM (Jabi et al., 2021) +45.83% +40.41% +25.79% ++ Our +47.20% +41.13% +26.76% +Table 5. Quantitative results for weakly-supervised classification +on STL10. 0.1, 0.05 and 0.01 correspond to different ratios of +labels used for supervision. “Only seeds” means that we only use +standard cross-entropy loss on labeled training data. +4. Conclusions +Our paper proposed a new self-labeling algorithm for dis- +criminative entropy clustering, but we also clarify several +important conceptual properties of this general methodol- +ogy. For example, we disproved a theoretical claim in a +recent TPAMI paper stating the equivalence between vari- +ance clustering (K-means) and discriminative entropy-based +clustering. We also demonstrate that standard formulations +of entropy clustering losses may lead to narrow decision mar- +gins. Unlike prior work on discriminative entropy clustering, +we show that classifier norm regularization is important for +margin maximization. +We also discussed several limitations of the existing self- +labeling formulations of entropy clustering and propose +a new loss addressing such limitations. In particular, we +replace the standard (forward) cross-entropy by the reverse +cross-entropy that we show is significantly more robust to +errors in estimated soft pseudo-labels. Our loss also uses +a strong formulation of the fairness constraint motivated +by a zero-avoiding version of KL divergence. 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In +International Conference on Learning Representations, +2015. + +Revisiting Discriminative Entropy Clustering and its relation to K-means +Tanaka, D., Ikami, D., Yamasaki, T., and Aizawa, K. Joint +optimization framework for learning with noisy labels. In +Proceedings of the IEEE conference on computer vision +and pattern recognition, pp. 5552–5560, 2018. +Torralba, A., Fergus, R., and Freeman, W. T. 80 million tiny +images: A large data set for nonparametric object and +scene recognition. IEEE transactions on pattern analysis +and machine intelligence, 30(11):1958–1970, 2008. +Xu, L., Neufeld, J., Larson, B., and Schuurmans, D. Maxi- +mum margin clustering. In Saul, L., Weiss, Y., and Bottou, +L. (eds.), Advances in Neural Information Processing Sys- +tems, volume 17. MIT Press, 2004. + +Revisiting Discriminative Entropy Clustering and its relation to K-means +A. Proof +Lemma A.1. Given fixed σi ∈ ∆K where i ∈ {1, ..., M} +and u ∈ ∆K, the objective +E(y) = − β +M +� +i +� +k +σk +i ln yk +i − λ +� +k +uk ln +� +i yk +i +M +is convex for y, where yi ∈ ∆K. +Proof. First, we rewrite E(y) +E(y) = − +� +k +� +β +M +� +i +σk +i ln yk +i + λuk ln +� +i yk +i +M +� +:= − +� +k +fk(yk) +(16) +Next, we prove that fk : RM +(0,1) → R is concave based on +the definition of concavity(Boyd & Vandenberghe, 2004) +for any k ∈ {1, ..., K}. Considering x = (1 − α)x1 + αx2 +where x1, x2 ∈ RM +(0,1) and α ∈ [0, 1], we have +fk(x) = +β +M +� +i +σk +i ln ((1 − α)x1i + αx2i)+ +λuk ln +� +i ((1 − α)x1i + αx2i) +M +≥ β +M +� +i +(1 − α)σk +i ln x1i + ασk +i ln x2i ++ λuk +� +(1 − α) ln +� +i x1i +M ++ α ln +� +i x2i +M +� += (1 − α)fk(x1) + αfk(x2) +The inequality uses Jensen’s inequality. Now that fk is +proved to be concave, −fk will be convex. Then E(y) +can be easily proved to be convex using the definition of +convexity with the similar steps above. +B. Our Algorithm +C. Loss Curve +D. Network Architecture +The network structure is VGG-style and adapted from (Ji +et al., 2019). +E. Dataset Summary +Table 7 indicates the number of (training) data and the input +size of each image for the unsupervised clustering. Training +and test sets are the same. +As for weakly-supervised classification on STL10, we use +5000 images for training and 8000 images for testing. We +Algorithm 1 Optimization for our loss +Input +:network parameters [v, w] and dataset +Output :network parameters [v∗, w∗] +for each epoch do +for each iteration do +Initialize y by the network output at current stage as +a warm start while not convergent do +Sk +i = +yk +i +� +j yk +j +yk +i = +σk +i +λMukSk +i +1+λM � +c ucSc +i +end +Update [w, v] using loss HB(σ, y) + γ ∥v∥2 via +stochastic gradient descent +end +end +Figure 5. Loss (10) curves for different update setting on y. This +is generated with just a linear classifier on MNIST. We use the +same initialization and run both for 50 epochs. The gray line has +an accuracy of 52.35% while the yellow one achieves 63%. +only keep a certain percentage of ground-truth labels for +each class of training data. The accuracy is calculated on +test set by comparing the hard-max of prediction to the +ground-truth. +F. Low-level Clustering +As for the experiments on MNIST (Lecun et al., 1998), we +transform the original image values linearly into [−1, 1] and +use the flattened images as input features. Note that here +we only use a linear classifier without training any features. +We employ stochastic gradient descent with learning rate +0.07 to update v in (4) and (10). We use the same (random) +intialization for both losses and run each 6 times up to 50 +epochs per run. We use 250 for batch size. We set γ = 0.01 +for both and use λ = 100 for (10) and λ = 1.3 for (4). +We fix the hyperparameter values for (9) and (4) +throughout the whole experimental sections. +We also conducted an ablation study on toy examples as +shown in Figure. 6. We use the normalized X-Y coordinates +of the data points as the input. We can see that each part +of our loss is necessary for obtaining a good result. Note + +loss: + update y on whole dataset once per epoch + update y on batch data per iteration +232 +231.5 +231 +Iteration +0 +2k +4k +6k +8k +10k +12kRevisiting Discriminative Entropy Clustering and its relation to K-means +Grey(28x28x1) +RGB(32x32x3) +RGB(96x96x3) +1xConv(5x5,s=1,p=2)@64 +1xConv(5x5,s=1,p=2)@32 +1xConv(5x5,s=2,p=2)@128 +1xMaxPool(2x2,s=2) +1xMaxPool(2x2,s=2) +1xMaxPool(2x2,s=2) +1xConv(5x5,s=1,p=2)@128 +1xConv(5x5,s=1,p=2)@64 +1xConv(5x5,s=2,p=2)@256 +1xMaxPool(2x2,s=2) +1xMaxPool(2x2,s=2) +1xMaxPool(2x2,s=2) +1xConv(5x5,s=1,p=2)@256 +1xConv(5x5,s=1,p=2)@128 +1xConv(5x5,s=2,p=2)@512 +1xMaxPool(2x2,s=2) +1xMaxPool(2x2,s=2) +1xMaxPool(2x2,s=2) +1xConv(5x5,s=1,p=2)@512 +1xConv(5x5,s=1,p=2)@256 +1xConv(5x5,s=2,p=2)@1024 +1xLinear(512x3x3,K) +1xLinear(256x4x4,K) +1xLinear(1024x1x1,K) +Table 6. Network architecture summary. s: stride; p: padding; K: +number of clusters. The first column is used on MNIST (Lecun +et al., 1998); the second one is used on CIFAR10/100 (Torralba +et al., 2008); the third one is used on STL10 (Coates et al., 2011). +Batch normalization is also applied after each Conv layer. ReLu is +adopted for non-linear activation function. +STL10 +CIFAR10 +CIFAR100-20 +MNIST +13000 +60000 +60000 +70000 +96x96x3 +32x32x3 +32x32x3 +28x28x1 +Table 7. Dataset summary for unsupervised clustering. +that, in Figure 6 (a), (c) of 3-label case, the clusters formed +are the same, but the decision boundaries which implies the +generalization are different. This emphasizes the importance +of including L2 norm of v to enforce maximum margin for +better generalization. +2 clusters +3 clusters +(a) γ = 0 +(b) λ = 0 +(c) full setting +Figure 6. “Shallow” ablation study on toy examples. +G. Deep Clustering +We add deep neural networks for learning features while +doing the clustering simultaneously. +We use four stan- +dard benchmark datasets: STL10 (Coates et al., 2011), CI- +FAR10/CIFAR100 (Torralba et al., 2008) and MNIST (Le- +cun et al., 1998). As for the architectures, we followed (Ji +et al., 2019) to use VGG11-like network structures whereas +we use it for both gray-scale and RGB images with some +adjustments as shown in Appendix. D. +We achieved the self-augmentation by setting σi += +Et[σ(v⊤fw(t(Xi))]. For each image, we generate two aug- +mentations sampled from “horizontal flip”, “rotation” and +“color distortion”. +We use Adam (Kingma & Ba, 2015) with learning rate 1e−4 +for optimizing the network parameters. We set batch size to +250 for CIFAR10, CIFAR100 and MNIST, and we use 160 +for STL10. In Table 4, we report the mean accuracy and Std +from 6 runs with different initializations while we use the +same initialization for all methods in each run. We still use +50 epochs for each run and all methods reach convergence +within 50 epochs. +As for other methods in Table 4, MI-D has the most com- +parable results to us, in part because our loss can be seen +as an approximation to the MI and we both update all vari- +ables per batch. SeLa achieves relatively better results on +other three datasets than MNIST, because it enforces a hard +constraint on “fairness” and MNIST is the only one out of +four sets that is not exactly balanced. In real world, the data +we collect is mostly not exactly balanced. This could be the +reason why such method is better for the unsupervised rep- +resentation learning where over-clustering can be employed +and real clusters become less important. MI-ADM only up- +dates the pseudo-labels once per epoch, thus easily leading +the network towards a trap of local minimum created by the +incorrect pseudo-labels through the forward cross-entropy +loss as illustrated in Figure 4. +H. Weakly-supervised Clustering +We use the same experimental settings as that in unsuper- +vised clustering except for two points: 1. We add cross- +entropy loss on labelled data; 2. We separate the training +data from test data while we use all the data for training and +test in unsupervised clustering. +While MI-ADM is the worst according to Table 4, it is +improved significantly in weakly-supervised setting. This +might be a sign that the advantage of more frequent update +on variables in unsupervised clustering is waning since the +seeds help the network keeping away from some bad local +minima. +Below is the result on CIFAR 10. +0.1 +0.05 +0.01 +Only seeds +58.77% +54.27% +39.01% ++ MI-D +65.54% +61.4% +46.97% ++ IIC +66.5% +61.17% +47.21% ++ SeLa +61.5% +58.35% +47.19% ++ MI-ADM +62.51% +57.05% +45.91% ++ Our +66.17% +61.59% +47.22% + +Revisiting Discriminative Entropy Clustering and its relation to K-means +I. Hyperparameter β +Below is an empirical justification for setting hyper- +parameter β = 1 in the loss (9). The first two terms in +(9) can be written as (1 − β)H(σ) + βH(σ, y). If β > 1 +then the negative entropy pushes predictions σ away from +one-hot solutions weakening the decisiveness. On the other +hand, if β < 1 then the loss is non-convex w.r.t σ that may +trap gradient descent in bad local minima, as illustrated by +the plots for y = (0.9, 0.1) in Figure 7 +Figure 7. (1 − β)H(σ) + βH(σ, y) + diff --git a/-tFIT4oBgHgl3EQf9Cv1/content/tmp_files/load_file.txt b/-tFIT4oBgHgl3EQf9Cv1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b9493a89a5a77a554dd743b9b7335a66e98fcb0 --- /dev/null +++ b/-tFIT4oBgHgl3EQf9Cv1/content/tmp_files/load_file.txt @@ -0,0 +1,1008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf,len=1007 +page_content='Revisiting Discriminative Entropy Clustering and its relation to K-means Zhongwen Zhang Yuri Boykov University of Waterloo {z889zhan, yboykov}@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='ca Abstract Maximization of mutual information between the model’s input and output is formally related to “decisiveness” and “fairness” of the softmax pre- dictions (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991), motivating such un- supervised entropy-based losses for discrimina- tive neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Recent self-labeling meth- ods based on such losses represent the state of the art in deep clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, some impor- tant properties of entropy clustering are not well- known or even misunderstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, we provide a counterexample to prior claims about equivalence to variance clustering (K-means) and point out technical mistakes in such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We discuss the fundamental differences between these discriminative and generative clustering ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Moreover, we show the susceptibility of standard entropy clustering to narrow margins and motivate an explicit margin maximization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also propose an improved self-labeling loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' it is robust to pseudo-labeling errors and enforces stronger fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We develop an EM algorithm for our loss that is significantly faster than the standard alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our results improve the state-of-the-art on standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Background and motivation Entropy-based loss functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' decisiveness and fairness, were proposed for network training (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010) and regularization (Grandvalet & Ben- gio, 2004) and are commonly used for unsupervised and weakly-supervised classification problems (Ghasedi Dizaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In particular, the state-of-the-art in unsupervised classification (Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) is achieved by self-labeling methods using extensions of decisiveness and fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The community pursues challenging applications of unsu- pervised classification using deep neural networks, but as we show in this paper, some important basic properties of entropy-based clustering are not well-understood or even examples of linear decision functions over X ∈ R2 kµ(X) = arg mink ∥X − µk∥ σv(X) = soft-max(v⊤X) (a) variance clustering (b) entropy clustering Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Variance vs entropy clustering - binary example (K = 2) for 2D data {Xi} ⊂ RN (N = 2) comparing linear methods of similar parametric complexity: (a) K-means [µk ∈ RN] and (b) entropy clustering based on a linear classifier using K-columns lin- ear discriminator matrix v = [vk ∈ RN] and soft-max predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Red and green colors in (a) and (b) illustrate optimal linear decision regions over X ∈ R2 produced by the decision functions kµ(X), σv(X) for parameters µ and v minimizing two losses: (a) com- pactness/variance of clusters � i ∥Xi−µki∥2 where ki = kµ(Xi) and (b) decisiveness and fairness of predictions � i H(σi)−H(¯σ) where H(·) is entropy function, σi = σv(Xi) and ¯σ = avg{σi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The decisions kµ(X) in (a) are hard and σv(X) in (b) are soft (distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The softness is visualized by transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The optimal results in (a) and (b) are analyzed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The result in (b) may require margin maximization term, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' understood wrongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Lapses of clarity call for simple illus- trative tests, but we did not find any basic low-level exam- ples of entropy clustering in prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We observe that decisiveness and fairness are general criteria applicable to any soft-max model, not necessarily deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, it should be possible to use them for unsupervised clustering even with a basic linear classifier using soft-max output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1(b) shows decision regions for an optimal linear classi- fier trained for 2D data without any supervision using only the standard decisiveness & fairness loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' It is natural to juxtapose such entropy-based linear clustering with the most popular linear clustering method, K-means, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='11405v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='LG] 26 Jan 2023 kμ(X)= 0 kμ(X)=1 compactness of clustersv（X）～ onehoto 0v(X) ～ onehot1 decisiveness & fairness of predictionsRevisiting Discriminative Entropy Clustering and its relation to K-means 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8 corruption level 15% 25% 35% 45% 55% 65% accuracy forward CE: H(y, ) forward CE: H(y, ) reverse CE: H( , y) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Robustness to noisy labels: reverse cross-entropy H(σ, y) vs standard cross-entropy H(y, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' These losses are used to train VGG-4 network on fully-supervised STL10 data with cor- rupted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The horizontal axis shows the level of corruption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' percentage η of training images where the correct ground truth labels were replaced by a random label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use soft target distributions ˜y = η ∗ u + (1 − η) ∗ y representing the mixture of one-hot distribution y for the observed corrupt label and the uniform distribution u, as recommended in (M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The vertical axis shows the test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Training with reverse cross-entropy is robust to high levels of labeling errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our paper provides both conceptual and algorithmic contri- butions briefly summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, our simple illus- trative example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1 works as a counterexample for the main theoretical claim of a recent TPAMI paper (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) wrongly stating the equivalence between the loss functions for discriminative entropy clustering and variance clustering, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We point out specific technical errors in their proof later in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our paper also dis- cusses the susceptibility of standard formulations of entropy clustering to narrow decision margins and how to avoid them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also propose a new formulation of entropy-based self-labeling loss for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' All standard self-labeling methods replace the entropy H(σ) of soft-max predictions σ (decisiveness) by cross-entropy H(y, σ) with pseudo-labels y representing extra hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In contrast, we pro- pose reverse cross-entropy H(σ, y) arguably demonstrating improved robustness to labeling errors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2, which are expected in estimated soft pseudo-labels y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that the second position inside the cross-entropy is natural for estimated distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' At the same time, the place of the first argument is natural for the network predictions σ since cross-entropy H(σ, y) is an upper-bound approximation for decisiveness H(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also propose a stronger formula- tion of the fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our new self-labeling loss addresses limitations of the standard formulations and it is amenable to an efficient EM solver derived in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The rest of this introductory section is organized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1 reviews the background and no- tation for entropy-based clustering with soft-max models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Then, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2 reviews the most closely related work using self-labeling loss formulations of entropy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We conclude the introduction by summarizing our main contributions and outlining the structure of the whole paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Background on discriminative entropy clustering The work of Bridle, Heading, and MacKay from 1991 (Bri- dle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991) formulated mutual information (MI) loss for unsupervised discriminative training of neural networks using probability-type outputs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' softmax σ : RK → ∆K mapping K logits lk ∈ R to a point in the probability simplex ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Such output σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' , σK) is often in- terpreted as a pseudo posterior1 over K classes, where σk = exp lk � i exp li is a scalar prediction for each class k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The unsupervised loss proposed in (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991) trains the model predictions to keep as much information about the input as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' They derived an estimate of MI as the difference between the average entropy of the output and the entropy of the average output Lmi := −MI(c, X) ≈ H(σ) − H(σ) (1) where c is a random variable representing class prediction, X represents the input, and the averaging is done over all input samples {Xi}M i=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' over M training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The derivation in (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991) assumes that soft- max represents the distribution Pr(c|X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, since softmax is not a true posterior, the right-hand side in (1) can be seen only as a pseudo MI loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In any case, (1) has a clear discriminative interpretation that stands on its own: H(σ) encourages “fair” predictions with a balanced support of all categories across the whole training dataset, while H(σ) encourages confident or “decisive” prediction at each data point implying that decision boundaries are away from the training examples (Grandvalet & Bengio, 2004), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our paper refers to unsupervised train- ing of discriminative soft-max models using predictions’ entropies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' see (1), as discriminative entropy clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This should not be confused with generative entropy clus- tering methods where the entropy is used as a measure of compactness for clusters’ density functions2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As mentioned earlier, discriminative clustering loss (1) can be applied to deep or shallow models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For clarity, this paper distinguishes parameters w of the representation layers of the network computing features fw(X) ∈ RN for any input X and the linear classifier parameters v of the output layer computing K-logit vector v⊤f for any feature f ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The overall network model is defined as σ(v⊤fw(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (2) 1The term pseudo emphasizes that discriminative training does not lead to the true Bayesian posteriors, in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', K-means minimizes clusters’ variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Their logarithms equal the cluster’s density entropies, assuming Gaussianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Revisiting Discriminative Entropy Clustering and its relation to K-means A special “shallow” case of the model in (2) is a basic linear discriminator σ(v⊤X) (3) directly operating on low-level input features f = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Op- timization of the loss (1) for the shallow model (3) is done only over linear classifier parameters v, but the deeper net- work model (2) is optimized over all network parameters [v, w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Typically, this is done via gradient descent or back- propagation (Rumelhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our simple 2D example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1(b) illustrates “decisive- ness” and “fairness” losses (1) in the context of a linear classifier (3) and compares with the standard “compactness” criterion optimized by K-means, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In this “shal- low” setting both clustering methods are linear and have similar parametric complexities, about K × N parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' K-means (a) finds balanced compact clusters of the least squared deviations or variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This can also be interpreted “generatively”, see (Kearns et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1997), as MLE-based fit- ting of two (isotropic) Gaussian densities, explaining the failure for non-isotropic clusters in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' To fix (a) “gener- atively”, one should use non-isotropic Gaussian densities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2-mode GMM would produce soft clusters similar to (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, this has costly parametric complexity - two extra covariance matrices to estimate and quadratic decision boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In contrast, there is no estimation of complex data density models in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Entropy-based loss (1) discrim- inatively trains a simple linear classifier (3) to produce a balanced (“fair”) decision boundary away from the data points (“decisiveness”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Later, we show that the “decisive- ness” may not be sufficient to avoid narrow decision margins without an extra margin maximization term, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In the context of deep models (2), the decision boundaries between the clusters of training data points {Xi} can be ar- bitrarily complex since the network learns high-dimensional non-linear representation map or embedding fw(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In this case, loss (1) is optimized with respect to both represen- tation w and classification v parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' To avoid overly complex clustering of the training data and to improve gen- erality, it is common to use self-augmentation techniques (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) maximize the mutual information between class predictions for input X and its augmentation counterpart X′ encouraging deep features invariant to augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' To reduce the complexity of the model, (Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010) proposed to combine entropy-based loss (1) with regular- ization of all network parameters interpreted as an isotropic Gaussian prior on these weights Lmi+decay = H(σ) − H(σ) + ∥[v, w]∥2 c= H(σ) + KL(σ ∥ u) + ∥[v, w]∥2 (4) where c= represents equality up to an additive constant and u is a uniform distribution over K classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Of course, mini- mizing the norm of network weights as above corresponds to the weight decay - a common default for network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The second formulation of the loss (4) uses KL divergence motivated in (Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010) by the possibility to gener- alize “fairness” to balancing with respect to any given target distribution different from the uniform u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Review of entropy-based self-labeling Optimization of losses (1) or (4) during network training is mostly done with standard gradient descent or backprop- agation (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, the difference between the two entropy terms implies non-convexity, which makes such losses chal- lenging for gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This motivates alternative formulations and optimization approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, it is common to extend the loss by incorporating auxiliary or hidden variables y representing pseudo-labels for unla- beled data points X, which are to be estimated jointly with optimization of the network parameters (Ghasedi Dizaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Typically, such self-labeling approaches to unsupervised network train- ing iterate optimization of the loss over pseudo-labels and network parameters, similarly to Lloyd’s algorithm for K- means or EM algorithm for Gaussian mixtures (Bishop, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' While the network parameters are still optimized via gradient descent, the pseudo-labels can be optimized via more powerful algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, (Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020) formulate self-labeling using the following constrained optimization problem with discrete pseudo-labels y tied to predictions by cross entropy function H(y, σ) Lce = H(y, σ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' y ∈ ∆K 0,1 and ¯y = u (5) where ∆K 0,1 are one-hot distributions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' corners of the probability simplex ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Training of the network is done by minimizing cross entropy H(y, σ), which is convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' σ, assuming fixed pseudo-labels y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Then, model predictions get fixed and cross-entropy is minimized w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t variables y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that cross-entropy H(y, σ) is linear with respect to y, and its minimum over simplex ∆K is achieved by one-hot distribution for a class label corresponding to arg max(σ) at each training example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, the balancing constraint ¯y = u converts minimization of cross-entropy over all data points into a non-trivial integer programming problem that can be approximately solved via optimal transport (Cuturi, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The cross-entropy in (5) encourages the network predictions σ to approximate the estimated one-hot target distributions y, which implies the decisiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Self-labeling methods for unsupervised clustering can also use soft pseudo-labels y ∈ ∆K as target distributions inside Revisiting Discriminative Entropy Clustering and its relation to K-means H(y, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In general, soft targets y are commonly used with cross-entropy functions H(y, σ), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' in the context of noisy labels (Tanaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Softened targets y can also assist network calibration (Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' M¨uller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) and improve generalization by reducing over-confidence (Pereyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In the con- text of unsupervised clustering, cross entropy H(y, σ) with soft pseudo-labels y approximates the decisiveness since it encourages σ ≈ y implying H(y, σ) ≈ H(y) ≈ H(σ) where the latter is the decisiveness term in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Inspired by (4), instead of the hard constraint ¯y = u used in (5), self-labeling losses can represent the fairness using KL di- vergence KL(¯y ∥ u), as in (Ghasedi Dizaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In particular, (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) formulates the following entropy-based self-labeling loss Lce+kl = H(y, σ) + KL(¯y ∥ u) (6) encouraging decisiveness and fairness, as discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Simi- larly to (5), the network parameters in loss (6) are trained by the standard cross-entropy term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' But, optimization over relaxed pseudo-labels y ∈ ∆K is relatively easy due to the convexity of KL divergence and linearity of cross-entropy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' While there is no closed-form solution, the authors offer an efficient approximate solver for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Iterating steps that estimate pseudo-labels y and optimize the model pa- rameters resembles Lloyd’s algorithm for K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The results in (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) also establish a formal relation between the loss (6) and the K-means objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our work is closely related to self-labeling loss (6) and the corresponding ADM algorithm proposed in (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Their inspiring approach is a good reference point for our self-labeling loss proposal (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' It also helps to illu- minate some problems with standard entropy-based losses and their limited understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In particular, we disagree with the main theoretical claim in (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) establishing a formal equivalence between K-means and “regularized” entropy-based clustering with soft-max models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In fact, our Figure 1 works as a simple 2D counterexample to their claim3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Also, they extend the entropy-based loss with the classifier regularization ∥v∥2, but this extra quadratic term is mainly used as a technical tool in their proof of algebraic similarity between their loss and the standard K-means loss4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In contrast to related prior work, we demonstrate that ∥v∥2 is needed in discriminative entropy clustering for margin maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 3The proof of Proposition 2 has a critical technical error - it ignores normalization for soft-max prediction in their equation (5), which is hidden via ∝ symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Such normalization is critical for pseudo-posterior models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 4Since they ignore normalization in the softmax prediction, then ln σ in the cross-entropy H(y, σ) turns into a linear term w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' logits v⊤x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Adding regularization ∥v∥2 to such loss allows them to create a quadratic form with respect to v that resembles squared errors loss in K-means, which is quadratic w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t means µk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Summary of contributions Our paper provides conceptual and algorithmic contribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First of all, our paper disproves the main theoretical claim (in the title) of a recent TPAMI paper (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) wrongly stating the equivalence between the stan- dard K-means loss and entropy-based clustering losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our Figure 1 provides a simple counterexample to the claim, but we also show specific technical errors in their proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Figure 1 helps to motivate entropy clustering with discrimi- native soft-max models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This general methodology is unde- servedly little-known to the broader ML community for two reasons: (A) it was previously presented only in the context of complex (non-linear, deep) softmax models obfuscating the basics and (B) because there is confusion even among the researchers who know about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Besides clarifying ear- lier claims about the relation to K-means, we also show that entropy-based losses may lead to narrow decision margins, which may contradict one common motivation for decisive- ness (Grandvalet & Bengio, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Unlike prior entropy clustering work, we motivate classifier norm regularization by demonstrating its importance for margin maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also discuss the limitations of the existing self-labeling formulations of entropy clustering and propose a new loss, as well as an efficient pseudo-labeling algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In par- ticular, we replace standard forward cross-entropy H(y, σ), where y are soft pseudo-labels, by the reverse cross-entropy H(σ, y) that is significantly more robust to errors in esti- mated soft pseudo-labels, see Figures 2 and 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our for- mulation of fairness is motivated by a zero-avoiding version of KL divergence enforcing stronger fairness, see Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We design a new EM algorithm with closed-form EM steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In part, our self-labeling formulation of entropy clus- tering is motivated by its amenability to an efficient EM solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our empirical results improve the state-of-the-art on many standard benchmarks for deep clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The rest of our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Section 2 motivates our new self-labeling loss for entropy clustering and derives our EM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Section 3 compares our ap- proach with the state-of-the-art entropy clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Conclusions are provided in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our entropy clustering approach We are focused on entropy-based losses for clustering with softmax models that typically enforce “decisiveness” and “fairness”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1 we argue that common for- mulations of such losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (1) or (5), may produce narrow classification margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We show that some explicit margin maximization constraints should be added, which motivates the classifier norm regularization ∥v∥2 similarly to SVM methods (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2 introduces our new entropy-based self-labeling loss incorporating strong fair- Revisiting Discriminative Entropy Clustering and its relation to K-means (a) γ = 0 (b) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='001 (c) γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='01 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Margin maximization term γ ∥v∥2 in our loss (7): low- level clustering results for the softmax linear classifier model (3) with N = 2 and different weights γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The dots represent data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The optimal softmax clustering of the data and the decision regions over the whole space are visualized by σ-weighted color trans- parency, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The “margin” is a weak-confidence “soft” region around the linear decision boundary lacking color-saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For small γ the classifier can “squeeze” a narrow-margin linear decision boundary just between the data points, while maintaining arbitrarily hard “decisiveness” on the data points themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' ness and reverse cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3 derives an effi- cient EM algorithm for an important sub-problem - estima- tion of pseudo-labels y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Margin maximization via norm regularization The average entropy term in (1), a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' “decisiveness”, is recommended in (Grandvalet & Bengio, 2004) as a general regularization term for semi-supervised problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' They argue that it produces decision boundaries away from all training examples, labeled or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This seems to suggest larger classification margins, which are good for general- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, the decisiveness may not automatically imply large margins if the norm of classifier v in pseudo posterior models (2, 3) is unrestricted, see Figure 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Tech- nically, this follows from the same arguments as in (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2004) where regularization of the classifier norm is formally related to the margin maximization in the context of their SVM approach to clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Interestingly, regularization of the norm for all network pa- rameters [v, w] is motivated in (4) differently (Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' But, since the classifier parameters v are included, coincidentally, it also leads to margin maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' On the other hand, many MI-based methods (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Ghasedi Dizaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020) do not have regularizer ∥v∥2 in their clustering loss, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' see (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' One may argue that practical implementations of these meth- ods implicitly benefit from the weight decay, which is om- nipresent in network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' It is also possible that gradient descent may implicitly restrict the classifier norm (Soudry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In any case, since margin maximization is important for clustering, ideally, it should not be left to chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, the norm regularization term ∥v∥2 should be explicitly present in any clustering loss for pseudo-posterior models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We extend MI loss (1) by combining it with the regulariza- tion of the classifier norm ∥v∥ encouraging margin maxi- mization, as shown in Figure 3 Lmi+mm := H(σ) − H(σ) + γ ∥v∥2 c= H(σ) + KL(σ ∥ u) + γ ∥v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (7) We note that (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) also extend their entropy- based loss (6) with the classifier regularization ∥v∥2, but this extra term is used mainly as a technical tool in relating their loss (6) to K-means, as detailed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' They do not discuss its relation to margin maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our self-labeling loss function Below we motivate and put forward some new ideas for entropy-based clustering losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, we observe that the entropy H(¯σ) in (1) is a weak formulation of the fairness constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Indeed, as clear from an equivalent formulation in (7), it is enforced by the reverse KL divergence for the average predictions ¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' It assigns a bounded penalty even for highly unbalanced solutions where ¯σk = 0 for some k, see the dashed red curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Compare this with the forward KL divergence KL(u ∥ σ), see the solid red curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We propose such zero-avoiding forward version of KL divergence as a strong fairness loss Lmi++ := H(σ) + λ KL(u ∥ σ) + γ ∥v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (8) We will derive our self-labeling loss directly from (8) using standard splitting technique (Boyd & Vandenberghe, 2004) to divide optimization of (8) into simpler sub-problems sep- arating the “decisiveness” and “fairness” terms, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Introducing auxiliary splitting variables y ∈ ∆K, one for each training example X, optimization of the loss (8) can be equivalently written as min v,w H(σ) + γ ∥v∥2 (decisiveness sub-problem) min y KL(u ∥ y) (fairness sub-problem) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' y = σ (consistency constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This constrained optimization problem can be formulated us- ing a Lagrangian function enforcing the equality constraint y = σ via the forward KL divergence for y (motivated below) Lour := H(σ) + β KL(σ ∥ y) + λ KL(u ∥ ¯y) + γ ∥v∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (9) The Lagrangian is optimized with respect to both the net- work parameters and latent variables y, but we treat the Lagrange multiplier β as a fixed hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, the constraint y = σ may not be satisfied exactly and the Lagrangian (9) works only an approximation of the loss (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Revisiting Discriminative Entropy Clustering and its relation to K-means (a) strong fairness KL(u∥¯σ) (b) reverse cross-entropy H(σ, y) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' “Forward” vs “reverse”: (a) KL-divergence and (b) cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Assuming binary classification K = 2, we can represent all possible probability distributions as points on the interval [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The solid curves in (a) illustrate our “strong” fairness constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' the forward KL-divergence KL(u∥¯σ) for the average prediction ¯σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We show two examples of volumetric prior u1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) (blue curve) and u2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For comparison, the dashed curves represent reverse KL-divergence KL(¯σ∥u) commonly used for fairness in the prior art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The solid curves in (b) show our reverse cross-entropy H(σ, y) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t the network prediction σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The dashed curves show the forward cross-entropy H(y, σ), which is standard in the prior art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The plots in (b) show examples for two fixed estimates of pseudo-labels y1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) (blue curves) and y2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) (red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The boundedness of H(σ, y) represents robustness to errors in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, our loss H(σ, y) turns off the training (sets zero-gradients) when the estimated confidence is highly uncertain, see y2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) (solid red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In contrast, the standard loss H(y, σ) trains the network to copy this uncertainty, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g observe the optimum σ for the dashed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Also, one can justify hyper-parameter β = 1 empirically, see Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Since H(σ) + KL(σ ∥ y) = H(σ ∥ y), we get the following self-labeling loss formulation Lour β = 1 = H(σ, y) + λ KL(u ∥ ¯y) + γ ∥v∥2 (10) where the reverse cross entropy H(σ, y) enforces both the decisiveness and consistency y ≈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' There are some notable differences between our loss (10) and existing self-labeling losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, consider the loss (6) proposed in (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our loss reverses the order of both the KL divergence and the cross-entropy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As explained earlier, our version of the KL divergence en- forces stronger fairness, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The reversal of the cross-entropy is motivated in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, it makes the training of network predictions σ robust to errors in noisy estimates y, see Figure 4(b), as the pseudo-labels y are not guaranteed to be accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' On the other hand, compared to the standard cross-entropy, it enforces stronger consistency of y with the predictions σ, which work as target distri- butions for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' pseudo-labels y, our loss (10) enforces stronger fairness and stronger consistency y ≈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The corresponding well-constrained optimization problem for y allows an efficient EM solver derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our EM algorithm for estimating pseudo-labels To optimize (10) with respect to y, basic Newton’s meth- ods (Kelley, 1995) can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Although the overall convergence rate of such second-order methods is fast, the calculation or approximation of the inverse Hessian is com- putationally costly as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This motivates us to derive the more efficient expectation-maximization (EM) algorithm (Bishop, 2006) for optimizing y as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Here we present a new efficient algorithm for optimizing our discriminative entropy-based loss (10) with respect to the pseudo-labels y when the model predictions are fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' σ and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Using the variational inference (Bishop, 2006), we derive a new EM algorithm introducing a dif- ferent type of latent variables, K distributions Sk ∈ ∆M representing normalized support for each cluster over M data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We refer to each vector Sk as a normalized cluster k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note the difference with distributions represented by pseudo-posteriors y ∈ ∆K showing support for each class at a given data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Since we explicitly use indi- vidual data points below, we will start to carefully index them by i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, we will use yi ∈ ∆K and σi ∈ ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Individual components of distribution Sk ∈ ∆M corresponding to data point i will be denoted by scalar Sk i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, we expand our loss (10) introducing the latent vari- ables Sk ∈ ∆M Lour c= H(σ, y) + λ H(u, ¯y) + γ ∥v∥2 (11) = H(σ, y) − λ � k uk ln � i Sk i yk i Sk i M + γ ∥v∥2 ≤ H(σ, y) − λ � k � i ukSk i ln yk i Sk i M + γ ∥v∥2 (12) Due to the convexity of negative log, we apply Jensen’s inequality to derive an upper bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (12), to Lour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Such Revisiting Discriminative Entropy Clustering and its relation to K-means a bound becomes tight when: Estep : Sk i = yk i � j yk j (13) Then, we fix Sk i as (13) and solve the Lagrangian of (12) with simplex constraint to update y as: Mstep : yk i = σk i + λMukSk i 1 + λM � c ucSc i (14) We run these two steps until convergence with respect to some predefined tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that the minimum y is guaranteed to be globally optimal since (11) is convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' y (Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The empirical convergence rate is within 15 steps on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The comparison of computation speed on synthetic data is shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' While the number of iterations to convergence is roughly the same as New- ton’s methods, our EM algorithm is much faster in terms of running time and is extremely easy to implement using the highly optimized built-in functions from the standard PyTorch library that supports GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' number of iterations running time in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (to convergence) (to convergence) K2 K20 K200 K2 K20 K200 Newton 3 3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8e−2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3e−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='7e−1 EM 2 2 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9e−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='0e−3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='0e−3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Comparison of our EM algorithm to Newton’s methods (Kelley, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' K2, K20 and K200 stand for the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Inspired by (Springenberg, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017), we also adapted our EM algorithm to allow for updating y within each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In fact, the mini-batch approximation of (11) is an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Considering the first two terms of (11), we can use Jensen’s inequality to get: H(σ, y) + λ H(u, ¯y) ≤ EB[HB(σ, y) + λ H(u, ¯yB)] (15) where B is the batch randomly sampled from the whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Now, we can apply our EM algorithm to update y in each batch, which is even more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Compared to other methods (Ghasedi Dizaji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) which also use the auxiliary vari- able y, we can efficiently update y on the fly while they only update once or just a few times per epoch due to the inefficiency to update y for the whole dataset per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Interestingly, we found that it is actually important to update y on the fly, which makes convergence faster and improves the performance significantly (Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use this “batch version” EM throughout all the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our full algorithm for the loss (10) is summarized in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Experimental results Our experiments start from pure clustering on fixed features to joint clustering with feature learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We have also com- pared different losses on weakly-supervised classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that our goal is comparing different losses together with their own optimization algorithms, thus we keeping our experimental setup as simple as possible to reduce the distraction factors for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Dataset For the clustering problem, we use four standard benchmarks: MNIST (Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1998), CIFAR10/100 (Torralba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2008) and STL10 (Coates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The training and test data are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As for the weakly- supervised setting, we conduct experiments on CIFAR10 and STL10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We split the data into training and test sets as suggested by the instructions for the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Evaluation As for the evaluation on clustering, we set the number of clusters to the number of ground-truth categories and we adopt the standard method (Kuhn, 1955) by finding the best one-to-one mapping between clusters and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use the accuracy as the measure for both unsupervised and weakly-supervised settings while the latter calculates the accuracy on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Clustering with fixed features In this section, we test our loss (10) with a simple linear classifier on MNIST (Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1998) by using the (fixed) original features of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We compare it to K-means and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The detailed experimental settings can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In Table 2, we report the mean accuracy K-means MI (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010) Our accuracy 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2% (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2%(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='7) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Comparison of different losses on MNIST without learn- ing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' and standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that discriminative clustering methods perform consistently much better than K-means (≥ 7%) while our approach achieves a bit higher accuracy but is more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Also, a low-level ablation study can be found in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' STL10 CIFAR10 CIFAR100-20 MNIST Kmeans 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='20%(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='78%(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='6) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='99%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='62%(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) MI-GD (Bridle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='56%(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='32%(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='59%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='92%(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='0) MI-ADM (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='28%(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='07%(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='70%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='15%(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='7) SeLa (Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='33%(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='31%(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='7) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='74%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='38%(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2) Our 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2%(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='48%(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2%(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Comparison of different methods on clustering with fixed features extracted from Resnet-50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The numbers are the average accuracy and the standard deviation over trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Besides using low-level features, we also compare our Revisiting Discriminative Entropy Clustering and its relation to K-means method against the state-of-the-art methods using fixed deep features generated by large models such as Resnet-50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We still use a one-layer linear classifier for all loss functions except for Kmeans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The coefficients γ for the margin maximization terms are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='009, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='02 for MNIST, CIFAR10, CIFAR100 and STL10 re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As illustrated in Figure 3, γ is important for the optimal decision boundary, especially when features are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' If we jointly learn the representation and cluster the data, we observed that the results are less sensitive to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that this backbone network could be trained together with the linear classifier even from the scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' However, we found that the clustering loss itself is not enough to generate reasonable features for the backbone network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, we keep the backbone network fixed and only train the linear classifier using different clustering loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Joint clustering and representation learning In this section, we train a deep network to jointly learn the features and cluster the data on the four standard benchmark datasets: STL10 (Coates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2011), CIFAR10/CIFAR100 (Torralba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2008) and MNIST (Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The only extra standard technique we add here is the self- augmentation, following (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This technique is important for en- forcing neural networks to learn augmentation-invariant fea- tures, which are often semantically meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' While (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) designed their loss directly based on such tech- nique, our loss and (Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) are more general for clustering without any guarantee to generate semantic clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Thus, for fair comparison and more reasonable results, we combine this augmentation technique into network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The exper- imental settings and more detailed discussion are given in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' From Table 4, it can be seen that our approach consistently achieves the best or the most competitive results in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' STL10 CIFAR10 CIFAR100-20 MNIST MI-D⋆ (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='28%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='39%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='7) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='90%(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3) IIC⋆ (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='12%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='7) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='58%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='6) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='51%(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3) SeLa§ (Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='99%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='16%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='34%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='86%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9) MI-ADM§ (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='37%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='27%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='6) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='02%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='75%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3) Our⋆,§ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='33%(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='16%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='09%(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='58%(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='8) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Quantitative results of accuracy for unsupervised cluster- ing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We only use the 20 coarse categories for CIFAR100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We reuse the code published by (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017) and implemented the optimization for loss of (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) according to the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' ⋆: all variables are updated for each batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' §: loss formula has pseudo-label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that we only use a very small network architecture (VGG4) here since we observed that more complex archi- tectures require more additional techniques to obtain rea- sonable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, Ji et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) also use auxiliary over-clustering, multiple heads, and more data to obtain high numbers on STL10 with ResNet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' To emphasize on the effects of different loss functions, we keep the experimental settings as simple as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Weakly-supervised classification We also test different methods over different levels of (very) weak supervision on STL10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In Table 5, we can see that our approach still shows very competitive results, especially with weaker supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' More details are given in Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' H including another test on CIFAR 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='01 Only seeds 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='27% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='26% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1% + MI-D (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2017) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='39% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='73% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='54% + IIC (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='73% 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='6% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='17% + SeLa (Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2020) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='84% 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='4% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='08% + MI-ADM (Jabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2021) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='83% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='41% 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='79% + Our 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='20% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='13% 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='76% Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Quantitative results for weakly-supervised classification on STL10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='05 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='01 correspond to different ratios of labels used for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' “Only seeds” means that we only use standard cross-entropy loss on labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Conclusions Our paper proposed a new self-labeling algorithm for dis- criminative entropy clustering, but we also clarify several important conceptual properties of this general methodol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For example, we disproved a theoretical claim in a recent TPAMI paper stating the equivalence between vari- ance clustering (K-means) and discriminative entropy-based clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also demonstrate that standard formulations of entropy clustering losses may lead to narrow decision mar- gins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Unlike prior work on discriminative entropy clustering, we show that classifier norm regularization is important for margin maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also discussed several limitations of the existing self- labeling formulations of entropy clustering and propose a new loss addressing such limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In particular, we replace the standard (forward) cross-entropy by the reverse cross-entropy that we show is significantly more robust to errors in estimated soft pseudo-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our loss also uses a strong formulation of the fairness constraint motivated by a zero-avoiding version of KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Moreover, we designed an efficient EM algorithm minimizing our loss w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' pseudo-labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' it is significantly faster than standard alternatives, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='g Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our empirical results improved the state-of-the-art on many standard benchmarks for deep clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Revisiting Discriminative Entropy Clustering and its relation to K-means References Asano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Rupprecht, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', and Vedaldi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Self-labelling via simultaneous clustering and representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In International Conference on Learning Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Bishop, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Pattern Recognition and Machine Learning.' 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categorical generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In International Conference on Learning Representations, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Revisiting Discriminative Entropy Clustering and its relation to K-means Tanaka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Ikami, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Yamasaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', and Aizawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Joint optimization framework for learning with noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 5552–5560, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Torralba, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Fergus, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', and Freeman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 80 million tiny images: A large data set for nonparametric object and scene recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' IEEE transactions on pattern analysis and machine intelligence, 30(11):1958–1970, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Xu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Neufeld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Larson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', and Schuurmans, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Maxi- mum margin clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In Saul, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', Weiss, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', and Bottou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' ), Advances in Neural Information Processing Sys- tems, volume 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' MIT Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Revisiting Discriminative Entropy Clustering and its relation to K-means A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Proof Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Given fixed σi ∈ ∆K where i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', M} and u ∈ ∆K, the objective E(y) = − β M � i � k σk i ln yk i − λ � k uk ln � i yk i M is convex for y, where yi ∈ ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' First, we rewrite E(y) E(y) = − � k � β M � i σk i ln yk i + λuk ln � i yk i M � := − � k fk(yk) (16) Next, we prove that fk : RM (0,1) → R is concave based on the definition of concavity(Boyd & Vandenberghe, 2004) for any k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Considering x = (1 − α)x1 + αx2 where x1, x2 ∈ RM (0,1) and α ∈ [0, 1], we have fk(x) = β M � i σk i ln ((1 − α)x1i + αx2i)+ λuk ln � i ((1 − α)x1i + αx2i) M ≥ β M � i (1 − α)σk i ln x1i + ασk i ln x2i + λuk � (1 − α) ln � i x1i M + α ln � i x2i M � = (1 − α)fk(x1) + αfk(x2) The inequality uses Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Now that fk is proved to be concave, −fk will be convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Then E(y) can be easily proved to be convex using the definition of convexity with the similar steps above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Our Algorithm C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Loss Curve D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Network Architecture The network structure is VGG-style and adapted from (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Dataset Summary Table 7 indicates the number of (training) data and the input size of each image for the unsupervised clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Training and test sets are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As for weakly-supervised classification on STL10, we use 5000 images for training and 8000 images for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We Algorithm 1 Optimization for our loss Input :network parameters [v, w] and dataset Output :network parameters [v∗, w∗] for each epoch do for each iteration do Initialize y by the network output at current stage as a warm start while not convergent do Sk i = yk i � j yk j yk i = σk i +λMukSk i 1+λM � c ucSc i end Update [w, v] using loss HB(σ, y) + γ ∥v∥2 via stochastic gradient descent end end Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Loss (10) curves for different update setting on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This is generated with just a linear classifier on MNIST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use the same initialization and run both for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The gray line has an accuracy of 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='35% while the yellow one achieves 63%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' only keep a certain percentage of ground-truth labels for each class of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The accuracy is calculated on test set by comparing the hard-max of prediction to the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Low-level Clustering As for the experiments on MNIST (Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1998), we transform the original image values linearly into [−1, 1] and use the flattened images as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note that here we only use a linear classifier without training any features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We employ stochastic gradient descent with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='07 to update v in (4) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use the same (random) intialization for both losses and run each 6 times up to 50 epochs per run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use 250 for batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We set γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='01 for both and use λ = 100 for (10) and λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='3 for (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We fix the hyperparameter values for (9) and (4) throughout the whole experimental sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We also conducted an ablation study on toy examples as shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use the normalized X-Y coordinates of the data points as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We can see that each part of our loss is necessary for obtaining a good result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Note loss: update y on whole dataset once per epoch update y on batch data per iteration 232 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='5 231 Iteration 0 2k 4k 6k 8k 10k 12kRevisiting Discriminative Entropy Clustering and its relation to K-means Grey(28x28x1) RGB(32x32x3) RGB(96x96x3) 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@64 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@32 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@128 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@128 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@64 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@256 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@256 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@128 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@512 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xMaxPool(2x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2) 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@512 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@256 1xConv(5x5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='s=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='p=2)@1024 1xLinear(512x3x3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='K) 1xLinear(256x4x4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='K) 1xLinear(1024x1x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='K) Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Network architecture summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' s: stride;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' p: padding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' K: number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The first column is used on MNIST (Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' the second one is used on CIFAR10/100 (Torralba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' the third one is used on STL10 (Coates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Batch normalization is also applied after each Conv layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' ReLu is adopted for non-linear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' STL10 CIFAR10 CIFAR100-20 MNIST 13000 60000 60000 70000 96x96x3 32x32x3 32x32x3 28x28x1 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Dataset summary for unsupervised clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' that, in Figure 6 (a), (c) of 3-label case, the clusters formed are the same, but the decision boundaries which implies the generalization are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This emphasizes the importance of including L2 norm of v to enforce maximum margin for better generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2 clusters 3 clusters (a) γ = 0 (b) λ = 0 (c) full setting Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' “Shallow” ablation study on toy examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Deep Clustering We add deep neural networks for learning features while doing the clustering simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use four stan- dard benchmark datasets: STL10 (Coates et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2011), CI- FAR10/CIFAR100 (Torralba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2008) and MNIST (Le- cun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As for the architectures, we followed (Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=', 2019) to use VGG11-like network structures whereas we use it for both gray-scale and RGB images with some adjustments as shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We achieved the self-augmentation by setting σi = Et[σ(v⊤fw(t(Xi))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' For each image, we generate two aug- mentations sampled from “horizontal flip”, “rotation” and “color distortion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We use Adam (Kingma & Ba, 2015) with learning rate 1e−4 for optimizing the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We set batch size to 250 for CIFAR10, CIFAR100 and MNIST, and we use 160 for STL10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In Table 4, we report the mean accuracy and Std from 6 runs with different initializations while we use the same initialization for all methods in each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We still use 50 epochs for each run and all methods reach convergence within 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' As for other methods in Table 4, MI-D has the most com- parable results to us, in part because our loss can be seen as an approximation to the MI and we both update all vari- ables per batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' SeLa achieves relatively better results on other three datasets than MNIST, because it enforces a hard constraint on “fairness” and MNIST is the only one out of four sets that is not exactly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' In real world, the data we collect is mostly not exactly balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This could be the reason why such method is better for the unsupervised rep- resentation learning where over-clustering can be employed and real clusters become less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' MI-ADM only up- dates the pseudo-labels once per epoch, thus easily leading the network towards a trap of local minimum created by the incorrect pseudo-labels through the forward cross-entropy loss as illustrated in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Weakly-supervised Clustering We use the same experimental settings as that in unsuper- vised clustering except for two points: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We add cross- entropy loss on labelled data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' We separate the training data from test data while we use all the data for training and test in unsupervised clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' While MI-ADM is the worst according to Table 4, it is improved significantly in weakly-supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' This might be a sign that the advantage of more frequent update on variables in unsupervised clustering is waning since the seeds help the network keeping away from some bad local minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Below is the result on CIFAR 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='01 Only seeds 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='77% 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='27% 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='01% + MI-D 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='54% 61.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='19% + MI-ADM 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='51% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='05% 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='91% + Our 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='17% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='59% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='22% Revisiting Discriminative Entropy Clustering and its relation to K-means I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' Hyperparameter β Below is an empirical justification for setting hyper- parameter β = 1 in the loss (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' The first two terms in (9) can be written as (1 − β)H(σ) + βH(σ, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' If β > 1 then the negative entropy pushes predictions σ away from one-hot solutions weakening the decisiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' On the other hand, if β < 1 then the loss is non-convex w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='t σ that may trap gradient descent in bad local minima, as illustrated by the plots for y = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content='1) in Figure 7 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} +page_content=' (1 − β)H(σ) + βH(σ, y)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFIT4oBgHgl3EQf9Cv1/content/2301.11405v1.pdf'} diff --git a/.gitattributes b/.gitattributes index 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Princeton University, USA +Fig. 1. Two players using the PokAR application. +We introduce PokAR, an augmented reality (AR) application to facilitate poker play. PokAR aims to alleviate three difficulties of +traditional poker by leveraging AR technology: (1) need to have physical poker chips, (2) complex rules of poker, (3) slow game pace +caused by laborious tasks. Despite the potential benefits of AR in poker, not much research has been done in the field. In fact, PokAR is +the first application to enable AR poker on a mobile device without requiring extra costly equipment. This has been done by creating +a Snapchat Lens 1 which can be used on most mobile devices. We evaluated this application by instructing 4 participant dyads to +use PokAR to engage in poker play and respond to survey questions about their experience. We found that most PokAR features +were positively received, AR did not significantly improve nor hinder socialization, PokAR slightly increased the game pace, and +participants had an overall enjoyable experience with the Lens. These findings led to three major conclusions: (1) AR has the potential +to augment and simplify traditional table games, (2) AR should not be used to replace traditional experiences, only augment them, (3) +Future work includes additional features like increased tactility and statistical annotations. +CCS Concepts: • Human-centered computing → Collaborative and social computing devices. +Additional Key Words and Phrases: connected lens, augmented reality, poker, co-located, interaction, socialization +ACM Reference Format: +Adam Gamba and Andrés Monroy-Hernández. 2023. PokAR: Facilitating Poker Play Through Augmented Reality. 1, 1 (January 2023), +11 pages. https://doi.org/XXXXXXX.XXXXXXX +1A Lens in Snapchat is an experience that utilizes augmented reality to transform the world around you [12]. +Authors’ address: Adam Gamba, agamba@princeton.edu; Andrés Monroy-Hernández, andresmh@princeton.edu, Princeton University, Princeton, New +Jersey, USA, 08544. +Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not +made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components +of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to +redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. +© 2023 Association for Computing Machinery. +Manuscript submitted to ACM +Manuscript submitted to ACM +1 +arXiv:2301.00505v1 [cs.HC] 2 Jan 2023 + +l：61:598 +old2 +Gamba and Monroy-Hernández +1 +INTRODUCTION +The goal of this project is to facilitate heads-up Texas hold’em poker play through augmented reality. Poker is +cumbersome to play in its current form, requiring players to have poker chips and knowledge of the complex rules to +play correctly. Without an experienced player to guide the game, new players often find it difficult to learn the rules +and play correctly [9]. Additionally, due to the burden of physical chips, it is difficult to play poker in many scenarios +(e.g., at the beach, while traveling, or camping). Finally, the game pace is often slowed due to poker’s complex rules and +the need for laborious tasks like counting chip stacks. +Augmented reality technology is well-equipped to solve these issues in three ways. Firstly, AR can eliminate the need +for physical poker chips by instead utilizing AR to render chips. Next, AR can help guide players through the complex +rules of poker by hinting at legal actions during gameplay. Finally, AR can help decrease the burden of laborious tasks +(like counting chips) and increase the game pace. PokAR helps alleviate these three issues, which we’ll discuss further +throughout this paper. +Poker is a popular game, with over 120 million players worldwide playing regularly online [7]. Texas hold’em is one +of the most popular poker variants. In this variant, players are dealt two private cards and five community cards, and +they battle to make the best hand or bluff opponents into folding. ’Heads-up’ poker is a term used to describe poker +played by just two players, head-to-head. In its current state, PokAR supports only heads-up Texas hold’em poker, but +with future work, it could be extended to more players and more variants. Throughout this paper, we will use the term +’poker’ to refer to heads-up Texas hold’em poker. +Poker is a classic example of a social, co-located game, since poker, by design, emphasizes in-person, co-located +interaction. Players often look at each other and speak to each other during a poker game, either to gain information or +to socialize. Also, poker forces players to focus on the same enablers, or "physical objects that trigger and are the focus +of the AR experience" [1]. These enablers, like playing cards and poker chips, can help guide an AR experience and +engage players more closely than in games that do not have a similar shared focus. For the above reasons, we chose to +augment poker in this study. +PokAR is not intended to replace traditional poker, rather it helps people play when traditional poker would be +difficult or impractical. The goal of AR applications should be to disappear completely and seamlessly immerse the user +in a realistic experience that combines reality with augmentation [16]. This disappearance frees users to utilize these +applications more effortlessly, allowing them to focus on new goals, beyond the application itself. +2 +RELATED WORK +While we have established AR as a possible solution to the aforementioned issues with poker, very little work has +been done concerning AR poker. Additionally, while AR is not yet a heavily explored area, researchers have argued that +“A Poker-Assistance-Software is an ideal test area for an AR Application with real added value,” with possible areas to +add value including automation and statistical estimations [15]. +Similar projects in the past have all relied on physical means to augment reality. For example, researchers used +overhead projectors to project all aspects of the poker game (e.g., cards, chips, etc.) onto a table [9]. Additionally, +researchers have used RFID playing cards to detect the dealt cards [9]. While this study succeeded in creating an AR +application to ease some of the same cumbersome aspects of poker tackled by PokAR (physical chips, complex rules, +slow game pace), it did so using a high-cost solution, which is impractical for most recreational use. Additionally, they +disregarded studying how this AR setup influenced social interactions in poker. +Manuscript submitted to ACM + +PokAR: Facilitating Poker Play Through Augmented Reality +3 +Furthermore, people have utilized virtual reality (VR) in the past to create commercial poker video games. One +example is PokerVR by Meta, which uses “expressive avatars built for reading tells with growing customizations” [6]. +While they may say this, VR poker applications still just employ static players’ avatars, which do not emphasize the +in-person, social nature of poker. +PokAR is the first project to enable AR poker without needing additional equipment other than a mobile device and +a regular deck of cards (like an overhead projector or a VR headset). This is a worthwhile problem because it is the +first application to enable AR poker at a low cost, since it only requires a few commonly-owned pieces of equipment +(mobile devices and playing cards). Additionally, utilizing AR over VR allows the gameplay to emphasize the social +aspects of co-location and increase socialization when compared to VR implementations. +Co-located gaming has been shown to lead to more effective and enjoyable gaming, as players can more easily +communicate and build social relationships [4]. One major reason for this is "out-of-the-game, game-related communi- +cation" [3]. By having the ability to converse about other topics while simultaneously being involved in a game with +another player, these players are given the opportunity to build a deeper connection. +Additionally, when comparing socialization in AR and VR, prior research tends to support increased socialization +in AR applications. AR games have the ability to "potentially enhance social communication and social interaction +between people" [10], whereas high-involvement in VR games could potentially isolate users socially and "negatively +affect their well-being" [5]. Thus, we chose to develop an AR application rather than a VR application to reap the social +benefits of shared, co-located experiences. +3 +POKAR SYSTEM +The PokAR Snapchat Lens allows users to play heads-up poker with another player on two mobile devices. AR +visual annotations include 3D models of poker chips which dynamically render with changing stack size, and 3D text +above the chip stacks denoting the size of each stack. 2D visual annotations include number of hands played, current +round, current dealer, previous action, amount to call, waiting message, and UI buttons with labels "Check," "Call," "Bet," +"Raise," and "Fold." All AR and 2D annotations render dynamically with changing game state, stack amounts, and legal +actions. PokAR implements five main features to help achieve its motivating goals. +• “3D AR Chips” - PokAR renders 3D models of chips to eliminate the requirement of needing physical poker +chips. +• “UI Action Buttons” - 2D buttons rendered on the player’s screen allows them to select among and perform legal +actions at the current state of the game. +• “Game Messages” - Messages provide additional information to both players about the actions of players, bet +amounts, and more throughout the game. +• “Counting Stacks” - A live count of the number of chips in all chip stacks is rendered above the 3D models, +eliminating the hassle of counting chips manually. +• “Awarding Pots” - After a winner is determined (through folding or at showdown), the chips are automatically +awarded to the winner, eliminating the hassle of moving chips manually. +3.1 +Approach +To achieve our motivating goals, we developed a Snapchat Lens which could be used on any mobile device to +facilitate poker play through AR. Besides having physical playing cards, players will play the game of poker in AR +Manuscript submitted to ACM + +4 +Gamba and Monroy-Hernández +Fig. 2. Side-by-side points of view of the same game of PokAR on two mobile devices. +by interacting with their augmented environment through a mobile device. We used the Snap Lens Studio IDE with +JavaScript for development, the Snapchat app for testing and deployment, and GitHub for version control. Code for this +project can be found at the link in Appendix D. Physical playing cards act as an enabler to ground the game in physical +reality. A demonstration of the completed application is shown in Fig 2. In the development of PokAR, we faced and +solved three major implementation subproblems. They will be discussed below. +3.2 +Subproblem 1: Modeling Poker in Code +The first implementation subproblem to solve was to figure out how to model a game of poker in code. By nature +of the rules of poker, the game is deterministic based on previous player actions within a betting round. Thus, the +game state can be modeled using a Deterministic Finite Automaton (DFA). The DFA for our application determines the +legal actions and/or termination state of the betting round, given previous actions within the betting round. At the +end of each betting round, one of two termination states is reached, dictating whether the hand has ended or players +will advance to the next betting round. Fig 3 and Fig 4 show a graphical representation of the DFAs we used in our +implementation. In both DFAs, the application begins with the Start state and terminates in either the endHand() or +advance() state. The double-headed arrow represents the possible cycle of betting, raising, reraising, etc., until one of the +players is eventually all-in. Player ’A’ is the one assigned ‘opponent’ at the start of a hand, and player ’B’ is the ‘dealer.’ +3.3 +Subproblem 2: Rendering 3D Objects Stably +The second implementation subproblem was to figure out how to render 3D objects in the world and effectively +track them. Originally the implementation used Snap’s World Tracking [14] functionality, and then its Surface Tracking +[14] functionality, with neither proving to be too accurate. Chip stacks are small and users expect them to stay in +Manuscript submitted to ACM + +5:49 +5:49 +79 +Hand #: 1 +Hand #:1 +Round: Preflop +Round:Preflop +Dealer:Opponent +Dealer: Me +Amount to Call: $1 +$99 +Opponent +$98 +Pot:ss +Waiting for opponent.. +1 +Z +3 +4 +5 +6 +7 +8 +9 +O +） +$ +& +@ +X +ABC +space +done +Send ChatPokAR: Facilitating Poker Play Through Augmented Reality +5 +Start +B Checks +B Bets +endhand() +A Checks +A Bets +A Folds +A Calls +advance() +B Folds +B Calls +Fig. 3. Pre-flop DFA (used before any community cards are dealt). +relatively the same position throughout a game. However, with World Tracking and Surface Tracking, chip stacks +would move throughout the room quite a bit if the mobile device’s camera was moved. +Then, we decided to use Marker Tracking [14], which uses a printed marker pattern to mark a position in the physical +world and allow the application to render objects relative to that position. Marker tracking proved to be very accurate +and stable for rendering 3D objects in AR. Chip stacks would no longer move throughout the room, as they were +grounded in a location in 3D space. Marker Tracking, however, is only a temporary solution while alternative tracking +solutions are improved with continued computer vision research. +3.4 +Subproblem 3: Connecting Multiple Players +The third implementation subproblem was to figure out how to connect two players within a single Snapchat Lens to +play together and share game data. To solve this problem, we designed an API to connect two different mobile devices +and allow them to send messages between each other, including updates on the game state and player actions. This API +is built on top of Snap’s Connected Lenses feature [11]. Thus, the two players will always observe the same data on +different devices in real time, unifying their gameplay experience. +Manuscript submitted to ACM + +6 +Gamba and Monroy-Hernández +Start +A Calls +A Raises +endhand() +B Checks +B Raises +B Folds +B Calls +advance() +A Folds +A Calls +A Folds +Fig. 4. Post-flop DFA (used once community cards have started to been dealt). +4 +EVALUATION +We recruited 8 participants who were found through a poker club on campus and recruited by email. We asked +participants to respond to a pre-study survey to learn about their individual experience with poker and its rules. This +survey can be found in Appendix B. We randomly paired participants into 4 dyads to utilize PokAR to play heads-up +poker for 25 minutes. Then, we asked them to respond to a post-study survey about their experience with the application. +In this survey, we asked participants about the benefits and detriments of particular PokAR features, the effects of AR +on socialization, the effects of AR on game pace, and the overall experience with PokAR. This survey can be found in +Appendix C. The study protocol above is described in more detail in Appendix A. +5 +RESULTS +Although participants had varying levels of poker expertise, they all had a self-reported understanding of the rules. +Based on the results of the pre-study survey, we grouped the 8 participants into three groups of varying experience +levels for analysis: Highly Experienced (play poker multiple times a week, 𝑛 = 2), Moderately Experienced (play poker +weekly to monthly, 𝑛 = 3), and Slightly Experienced (play poker yearly or less, 𝑛 = 3). +Manuscript submitted to ACM + +PokAR: Facilitating Poker Play Through Augmented Reality +7 +5.1 +Evaluation of Features +We asked participants to rate each of the five major PokAR features on a scale of 1 (detrimental) to 5 (beneficial) in +terms of its effectiveness compared to the corresponding object/action in real-life poker. Each feature earned an average +score > 3 (leaning beneficial) among all participants. Specifically, "3D AR Chips" scored a 4, "UI Action Buttons" scored +a 4.25, "Game Messages" scored a 3.875, "Counting Stacks" scored a 4.375, and "Awarding Pots" scored a 4.25. +Notably, the only features that scored < 3 (leaning detrimental) were “3D AR Chips,” “UI Action Buttons,” and +“Game Messages” for the Highly Experienced subgroup of participants. This could be explained by the fact that all +three of these features are intended to alleviate the requirement of knowing the complex rules of poker. However, in +the pre-study survey, all members of this subgroup answered that they play poker quite often and they confidently +understand all the rules, so these features likely just got in the way of their gameplay. The features of “Counting Stacks” +and “Awarding Pots” were, however, positively received by all three subgroups of participants. +5.2 +Evaluation of Socialization +The average response to the survey question: “How much did AR affect the in-person social aspects of the game of +poker?” was a 3.25 on a scale of 1 (negatively) to 5 (positively), meaning that AR neither significantly improved nor +impaired the in-person social aspects of poker. This is a beneficial result, as one of PokAR’s goals was to supplement the +game of poker. We did not implement social-related features intended to improve socialization, but this result supports +the claim that the AR features of PokAR did not impair socialization. In other words, players are utilizing PokAR as a +tool to enable poker play, which does not get in the way of the traditional social interactions at a poker table. +Additionally, multiple participants noted that AR did not heavily influence socialization. P1 stated that the experience +was “no different, we could still talk and converse,” and P5 stated that AR “Didn’t affect socialization because everything +was still in person.” +5.3 +Evaluation of Game Pace +The average response to the survey question: “How did augmented reality affect the game pace of poker?” was a +3.75 on a scale of 1 (slowed the game) to 5 (sped up the game), meaning that AR slightly increased the game pace of +poker. In this study, the average game pace was 40.3 hands/hour (67.2 hands/hour for the Highly Experienced subgroup). +Comparatively, “A typical live poker game will deal 25-30 per hour,” assuming 9 players [2]. This section requires +additional study, including a control session of each participant group playing traditional poker to compare the game +pace with AR poker. +5.4 +Evaluation of Overall Experience +On average, participants rated their overall experience at a 4.5/5, overwhelmingly positive. P2 stated that “It’s quicker, +but annoying to hold the phone up.” P6 stated that “It was a different experience which took a little getting used to, but I +enjoyed it.” P4 stated that it “Felt cool to have the chips tracked for you. Definitely could see myself using it on a camping +trip or during traveling.” P5 stated that “It was cool, because sometimes I’d like to play poker but sometimes have no chips!” +6 +CONCLUSION +Through developing a complete AR application and studying how people utilize it, we have generated three main +conclusions. +Manuscript submitted to ACM + +8 +Gamba and Monroy-Hernández +6.1 +AR Has the Potential to Augment and Simplify Traditional Table Games +After our work throughout this semester, we are confident that AR as a technology can and will be used in the future +to augment and simplify traditional table games, like poker. While the technology is currently in a primitive state, it is +continually evolving and progressing. Several participants noted that the gameplay of PokAR was clunky since they +had to constantly hold their mobile devices up to see the game. However, with improved AR technology, this annoyance +will begin to fade away. For example, AR glasses like Spectacles will eliminate the need to hold up a mobile device [13]. +This study revealed promising results concerning the future potential of AR in games. For instance, the use of AR did +not hinder socialization, and participants had a positive overall experience. PokAR features meant to facilitate gameplay +were positively received by most players, and options to disable disruptive features would alleviate the rest. +6.2 +AR Should Not Be Used to Replace Traditional Experiences; It Should Be Used to Augment Them +This conclusion stems from the fact that AR technology has innate limitations compared to the physical world. For +instance, AR experiences are less tactile than physical world experiences. While software tricks exist to improve the +tactility of AR experiences (like hand tracking, which enables object manipulation), it will never feel quite like the +physical world. For example, several participants noted that PokAR was missing one important aspect of traditional +poker: chip shuffling. Chip shuffling is a common fidgeting technique among poker players in which they use one hand +to rearrange a stack of chips. Shuffling is almost unanimous among poker players and is commonly used to pass time +and cure boredom during long poker sessions. While AR could simulate chip shuffling, it could never reproduce the +experience perfectly. +Early intuition about this conclusion is one reason we decided to utilize physical playing cards in PokAR. If playing +cards were virtual, players would be playing an online poker game in which in-person social interactions were minimal. +Players wouldn’t even need to be co-located to play PokAR anymore. This would be a case of using AR to replace +a traditional game experience. Instead, we decided to use physical cards and virtual chips in PokAR to afford some +physical-world tactility to players and streamline some of the more annoying and time-consuming aspects of poker, +like counting chips. +As mentioned in the introduction, PokAR is not intended to replace traditional poker, only augment it. This is due to +the inherent limitations of AR technology. More broadly, AR should not be used to replace traditional experiences, only +augment them. Augmentations should be deliberately planned and carefully implemented to ensure that they do not +take over the spirit of the game. Go too far with augmentation, and you approach the virtual reality world and lose out +on social interaction. +6.3 +Future Work +Our work on PokAR has revealed possible directions for future study and enhancements to the application. Firstly, +due to time constraints, not all features of poker were able to be added. PokAR is currently limited to just two players. +This design choice was made to reduce the project’s complexity, but this limit should be increased to the accepted limit +of nine players to better simulate traditional poker. Additionally, the option to chop pots (split pots equally between +tied players) is currently not implemented. The option to run it multiple times (deal remaining cards multiple times in +an all-in situation and award the pot proportionally to winners) is also not yet implemented. +PokAR would benefit from increased tactility, which is why we believe that it is a necessary direction for future +work. Increased tactility could come in two forms, with the first being the ability to grab AR chips and manipulate them +Manuscript submitted to ACM + +PokAR: Facilitating Poker Play Through Augmented Reality +9 +with your hands. This feature would let players bet more realistically (rather than just clicking a button) or could help +simulate chip shuffling (which was mentioned earlier as a lacking aspect). The other way to increase tactility would be +to utilize hand gestures, rather than UI buttons, to signal actions. For example, players could tap the table with a fist to +signal a ‘check,’ as in traditional poker. These features would further increase the immersion of PokAR. +Finally, AR could be utilized to provide helpful statistical annotations for players. Possible annotations could include +the probability of winning or the probability of making a certain hand. To implement this feature, one must first +implement a computer vision model to recognize and classify playing cards. This has been done in the past with high +accuracy (> 99%) [8]. This feature would further reduce the mental load on players and help them play and learn poker +more effectively. +Manuscript submitted to ACM + +10 +Gamba and Monroy-Hernández +REFERENCES +[1] Ella Dagan, Ana Cárdenas Gasca, Ava Robinson, Anwar Noriega, Yu Jiang Tham, Rajan Vaish, and Andrés Monroy-Hernández. 2022. Project IRL: +Playful Co-Located Interactions with Mobile Augmented Reality. Proceedings of the ACM on Human-Computer Interaction 6, CSCW1 (March 2022), +1–27. https://doi.org/10.1145/3512909 arXiv:2201.02558 [cs]. +[2] Geoffrey Fisk. 2020. How Many Hands Are Played Per Hour in Live Poker Games? https://upswingpoker.com/hands-per-hour-live-poker-vs-online/ +[3] Christothea Herodotou. 2010. Social Praxis Within and Around Online Gaming: The Case of World of Warcraft. In 2010 Third IEEE International +Conference on Digital Game and Intelligent Toy Enhanced Learning. 10–22. +[4] Christothea Herodotou, Niall Winters, and Maria Kambouri. 2015. An Iterative, Multidisciplinary Approach to Studying Digital Play Motivation: +The Model of Game Motivation. Games and Culture 10, 3 (May 2015), 249–268. https://doi.org/10.1177/1555412014557633 +[5] Hyun-Woo Lee, Sanghoon Kim, and Jun-Phil Uhm. 2021. Social Virtual Reality (VR) Involvement Affects Depression When Social Connectedness +and Self-Esteem Are Low: A Moderated Mediation on Well-Being. Frontiers in Psychology 12 (2021). https://www.frontiersin.org/articles/10.3389/ +fpsyg.2021.753019 +[6] Meta. 2019. Poker VR - Multi Table Tournaments on Oculus Quest. https://www.oculus.com/experiences/quest/2257223740990488/ +[7] Fast Offshore. 2021. Online poker sector overview for 2021: Stats, key drivers and more. https://fastoffshore.com/2021/10/online-poker-sector- +overview-2021/ +[8] Arjun Rohlfing-Das. 2020. Image Classification for Playing Cards. https://medium.com/swlh/image-classification-for-playing-cards-26d660f3149e +[9] Hiroyuki Sakuma, Tetsuo Yamabe, and Tatsuo Nakajima. 2012. Enhancing Traditional Games with Augmented Reality Technologies. In 2012 9th +International Conference on Ubiquitous Intelligence and Computing and 9th International Conference on Autonomic and Trusted Computing. 822–825. +https://doi.org/10.1109/UIC-ATC.2012.95 +[10] Nina Savela, Atte Oksanen, Markus Kaakinen, Marius Noreikis, and Yu Xiao. 2020. Does Augmented Reality Affect Sociability, Entertainment, and +Learning? A Field Experiment. Applied Sciences 10, 4 (Jan. 2020), 1392. https://doi.org/10.3390/app10041392 Number: 4 Publisher: Multidisciplinary +Digital Publishing Institute. +[11] Snap. 2022. Connected Lenses Overview | Docs. https://docs.snap.com/lens-studio/references/guides/lens-features/connected-lenses/connected- +lenses-overview +[12] Snap. 2022. How do I use Lenses on Snapchat? https://support.snapchat.com/en-US/a/face-world-lenses +[13] Snap. 2022. Spectacles by Snap Inc. • The Next Generation of Spectacles. https://www.spectacles.com/ +[14] Snap. 2022. Tracking Modes | Docs. https://docs.snap.com/lens-studio/references/guides/lens-features/tracking/world/tracking-modes +[15] Christoph Thul. 2013. PokerTool - Entwicklung und Implementierung einer AR-Android-Anwendung für Wahrscheinlichkeitsberechnungen bei +Texas Holdem Poker. (Sept. 2013). https://kola.opus.hbz-nrw.de/opus45-kola/frontdoor/index/index/docId/769 +[16] Mark Weiser. 1991. The Computer for the 21st Century. (1991). +Manuscript submitted to ACM + +PokAR: Facilitating Poker Play Through Augmented Reality +11 +A +STUDY PROTOCOL +Below is the process we asked participants to follow during the study: +(1) Participants were found through a poker club on campus and recruited by email. +(2) Participants were asked to respond to the pre-study survey. +(3) The participants were randomly paired into dyads for heads-up poker play. +(4) During the study: +(a) Participants were asked to download Snapchat (if necessary) and scan a code to gain access to the PokAR +Snapchat Lens. +(b) Participants were asked to sign consent forms. +(c) Participants were asked to use PokAR to play heads-up poker (without using real-world money) for 25 minutes. +(d) We took notes on comments, reactions, game pace, frustrations, etc. We took photos and videos throughout. +We also answered questions about the application when asked, but we avoided guiding the players. +(5) After play, participants were asked to respond to the post-study survey. +B +PRE-STUDY SURVEY +Below are the questions asked during the pre-study survey. +• How well do you know the rules of Heads-Up Texas Hold’em Poker? +• How often do you play poker? +C +POST-STUDY SURVEY +Below are the questions asked during the post-study survey. +• Please rate each of the following PokAR features in terms of its effectiveness when compared to the corresponding +object/action in real-life Texas hold’em poker? +– 3D AR Chips +– UI Action Buttons +– Game Messages +– Counting Stacks +– Awarding Pots +• How much did AR affect the in-person social aspects of the game of poker? +• How did augmented reality affect the game pace of poker (# of hands played / unit time)? Ignore the first few +hands in which you were learning the application. +• Overall, how would you describe your experience with PokAR? +D +CODE REPOSITORY +The code for this project can be found at the following GitHub repository: https://github.com/adamgamba/PokAR. +Manuscript submitted to ACM + diff --git a/0tAyT4oBgHgl3EQfoPjL/content/tmp_files/load_file.txt b/0tAyT4oBgHgl3EQfoPjL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ddf58d6918494b0a4ece0a0f44c7b4bbcbe6e921 --- /dev/null +++ b/0tAyT4oBgHgl3EQfoPjL/content/tmp_files/load_file.txt @@ -0,0 +1,343 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf,len=342 +page_content='PokAR: Facilitating Poker Play Through Augmented Reality ADAM GAMBA and ANDRÉS MONROY-HERNÁNDEZ, Princeton University, USA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Two players using the PokAR application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We introduce PokAR, an augmented reality (AR) application to facilitate poker play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR aims to alleviate three difficulties of traditional poker by leveraging AR technology: (1) need to have physical poker chips, (2) complex rules of poker, (3) slow game pace caused by laborious tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Despite the potential benefits of AR in poker, not much research has been done in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In fact, PokAR is the first application to enable AR poker on a mobile device without requiring extra costly equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This has been done by creating a Snapchat Lens 1 which can be used on most mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We evaluated this application by instructing 4 participant dyads to use PokAR to engage in poker play and respond to survey questions about their experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We found that most PokAR features were positively received, AR did not significantly improve nor hinder socialization, PokAR slightly increased the game pace, and participants had an overall enjoyable experience with the Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' These findings led to three major conclusions: (1) AR has the potential to augment and simplify traditional table games, (2) AR should not be used to replace traditional experiences, only augment them, (3) Future work includes additional features like increased tactility and statistical annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' CCS Concepts: • Human-centered computing → Collaborative and social computing devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additional Key Words and Phrases: connected lens, augmented reality, poker, co-located, interaction, socialization ACM Reference Format: Adam Gamba and Andrés Monroy-Hernández.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR: Facilitating Poker Play Through Augmented Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 1, 1 (January 2023), 11 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='XXXXXXX 1A Lens in Snapchat is an experience that utilizes augmented reality to transform the world around you [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Authors’ address: Adam Gamba, agamba@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Andrés Monroy-Hernández, andresmh@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='edu, Princeton University, Princeton, New Jersey, USA, 08544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Permission to make digital or hard 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM Manuscript submitted to ACM 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='00505v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='HC] 2 Jan 2023 l：61:598 old2 Gamba and Monroy-Hernández 1 INTRODUCTION The goal of this project is to facilitate heads-up Texas hold’em poker play through augmented reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Poker is cumbersome to play in its current form, requiring players to have poker chips and knowledge of the complex rules to play correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Without an experienced player to guide the game, new players often find it difficult to learn the rules and play correctly [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, due to the burden of physical chips, it is difficult to play poker in many scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=', at the beach, while traveling, or camping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Finally, the game pace is often slowed due to poker’s complex rules and the need for laborious tasks like counting chip stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Augmented reality technology is well-equipped to solve these issues in three ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Firstly, AR can eliminate the need for physical poker chips by instead utilizing AR to render chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Next, AR can help guide players through the complex rules of poker by hinting at legal actions during gameplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Finally, AR can help decrease the burden of laborious tasks (like counting chips) and increase the game pace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR helps alleviate these three issues, which we’ll discuss further throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Poker is a popular game, with over 120 million players worldwide playing regularly online [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Texas hold’em is one of the most popular poker variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In this variant, players are dealt two private cards and five community cards, and they battle to make the best hand or bluff opponents into folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' ’Heads-up’ poker is a term used to describe poker played by just two players, head-to-head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In its current state, PokAR supports only heads-up Texas hold’em poker, but with future work, it could be extended to more players and more variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Throughout this paper, we will use the term ’poker’ to refer to heads-up Texas hold’em poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Poker is a classic example of a social, co-located game, since poker, by design, emphasizes in-person, co-located interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Players often look at each other and speak to each other during a poker game, either to gain information or to socialize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Also, poker forces players to focus on the same enablers, or "physical objects that trigger and are the focus of the AR experience" [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' These enablers, like playing cards and poker chips, can help guide an AR experience and engage players more closely than in games that do not have a similar shared focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For the above reasons, we chose to augment poker in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR is not intended to replace traditional poker, rather it helps people play when traditional poker would be difficult or impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The goal of AR applications should be to disappear completely and seamlessly immerse the user in a realistic experience that combines reality with augmentation [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This disappearance frees users to utilize these applications more effortlessly, allowing them to focus on new goals, beyond the application itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2 RELATED WORK While we have established AR as a possible solution to the aforementioned issues with poker, very little work has been done concerning AR poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, while AR is not yet a heavily explored area, researchers have argued that “A Poker-Assistance-Software is an ideal test area for an AR Application with real added value,” with possible areas to add value including automation and statistical estimations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Similar projects in the past have all relied on physical means to augment reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For example, researchers used overhead projectors to project all aspects of the poker game (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=', cards, chips, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=') onto a table [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, researchers have used RFID playing cards to detect the dealt cards [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' While this study succeeded in creating an AR application to ease some of the same cumbersome aspects of poker tackled by PokAR (physical chips, complex rules, slow game pace), it did so using a high-cost solution, which is impractical for most recreational use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, they disregarded studying how this AR setup influenced social interactions in poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM PokAR: Facilitating Poker Play Through Augmented Reality 3 Furthermore, people have utilized virtual reality (VR) in the past to create commercial poker video games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' One example is PokerVR by Meta, which uses “expressive avatars built for reading tells with growing customizations” [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' While they may say this, VR poker applications still just employ static players’ avatars, which do not emphasize the in-person, social nature of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR is the first project to enable AR poker without needing additional equipment other than a mobile device and a regular deck of cards (like an overhead projector or a VR headset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This is a worthwhile problem because it is the first application to enable AR poker at a low cost, since it only requires a few commonly-owned pieces of equipment (mobile devices and playing cards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, utilizing AR over VR allows the gameplay to emphasize the social aspects of co-location and increase socialization when compared to VR implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Co-located gaming has been shown to lead to more effective and enjoyable gaming, as players can more easily communicate and build social relationships [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' One major reason for this is "out-of-the-game, game-related communi- cation" [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' By having the ability to converse about other topics while simultaneously being involved in a game with another player, these players are given the opportunity to build a deeper connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, when comparing socialization in AR and VR, prior research tends to support increased socialization in AR applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' AR games have the ability to "potentially enhance social communication and social interaction between people" [10], whereas high-involvement in VR games could potentially isolate users socially and "negatively affect their well-being" [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Thus, we chose to develop an AR application rather than a VR application to reap the social benefits of shared, co-located experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 3 POKAR SYSTEM The PokAR Snapchat Lens allows users to play heads-up poker with another player on two mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' AR visual annotations include 3D models of poker chips which dynamically render with changing stack size, and 3D text above the chip stacks denoting the size of each stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2D visual annotations include number of hands played, current round, current dealer, previous action, amount to call, waiting message, and UI buttons with labels "Check," "Call," "Bet," "Raise," and "Fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='" All AR and 2D annotations render dynamically with changing game state, stack amounts, and legal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR implements five main features to help achieve its motivating goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' “3D AR Chips” - PokAR renders 3D models of chips to eliminate the requirement of needing physical poker chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' “UI Action Buttons” - 2D buttons rendered on the player’s screen allows them to select among and perform legal actions at the current state of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' “Game Messages” - Messages provide additional information to both players about the actions of players, bet amounts, and more throughout the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' “Counting Stacks” - A live count of the number of chips in all chip stacks is rendered above the 3D models, eliminating the hassle of counting chips manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' “Awarding Pots” - After a winner is determined (through folding or at showdown), the chips are automatically awarded to the winner, eliminating the hassle of moving chips manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='1 Approach To achieve our motivating goals, we developed a Snapchat Lens which could be used on any mobile device to facilitate poker play through AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Besides having physical playing cards, players will play the game of poker in AR Manuscript submitted to ACM 4 Gamba and Monroy-Hernández Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Side-by-side points of view of the same game of PokAR on two mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' by interacting with their augmented environment through a mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We used the Snap Lens Studio IDE with JavaScript for development, the Snapchat app for testing and deployment, and GitHub for version control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Code for this project can be found at the link in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Physical playing cards act as an enabler to ground the game in physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' A demonstration of the completed application is shown in Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In the development of PokAR, we faced and solved three major implementation subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' They will be discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='2 Subproblem 1: Modeling Poker in Code The first implementation subproblem to solve was to figure out how to model a game of poker in code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' By nature of the rules of poker, the game is deterministic based on previous player actions within a betting round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Thus, the game state can be modeled using a Deterministic Finite Automaton (DFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The DFA for our application determines the legal actions and/or termination state of the betting round, given previous actions within the betting round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' At the end of each betting round, one of two termination states is reached, dictating whether the hand has ended or players will advance to the next betting round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Fig 3 and Fig 4 show a graphical representation of the DFAs we used in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In both DFAs, the application begins with the Start state and terminates in either the endHand() or advance() state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The double-headed arrow represents the possible cycle of betting, raising, reraising, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=', until one of the players is eventually all-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Player ’A’ is the one assigned ‘opponent’ at the start of a hand, and player ’B’ is the ‘dealer.’ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='3 Subproblem 2: Rendering 3D Objects Stably The second implementation subproblem was to figure out how to render 3D objects in the world and effectively track them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Originally the implementation used Snap’s World Tracking [14] functionality, and then its Surface Tracking [14] functionality, with neither proving to be too accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Chip stacks are small and users expect them to stay in Manuscript submitted to ACM 5:49 5:49 79 Hand #: 1 Hand #:1 Round: Preflop Round:Preflop Dealer:Opponent Dealer: Me Amount to Call: $1 $99 Opponent $98 Pot:ss Waiting for opponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='. 1 Z 3 4 5 6 7 8 9 O ） $ & @ X ABC space done Send ChatPokAR: Facilitating Poker Play Through Augmented Reality 5 Start B Checks B Bets endhand() A Checks A Bets A Folds A Calls advance() B Folds B Calls Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Pre-flop DFA (used before any community cards are dealt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' relatively the same position throughout a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' However, with World Tracking and Surface Tracking, chip stacks would move throughout the room quite a bit if the mobile device’s camera was moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Then, we decided to use Marker Tracking [14], which uses a printed marker pattern to mark a position in the physical world and allow the application to render objects relative to that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Marker tracking proved to be very accurate and stable for rendering 3D objects in AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Chip stacks would no longer move throughout the room, as they were grounded in a location in 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Marker Tracking, however, is only a temporary solution while alternative tracking solutions are improved with continued computer vision research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='4 Subproblem 3: Connecting Multiple Players The third implementation subproblem was to figure out how to connect two players within a single Snapchat Lens to play together and share game data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' To solve this problem, we designed an API to connect two different mobile devices and allow them to send messages between each other, including updates on the game state and player actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This API is built on top of Snap’s Connected Lenses feature [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Thus, the two players will always observe the same data on different devices in real time, unifying their gameplay experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM 6 Gamba and Monroy-Hernández Start A Calls A Raises endhand() B Checks B Raises B Folds B Calls advance() A Folds A Calls A Folds Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Post-flop DFA (used once community cards have started to been dealt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 4 EVALUATION We recruited 8 participants who were found through a poker club on campus and recruited by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We asked participants to respond to a pre-study survey to learn about their individual experience with poker and its rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This survey can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We randomly paired participants into 4 dyads to utilize PokAR to play heads-up poker for 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Then, we asked them to respond to a post-study survey about their experience with the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In this survey, we asked participants about the benefits and detriments of particular PokAR features, the effects of AR on socialization, the effects of AR on game pace, and the overall experience with PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This survey can be found in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The study protocol above is described in more detail in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 5 RESULTS Although participants had varying levels of poker expertise, they all had a self-reported understanding of the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Based on the results of the pre-study survey, we grouped the 8 participants into three groups of varying experience levels for analysis: Highly Experienced (play poker multiple times a week, 𝑛 = 2), Moderately Experienced (play poker weekly to monthly, 𝑛 = 3), and Slightly Experienced (play poker yearly or less, 𝑛 = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM PokAR: Facilitating Poker Play Through Augmented Reality 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='1 Evaluation of Features We asked participants to rate each of the five major PokAR features on a scale of 1 (detrimental) to 5 (beneficial) in terms of its effectiveness compared to the corresponding object/action in real-life poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Each feature earned an average score > 3 (leaning beneficial) among all participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Specifically, "3D AR Chips" scored a 4, "UI Action Buttons" scored a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='25, "Game Messages" scored a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='875, "Counting Stacks" scored a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='375, and "Awarding Pots" scored a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Notably, the only features that scored < 3 (leaning detrimental) were “3D AR Chips,” “UI Action Buttons,” and “Game Messages” for the Highly Experienced subgroup of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This could be explained by the fact that all three of these features are intended to alleviate the requirement of knowing the complex rules of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' However, in the pre-study survey, all members of this subgroup answered that they play poker quite often and they confidently understand all the rules, so these features likely just got in the way of their gameplay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The features of “Counting Stacks” and “Awarding Pots” were, however, positively received by all three subgroups of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='2 Evaluation of Socialization The average response to the survey question: “How much did AR affect the in-person social aspects of the game of poker?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' was a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='25 on a scale of 1 (negatively) to 5 (positively), meaning that AR neither significantly improved nor impaired the in-person social aspects of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This is a beneficial result, as one of PokAR’s goals was to supplement the game of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We did not implement social-related features intended to improve socialization, but this result supports the claim that the AR features of PokAR did not impair socialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In other words, players are utilizing PokAR as a tool to enable poker play, which does not get in the way of the traditional social interactions at a poker table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, multiple participants noted that AR did not heavily influence socialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' P1 stated that the experience was “no different, we could still talk and converse,” and P5 stated that AR “Didn’t affect socialization because everything was still in person.” 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='3 Evaluation of Game Pace The average response to the survey question: “How did augmented reality affect the game pace of poker?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' was a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='75 on a scale of 1 (slowed the game) to 5 (sped up the game), meaning that AR slightly increased the game pace of poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' In this study, the average game pace was 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='3 hands/hour (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='2 hands/hour for the Highly Experienced subgroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Comparatively, “A typical live poker game will deal 25-30 per hour,” assuming 9 players [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This section requires additional study, including a control session of each participant group playing traditional poker to compare the game pace with AR poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='4 Evaluation of Overall Experience On average, participants rated their overall experience at a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='5/5, overwhelmingly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' P2 stated that “It’s quicker, but annoying to hold the phone up.” P6 stated that “It was a different experience which took a little getting used to, but I enjoyed it.” P4 stated that it “Felt cool to have the chips tracked for you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Definitely could see myself using it on a camping trip or during traveling.” P5 stated that “It was cool, because sometimes I’d like to play poker but sometimes have no chips!”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 6 CONCLUSION Through developing a complete AR application and studying how people utilize it, we have generated three main conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM 8 Gamba and Monroy-Hernández 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='1 AR Has the Potential to Augment and Simplify Traditional Table Games After our work throughout this semester, we are confident that AR as a technology can and will be used in the future to augment and simplify traditional table games, like poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' While the technology is currently in a primitive state, it is continually evolving and progressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Several participants noted that the gameplay of PokAR was clunky since they had to constantly hold their mobile devices up to see the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' However, with improved AR technology, this annoyance will begin to fade away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For example, AR glasses like Spectacles will eliminate the need to hold up a mobile device [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This study revealed promising results concerning the future potential of AR in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For instance, the use of AR did not hinder socialization, and participants had a positive overall experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR features meant to facilitate gameplay were positively received by most players, and options to disable disruptive features would alleviate the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='2 AR Should Not Be Used to Replace Traditional Experiences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' It Should Be Used to Augment Them This conclusion stems from the fact that AR technology has innate limitations compared to the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For instance, AR experiences are less tactile than physical world experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' While software tricks exist to improve the tactility of AR experiences (like hand tracking, which enables object manipulation), it will never feel quite like the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For example, several participants noted that PokAR was missing one important aspect of traditional poker: chip shuffling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Chip shuffling is a common fidgeting technique among poker players in which they use one hand to rearrange a stack of chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Shuffling is almost unanimous among poker players and is commonly used to pass time and cure boredom during long poker sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' While AR could simulate chip shuffling, it could never reproduce the experience perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Early intuition about this conclusion is one reason we decided to utilize physical playing cards in PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' If playing cards were virtual, players would be playing an online poker game in which in-person social interactions were minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Players wouldn’t even need to be co-located to play PokAR anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This would be a case of using AR to replace a traditional game experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Instead, we decided to use physical cards and virtual chips in PokAR to afford some physical-world tactility to players and streamline some of the more annoying and time-consuming aspects of poker, like counting chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' As mentioned in the introduction, PokAR is not intended to replace traditional poker, only augment it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This is due to the inherent limitations of AR technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' More broadly, AR should not be used to replace traditional experiences, only augment them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Augmentations should be deliberately planned and carefully implemented to ensure that they do not take over the spirit of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Go too far with augmentation, and you approach the virtual reality world and lose out on social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='3 Future Work Our work on PokAR has revealed possible directions for future study and enhancements to the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Firstly, due to time constraints, not all features of poker were able to be added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR is currently limited to just two players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This design choice was made to reduce the project’s complexity, but this limit should be increased to the accepted limit of nine players to better simulate traditional poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Additionally, the option to chop pots (split pots equally between tied players) is currently not implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The option to run it multiple times (deal remaining cards multiple times in an all-in situation and award the pot proportionally to winners) is also not yet implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokAR would benefit from increased tactility, which is why we believe that it is a necessary direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Increased tactility could come in two forms, with the first being the ability to grab AR chips and manipulate them Manuscript submitted to ACM PokAR: Facilitating Poker Play Through Augmented Reality 9 with your hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This feature would let players bet more realistically (rather than just clicking a button) or could help simulate chip shuffling (which was mentioned earlier as a lacking aspect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The other way to increase tactility would be to utilize hand gestures, rather than UI buttons, to signal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' For example, players could tap the table with a fist to signal a ‘check,’ as in traditional poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' These features would further increase the immersion of PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Finally, AR could be utilized to provide helpful statistical annotations for players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Possible annotations could include the probability of winning or the probability of making a certain hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' To implement this feature, one must first implement a computer vision model to recognize and classify playing cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This has been done in the past with high accuracy (> 99%) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' This feature would further reduce the mental load on players and help them play and learn poker more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM 10 Gamba and Monroy-Hernández REFERENCES [1] Ella Dagan, Ana Cárdenas Gasca, Ava Robinson, Anwar Noriega, Yu Jiang Tham, Rajan Vaish, and Andrés Monroy-Hernández.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Project IRL: Playful Co-Located Interactions with Mobile Augmented Reality.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='com/lens-studio/references/guides/lens-features/connected-lenses/connected- lenses-overview [12] Snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' How do I use Lenses on Snapchat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' https://support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='snapchat.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='com/ [14] Snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Tracking Modes | Docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='snap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='com/lens-studio/references/guides/lens-features/tracking/world/tracking-modes [15] Christoph Thul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' PokerTool - Entwicklung und Implementierung einer AR-Android-Anwendung für Wahrscheinlichkeitsberechnungen bei Texas Holdem Poker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' https://kola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='opus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='hbz-nrw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='de/opus45-kola/frontdoor/index/index/docId/769 [16] Mark Weiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' The Computer for the 21st Century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM PokAR: Facilitating Poker Play Through Augmented Reality 11 A STUDY PROTOCOL Below is the process we asked participants to follow during the study: (1) Participants were found through a poker club on campus and recruited by email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (2) Participants were asked to respond to the pre-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (3) The participants were randomly paired into dyads for heads-up poker play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (4) During the study: (a) Participants were asked to download Snapchat (if necessary) and scan a code to gain access to the PokAR Snapchat Lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (b) Participants were asked to sign consent forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (c) Participants were asked to use PokAR to play heads-up poker (without using real-world money) for 25 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (d) We took notes on comments, reactions, game pace, frustrations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We took photos and videos throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' We also answered questions about the application when asked, but we avoided guiding the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' (5) After play, participants were asked to respond to the post-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' B PRE-STUDY SURVEY Below are the questions asked during the pre-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' How well do you know the rules of Heads-Up Texas Hold’em Poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' How often do you play poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' C POST-STUDY SURVEY Below are the questions asked during the post-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Please rate each of the following PokAR features in terms of its effectiveness when compared to the corresponding object/action in real-life Texas hold’em poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' – 3D AR Chips – UI Action Buttons – Game Messages – Counting Stacks – Awarding Pots How much did AR affect the in-person social aspects of the game of poker?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' How did augmented reality affect the game pace of poker (# of hands played / unit time)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Ignore the first few hands in which you were learning the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Overall, how would you describe your experience with PokAR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' D CODE REPOSITORY The code for this project can be found at the following GitHub repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content='com/adamgamba/PokAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} +page_content=' Manuscript submitted to ACM' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQfoPjL/content/2301.00505v1.pdf'} diff --git a/19E2T4oBgHgl3EQfiwfE/vector_store/index.pkl b/19E2T4oBgHgl3EQfiwfE/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..dbe408505187fbb16b85aecbf66109e228cf91bc --- /dev/null +++ b/19E2T4oBgHgl3EQfiwfE/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:76d609a55f97acd9c9112ef5a22af45c952d34d5e3baf8dd1600ce6b99c44230 +size 272976 diff --git a/1NAyT4oBgHgl3EQfbff_/content/tmp_files/2301.00266v1.pdf.txt b/1NAyT4oBgHgl3EQfbff_/content/tmp_files/2301.00266v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7ff2830e8601bd35459573074b06832085eca74 --- /dev/null +++ b/1NAyT4oBgHgl3EQfbff_/content/tmp_files/2301.00266v1.pdf.txt @@ -0,0 +1,9270 @@ +arXiv:2301.00266v1 [math.SG] 31 Dec 2022 +Action-angle coordinates and KAM theory for +singular symplectic manifolds +Eva Miranda +Arnau Planas +Laboratory of Geometry and Dynamical Systems, Department of +Mathematics & IMTech, Universitat Polit`ecnica de Catalunya, Barcelona +and CRM, Centre de Recerca Matem`atica, Bellaterra +Current address: UPC-Edifici P, Avinguda del Doctor Maran´on, 44-50, 08028, +Barcelona, Spain +Email address: evamiranda@upc.edu +Department of Mathematics, Universitat Polit`ecnica de Catalunya, +Barcelona +Email address: arnau.planas.bahi@gmail.com + +2020 Mathematics Subject Classification. +53D05, 53D20, 70H08, 37J35, 37J40 ; +37J39, 58D19 +Key words and phrases. amsbook, AMS-LATEX +Eva Miranda is supported by the Catalan Institution for Research and Advanced +Studies via an ICREA Academia Prize 2016 and ICREA Academia Prize 2021. +Eva Miranda is also supported by the Spanish State Research Agency, through the +Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence +in R&D (project CEX2020-001084-M). Eva Miranda also acknowledges partial +support from the grant “Computational, dynamical and geometrical complexity in +fluid dynamics”, Ayudas Fundaci´on BBVA a Proyectos de Investigaci´on Cient´ıfica +2021. Both authors are supported by the project PID2019-103849GB-I00 of the +Spanish State Agency AEI /10.13039/501100011033. +Dedicated to the memory of Amelia Galcer´an Sorribes. + +Contents +Preface +vii +Part 1. +Introduction and preliminaries +1 +Chapter 1. +Introduction +3 +1.1. +Structure and results of this monograph +3 +Chapter 2. +A primer on singular symplectic manifolds +7 +2.1. +b-Poisson manifolds +7 +2.2. +On bm-Symplectic manifolds +10 +2.3. +Desingularizing bm-Poisson manifolds +12 +Chapter 3. +A crash course on KAM theory +15 +Part 2. +Action-angle coordinates and cotangent models +19 +Chapter 4. +An action-angle theorem for bm-symplectic manifolds +23 +4.1. +Basic definitions +23 +4.2. +On bm-integrable systems +24 +4.3. +Examples of bm-integrable systems +26 +4.4. +Looking for a toric action +29 +4.5. +Action-angle coordinates on bm-symplectic +manifolds +32 +Chapter 5. +Reformulating the action-angle coordinate via cotangent lifts +37 +5.1. +Cotangent lifts and Arnold-Liouville-Mineur in Symplectic Geometry +37 +5.2. +The case of bm-symplectic manifolds +38 +Part 3. +A KAM theorem for bm-symplectic manifolds +41 +Chapter 6. +A new KAM theorem +43 +6.1. +On the structure of the proof +43 +6.2. +Technical results +50 +6.3. +A KAM theorem on bm-symplectic manifolds +69 +Chapter 7. +Desingularization of bm-integrable systems +93 +Chapter 8. +Desingularization of the KAM theorem on bm-symplectic +manifolds +97 +Chapter 9. +Potential applications to Celestial mechanics +101 +9.1. +The Kepler Problem +101 +v + +vi +CONTENTS +9.2. +The Problem of Two Fixed Centers +102 +9.3. +Double Collision and McGehee coordinates +103 +9.4. +The restricted three-body problem +105 +Bibliography +107 + +Preface +I confess I envy the planets — +they’ve got their own orbits and +nothing stands on their way. +Intermezzo, +Mykhailo +Kotsi- +ubynsky. +This monograph explores classification and perturbation problems for inte- +grable systems on a class of Poisson manifolds called bm-Poisson manifolds. This +is the first class of Poisson manifolds for which perturbation theory is established +outside the symplectic category. Even if the class of bm-Poisson manifolds is not +ample enough to represent the wild set of Poisson manifolds, this investigation can +be seen as a first step for the study of perturbation theory for general Poisson man- +ifolds. In view of the work of the first author with Nest, the theorems established +in this monograph constitute more than a mild generalization in Poisson Geometry +and, this toy example, sets the path to consider KAM theory in the general realm +of Poisson manifolds. Reduction theorems and bm-symplectic manifolds have been +recently explored in [MM22]. This monograph contributes to the theory opening +the investigation of perturbation theory on these manifolds thus completing other +facets in the study of their dynamics as the recent work on the Arnold conjecture +[BMO22]. +Symplectic geometry has been the common language of physics as the position- +momentum tandem can be modelled over a cotangent bundle. Cotangent bundles +are naturally endowed with a symplectic form which is a non-degenerate closed +2-form. The symplectic form of the cotangent bundle is given by the differential of +the Liouville one-form. +bm-Poisson manifolds are manifolds that are symplectic away from a hyper- +surface along which they satisfy some transversality properties. They often model +problems on symplectic manifolds with boundary such as the study of their de- +formation quantization and celestial mechanics. As on the complementary of the +critical set the manifolds are symplectic, extending the investigation of Hamiltonian +dynamics to this realm is key to understand Hamiltonian Dynamics on compact- +ification of symplectic manifolds. Several regularization transformations used in +celestial mechanics (as McGehee or Moser regularization) provide examples of such +compactifications. +One of the interesting properties of bm-Poisson manifolds is that their investi- +gation can be achieved considering the language of bm-forms. That is to say, we can +work with forms that are symplectic away from the critical set and admit a smooth +extension as a form over a Lie algebroid generalizing De Rham forms as form over +the standard Lie algebroid of the tangent bundle of the manifold. +To consider +bm-forms the standard tangent bundle is replaced by the bm-tangent bundle. This +vii + +viii +PREFACE +allows us to mimic symplectic geometry by replacing the cotangent bundle by the +dual of the bm-tangent bundle. However, Poisson geometry leaves its footprint and +new invariants which can be identified as the modular class of the Poisson structure +arise already at the semilocal level. +Contrary to the initial expectations, several of the results for bm-symplectic +manifolds do not resemble the b-case so far. Considering these more general singu- +larities yields a better understanding of the general Poisson case and the different +levels of complexity. As an illustration of this phenomena: in the study of quan- +tization of those systems an interesting pattern makes the quantization radically +different in the even and odd case [GMW18b, GMW21] and the resulting model +is finite-dimensional in the b-case. Understanding how the different degrees m are +related is a hard task: The desingularization technique introduced by Guillemin- +Miranda-Weitsman in [GMW17] turned out to have important applications in the +investigation of complexity properties of toric bm-symplectic manifolds [GMW18a] +and to the study of the Arnold conjecture in this set-up [BMO22]. In this mono- +graph we explore a new facet of these manifolds: that of perturbation theory. +In the second part of the monograph we consider integrable systems on these +manifolds turn out to have associated generalized Hamiltonian actions of tori in +a neighbourhood of a Liouville torus. We use this generalized Hamiltonian group +action to prove existence of action-angle coordinates in a neighborhood of a Liouville +torus. The action-angle coordinate theorem that we prove gives a semilocal normal +form in the neighbourhood of a Liouville torus for the bm-symplectic structure +which depends on the modular weight of the connected component of the critical +set in which the Liouville torus is lying and the modular weights of the associated +toric action. This action-angle theorem allows us to identify a neighborhood of +the Liouville torus with the bm-cotangent lift of the action of a torus acting by +translations on itself. This interpretation of the action-angle theorem as cotangent +lift allows us to identify the modular weight as their only semilocal invariant. In +doing so, we compare this action-angle coordinate theorem with the classical action- +angle coordinate theorems for symplectic manifolds and an action-angle theorem +for folded symplectic manifolds ([CM22]). +In part 3 of the monograph we study perturbation theory in this new set-up +and examine some potential applications to physical systems. In particular, we +prove a KAM theorem for bm-Poisson manifolds which clearly refines and improves +the one obtained for b-Poisson manifolds in [KMS16a]. As an outcome of this +result together with the extension of the desingularization techniques of Guillemin- +Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem +for folded symplectic manifolds where KAM theory has never been considered be- +fore. In the way, we also obtain a brand-new KAM theorem for symplectic manifolds +where the perturbation keeps track of a distinguished hypersurface. In celestial me- +chanics, this distinguished hypersurface can be the line at infinity or the collision +set. +Barcelona, December 2022, Eva Miranda and Arnau Planas + +Part 1 +Introduction and preliminaries + + +CHAPTER 1 +Introduction +Both symplectic and Poisson geometry emerge from the study of classical me- +chanics. Both are broad fields widely studied and with powerful results. But as +Poisson structures are far more general than the symplectic ones, most outstanding +results in symplectic geometry do not translate well to Poisson manifolds. Here is +where bm-Poisson structures come to play. bm-Poisson structures (or bm-symplectic +structures) lie somewhere between these two worlds. They extend symplectic struc- +tures but in a really controlled way. This is why fundamental results in symplectic +geometry still work in bm-symplectic geometry. However, an adaptation of these +theories like deformation or Moser theory requires some work (see [GMPS15a] +and others). +The study of bm-Poisson geometry sparked from the study of symplectic man- +ifold with boundary [Mel93a]. In the last years the interest in this field increased +after the classification result for b-Poisson structures obtained in [Rad02]. Later +on, [GMP14] translated these structures to the language of forms and started +applying symplectic tools to study them. A lot of papers in the following years +studied different aspects of these structures: [GMP10], [GMP14], [GMPS15b], +[GMW17], [MOT14] and [GLPR17] are some examples. +Inspired by the study of manifolds with boundary, we work on a pair of mani- +folds (M, Z) where Z is an hypersurface and call this pair b-manifold +In this context, [Sco16] generalized the b-symplectic forms by allowing higher +degrees of degeneracy of the Poisson structures. +The bm-symplectic structures +inherit most of the properties of b-symplectic structures. This booklet focuses on +different aspects of the investigation of bm-symplectic structures covering mainly +integrable systems and KAM theory. First, we present some preliminary notions +necessary to address the problem of perturbation. +We present an action-angle +theorem for bm-Poisson structures and state and prove the KAM theory equivalent +in manifolds with bm-symplectic structures. +1.1. Structure and results of this monograph +1.1.1. Part 1: Introduction and Preliminaries. In the preliminaries, we +give the basic notions that lead to the questions we are addressing in this booklet. +In the first part, we introduce the concept of b-Poisson manifolds or b-symplectic +manifolds, a class of Poisson manifold which is symplectic outside a critical hyper- +surface. It study comes motivated by the investigation of manifolds with boundary. +Next, we talk about a generalization of these structures, that allow a higher degree +of degeneracy of the structure: the bm-symplectic structures. These structures are +the main focus of our investigations. A key concept that will play an important +work in this book is the study of the desingularization of these singular structures. +3 + +4 +1. INTRODUCTION +Finally, we give a short introduction to KAM theory, a theory that will be gener- +alized in the setting of bm-manifolds in the last chapter. +Motivation comes from several examples of singular symplectic structures ap- +pearing naturally in classical problems of celestial mechanics which are discussed +on the last chapter of the monograph. We also describe the difficulties of finding +these examples, and the subtleties of dealing with these singular structures in the +exploration of conservative systems. +1.1.2. Part 2: Action-angle coordinates and cotangent models for bm- +integrable systems. In this Chapter we define the concept of bm-functions and +bm-integrable systems. We present several examples of bm-integrable systems that +come from classical mechanics. After all this we present a version of the action-angle +theorem for bm-symplectic manifolds. +Theorem A. Let (M, x, ω, F) be a bm-integrable system, where F = (f1 = +a0 log(x)+�m−1 +j=1 aj 1 +xj , f2, . . . , fn). Let m ∈ Z be a regular point, and such that the +integral manifold through m is compact. Let Fm be the Liouville torus through m. +Then, there exists a neighborhood U of Fm and coordinates (θ1, . . . , θn, σ1, . . . , σn) : +U → Tn × Bn such that: +(1) We can find an equivalent integrable system F = (f1 = a′ +0 log(x) + +�m−1 +j=1 a′ +j +1 +xj ) such that a′ +0, . . . , a′ +m−1 ∈ R, +(2) +ω|U = + + +m +� +j=1 +c′ +j +c +σj +1 +dσ1 ∧ dθn + + + +n +� +i=2 +dσi ∧ dθi +where c is the modular period and c′ +j = −(j − 1)a′ +j−1, also +(3) the coordinates σ1, . . . , σn depend only on fn, . . . fn. +1.1.3. Part 3: KAM theory on bm-symplectic manifolds and applica- +tions to Celestial Mechanics. In this chapter we provide several KAM theorem +for (singular) symplectic manifolds including bm-symplectic manifolds. +We begin by considering perturbation theory for bm-symplectic manifolds. Then +we give an outline of how to construct the bm-symplectomorphism that will be the +main character of the proof of the KAM theorem for bm-symplectic manifolds. After +this, we show some technical results that are needed for the proof. These technical +results even if quite similar to the standard KAM equivalents, have some subtleties +that need to be addressed. We end the chapter with the proof of the bm-KAM +theorem and several applications to establish KAM theorems in other singular sit- +uations (folded symplectic manifolds) and on symplectic manifolds with prescribed +invariant hypersurfaces. +The first KAM theorem is the following: +Theorem B. Let G ⊂ Rn, n ≥ 2 be a compact set. Let H(φ, I) = ˆh(I) + +f(φ, I), where ˆh is a bm-function ˆh(I) = h(I) + q0 log(I1) + �m−1 +i=1 +qi +Ii +1 defined on +Dρ(G), with h(I) and f(φ, I) analytic. Let ˆu = ∂ˆh +∂I and u = ∂h +∂I . Assume | ∂u +∂I |G,ρ2 ≤ +M, |u|ξ ≤ L. Assume that u is µ non-degenerate (| ∂u +∂I | ≥ µ|v| for some µ ∈ R+ and +I ∈ G. Take a = 16M. Assume that u is one-to-one on G and its range F = u(G) +is a D-set. Let τ > n − 1, γ > 0 and 0 < ν < 1. Let + +1.1. STRUCTURE AND RESULTS OF THIS MONOGRAPH +5 +(1) +(1.1) +ε := ∥f∥G,ρ ≤ +ν2µ2ˆρ2τ+2 +24τ+32L6M 3 γ2, +(2) +(1.2) +γ ≤ min(8LMρ2 +ν ˆρτ+1 , L +K′ ) +(3) +(1.3) +µ ≤ min(2τ+5L2M, 27ρ1L4Kτ+1, βντ+122τ+1ρτ +1), +where ˆρ := min +� +νρ1 +12(τ+2), 1 +� +. Define the set ˆG = ˆGγ := {I ∈ G− 2γ +µ |u(I) is τ, γ, c, ˆq− +Dioph.}. Then, there exists a real continuous map T : W ρ1 +4 (Tn) × ˆG → Dρ(G) an- +alytic with respect the angular variables such that +(1) For all I ∈ ˆG the set T (Tn ×{I}) is an invariant torus of H, its frequency +vector is equal to u(I). +(2) Writing T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)) with estimates +|Tφ(φ, I)| ≤ 22τ+15ML2 +ν2ˆρ2τ+1 +ε +γ2 +|TI(φ, I))| ≤ 210+τL(1 + M) +ν ˆρτ+1 +ε +γ +(3) meas[(Tn ×G)\T (Tn× ˆG)] ≤ Cγ where C is a really complicated constant +depending on n, µ, D, diamF, M, τ, ρ1, ρ2, K and L. +Also, we obtain a way to associate a standard symplectic integrable system or +a folded integrable system to a bm-integrable system, depending on the parity of m. +This is done in such a way that the dynamics of the desingularized system are the +same than the dynamics of the original one. So it defines a honest desingularization +of the integrable system. +Theorem C. The desingularization transforms a bm-integrable system into an +integrable system on a symplectic manifold for even m. +For m odd, the desin- +gularization associates to it a folded integrable system. +The integrable systems +satisfy: +Xω +fj = Xωǫ +fjǫ. +This allows us to obtain two new KAM theorems using this desingularization +combined with the former bm-KAM theorem. The first of these theorems is a KAM +theorem for standard symplectic manifolds, where the perturbation has a particu- +lar expression. This result is more restrictive than the standard KAM theorem but +allow us to guarantee that the perturbations leave a given hypersurface invariant. +This means that the tori belonging to that hypersurface remain on the hypersur- +face after the perturbation. This can be interesting for a number of reasons and +situations such as problems in Celestial mechanics where it is convenient to keep +track of a particular hypersurface such as the line at infinity. The higher order +singularities allow to consider perturbations that are tangent to the hypersurface +up to a certain order. + +6 +1. INTRODUCTION +Theorem D. Consider a neighborhood of a Liouville torus of an integrable +system Fε as in 8.1 of a symplectic manifold (M, ωε) semilocally endowed with +coordinates (I, φ), where φ are the angular coordinates of the torus, with ωε = +c′dI1 ∧dφi +�n +j=1 dIj ∧dφj. Let H = (m−1)cm−1c′I1 +h(˜I)+R(˜I, ˜φ) be a nearly +integrable system where +� +˜I1 += +c′ Im+1 +1 +m+1 , +˜φ1 += +c′Im +1 φ1, +and +� ˜I += +(˜I1, I2, . . . , In), +˜φ += +(˜φ1, φ2, . . . , φn). +Then the results for the bm-KAM theorem 6.3 applied to Hsing = +1 +I2k−1 +1 ++ h(I) + +R(I, φ) hold also for this desingularized system. +The second one is a KAM theorem for folded-symplectic manifolds, where KAM +theory has not been considered to-date. +Theorem E. Consider a neighborhood of a Liouville torus of an integrable +system Fε as in 8.2 of a folded symplectic manifold (M, ωε) semilocally endowed +with coordinates (I, φ), where φ are the angular coordinates of the Torus, with +ωε = 2cI1dI1 ∧ dφ1 + �m +j=2 dIj ∧ dφj. Let H = (m − 1)cm−1cI2 +1 + h(˜I) + R(˜I, ˜φ) a +nearly integrable system with� +˜I1 += +2c Im+2 +1 +m+2 , +˜φ1 += +2cIm+1 +1 +φ1, +and +� ˜I += +(˜I1, I2, . . . , In), +˜φ += +(˜φ1, φ2, . . . , φn). +Then the results for the bm-KAM theorem 6.3 applied to Hsing = +1 +I2k +1 +h(I)+R(I, φ) +also hold for this desingularized system. +Last but not least, we illustrate the connection between bm-symplectic struc- +tures and classical mechanics by providing several examples. +Several potential +applications to celestial mechanics are discussed. + +CHAPTER 2 +A primer on singular symplectic manifolds +In this first chapter of the booklet we introduce basic notions on singular sym- +plectic structures, as well as some concepts on standard KAM theory. Those are +the two main pillars of this monograph. +Let M be a smooth manifold, a Poisson structure on M is a bilinear map +{·, ·} : C∞(M) × C∞(M) → C∞(M) which is skew-symmetric and satisfies both +the Jacobi identity and the Leibniz rule. It is possible to express {f, g} in terms +of a bivector field via the following equality {f, g} = Π(df ∧ dg) with Π a section +of Λ2(T M). Π is the associated Poisson bivector. We will use indistinctively +the terminology of Poisson structure when referring to the bracket or the Poisson +bivector. +A b-Poisson bivector field on a manifold M 2n is a Poisson bivector such that +the map +(2.1) +F : M → +2n +� +T M : p �→ (Π(p))n +is transverse to the zero section. Then, a pair (M, Π) is called a b-Poisson man- +ifold and the vanishing set Z of F is called the critical hypersurface. Observe +that Z is an embedded hypersurface. +This class of Poisson structures was studied by Radko [Rad02] in dimension +two and considered in numerous papers in the last years: [GMP10], [GMP14], +[GMPS15b], [GMW17], [MOT14] and [GLPR17] among others. +2.1. b-Poisson manifolds +Next, we recall classification theorem of b-Poisson surfaces as presented by Olga +Radko and the cohomological re-statement and proof given by Guillemin, Miranda +and Pires in [GMP14]. +In what follows, (M, Π) will be a closed smooth surface with a b-Poisson struc- +ture on it, and Z its critical hypersurface. +Let h be the distance function to Z as in [MOT14]1. +Definition 2.1. The Liouville volume of (M, Π) is the following limit: +V (Π) := limǫ→0 +� +|h|>ǫ ωn2. +The previous limit exists and it is independent of the choice of the defining +function h of Z (see [Rad02] for the proof). +Definition 2.2. For any (M, Π) oriented Poisson manifold, let Ω be a volume +form on it, and let uf denote the Hamiltonian vector field of a smooth function +1Notice the difference with [Rad02] where h is assumed to be a global defining function. +2For surfaces n = 1. +7 + +8 +2. A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS +f : M → R. The modular vector field XΩ is the derivation defined as follows: +f �→ Luf Ω +Ω +. +Definition 2.3. Given γ a connected component of the critical set Z(Π) of a +closed b-Poisson manifold (M, Π), the modular period of Π around γ is defined +as: +Tγ(Π) := period of XΩ|γ. +Remark 2.4. The modular vector field XΩ of the b-Poisson manifold (M, Z) +does not depend at Z on the choice of Ω because for different choices for volume +form the difference of modular vector fields is a Hamiltonian vector field. Observe +that this Hamiltonian vector field vanishes on the critical set as Π vanishes there +too. +Definition 2.5. Let Mn(M) = Cn(M)/ ∼ where Cn(M) is the space of dis- +joint oriented curves and ∼ identifies two sets of curves if there is an orientation- +preserving diffeomorphism mapping the first one to the second one and preserving +the orientations of the curves. +The following theorem classifies b-symplectic structures on surfaces using these +invariants: +Theorem 2.6 (Radko [Rad02]). Consider two b-Poisson structures Π, Π′ on +a closed orientable surface M. Denote its critical hypersurfaces by Z and Z′. These +two b-Poisson structures are globally equivalent (there exists a global orientation +preserving diffeomorphism sending Π to Π′) if and only if the following coincide: +• the equivalence classes of [Z] and [Z′] ∈ Mn(M), +• their modular periods around the connected components of Z and Z′, +• their Liouville volume. +An appropriate formalism to deal with these structures was introduced in +[GMP10]. +Definition 2.7. A b-manifold3 is a pair (M, Z) of a manifold and an embed- +ded hypersurface. +In this way, the concept of b-manifold previously introduced by Melrose is +generalized to consider additional geometric structures on the manifold. +Definition 2.8. A b-vector field on a b-manifold (M, Z) is a vector field +tangent to the hypersurface Z at every point p ∈ Z. +Definition 2.9. A b-map from (M, Z) to (M ′, Z′) is a smooth map φ : M → +M ′ such that φ−1(Z′) = Z and φ is transverse to Z′. +Observe that if x is a local defining function for Z and (x, x1, . . . , xn−1) are +local coordinates in a neighborhood of p ∈ Z then the C∞(M)-module of b-vector +fields has the following local basis +(2.2) +{x ∂ +∂x, ∂ +∂x1 +, . . . , +∂ +∂xn−1 +}. +3The ‘b’ of b-manifolds stands for ‘boundary’, as initially considered by Melrose (Chapter 2 +of [Mel93b]) for the study of pseudo-differential operators on manifolds with boundary. + +2.1. b-POISSON MANIFOLDS +9 +Figure 1. Artistic representation of a b-function on a b-manifold +near the critical hypersurface. +In contrast to [GMP10], in this monograph we are not requiring the existence +of a global defining function for Z and orientability of M. However, we require the +existence of a defining function in a neighborhood of each point of Z. By relaxing +this condition, the normal bundle of Z need not be trivial. +Given (M, Z) a b-manifold, [GMP10] shows that there exists a vector bundle, +denoted by bT M whose smooth sections are b-vector fields. This bundle is called +the b-tangent bundle of (M, Z). +The b-cotangent bundle bT ∗M is defined using duality. A b-form is a section +of the b-cotangent bundle. Around a point p ∈ Z the C∞(M)-module of these +sections has the following local basis: +(2.3) +{ 1 +xdx, dx1, . . . , dxn−1}. +In the same way we define a b-form of degree k to be a section of the bundle +�k(bT ∗M), the set of these forms is denoted bΩk(M). Denoting by f the distance +function4 to the critical hypersurface Z, we may write the following decomposition +as in [GMP10] for any ω ∈b Ωk(M) : +(2.4) +ω = α ∧ df +f + β, with α ∈ Ωk−1(M) and β ∈ Ωk(M). +This decomposition allows to extend the differential of the de Rham complex +d to bΩ(M) by setting dω = dα ∧ df +f + dβ. +Degree 0 functions are called b-functions and and near Z can be written as +c log |x| + g, +where c ∈ R, g ∈ C∞, and x is a local defining function. +The associated cohomology is called b-cohomology and it is denoted by bH∗(M). +Definition 2.10. A b-symplectic form on a b-manifold (M 2n, Z) is defined +as a non-degenerate closed b-form of degree 2 (i.e., ωp is of maximal rank as an +element of Λ2( bT ∗ +p M) for all p ∈ M). +The notion of b-symplectic forms is dual to the notion of b-Poisson structures. +The advantage of using forms rather than bivector fields is that symplectic tools +can be ‘easily’ exported. +4Originally in [GMP10] f stands for a global function, but for non-orientable manifolds we +may use the distance function instead. + +10 +2. A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS +Radko’s classification theorem [Rad02] can be translated into this language. +This translation was already formulated in [GMP10]: +Theorem 2.11 (Radko’s theorem in b-cohomological language, [GMP14]). +Let S be a closed orientable surface and let ω0 and ω1 be two b-symplectic forms on +(S, Z) defining the same b-cohomology class (i.e.,[ω0] = [ω1]). Then there exists a +diffeomorphism φ : S → S such that φ∗ω1 = ω0. +2.2. On bm-Symplectic manifolds +2.2.1. Basic definitions. By relaxing the transversality condition allowing +higher order singularities ([Arn89] and [AA81]) we may consider other symplectic +structures with singularities as done by Scott [Sco16] with bm-symplectic struc- +tures. +Let m be a positive integer a bm-manifold is a b-manifold (M, Z) together +with a bm-tangent bundle attached to it. The bm-tangent bundle is (by Serre-Swan +theorem [Swa62]) a vector bundle, bmT M whose sections are given by, +Γ(bmT M) = {v ∈ Γ(T M) : v(x) +vanishes to order m at Z}, +where x is a defining function for the critical set Z in a neighborhood of each +connected component of Z and can be defined as x : M \ Z → (0, ∞), x ∈ C∞(M) +such that: +• x(p) = d(p) a distance function from p to Z for p : d(p) ≤ 1/2 +• x(p) = 1 on M \ {p ∈ M such that d(p) < 1}.5 +(This definition of x allows us to extend the construction in [Sco16] to the non- +orientable case as in [MOT14].) We may define the notion of a bm-map as a map +in this category (see [Sco16]). +The sections of this bundle are referred to as bm-vector fields and their flows +define bm-maps. In local coordinates, the sections of the bm-tangent bundle are +generated by: +(2.5) +{xm ∂ +∂x, ∂ +∂x1 +, . . . , +∂ +∂xn−1 +}. +Proceeding mutatis mutandis as in the b-case one defines the bm-cotangent +bundle (bmT ∗M), the bm-de Rham complex and the bm-symplectic structures. +A Laurent Series of a closed bm-form ω is a decomposition of ω in a tubular +neighborhood U of Z of the form +(2.6) +ω = dx +xm ∧ ( +m−1 +� +i=0 +π∗(αi)xi) + β +with π : U → Z the projection of the tubular neighborhood onto Z, αi a closed +smooth de Rham form on Z and β a de Rham form on M. +In [Sco16] it is proved that in a neighborhood of Z, every closed bm-form ω +can be written in a Laurent form of type (2.6) having fixed a (semi)local defining +function. +bm-Cohomology is related to de Rham cohomology via the following theorem: +5Then a bm-manifold will be a triple (M, Z, x), but for the sake of simplicity we refer to it +as a pair (M, Z) and we tacitly assume that the function x is fixed. + +2.2. ON bm-SYMPLECTIC MANIFOLDS +11 +Theorem 2.12 (bm-Mazzeo-Melrose, [Sco16]). Let (M, Z) be a bm-manifold, +then: +(2.7) +bmHp(M) ∼= Hp(M) ⊕ (Hp−1(Z))m. +The isomorphism constructed in the proof of the theorem above is non-canonical +(see [Sco16]). +The Moser path method can be generalized to bm-symplectic structures (see +[MS21] for the generalization from surfaces in [Sco16] to general manifolds): +Theorem 2.13 (Moser path method). Let ωt be a path of bm-symplectic +forms defining the same bm-cohomology class [ωt] on (M 2n, Z) with M 2n closed +and orientable then there exist a bm-symplectomorphism ϕ : (M 2n, Z) −→ (M 2n, Z) +such that ϕ∗(ω1) = ω0. +An outstanding consequence of Moser path method is a global classification +of closed orientable bm-symplectic surfaces `a la Radko in terms of bm-cohomology +classes. +Theorem 2.14 (Classification of closed orientable bm-surfaces, [Sco16]). +Let ω0 and ω1 be two bm-symplectic forms on a closed orientable connected bm- +surface (S, Z). Then, the following conditions are equivalent: +• their bm-cohomology classes coincide [ω0] = [ω1], +• the surfaces are globally bm-symplectomorphic, +• the Liouville volumes of ω0 and ω1 and the numbers +� +γ +αi +for all connected components γ ⊆ Z and all 1 ≤ i ≤ m coincide (where +αi are the one-forms appearing in the Laurent decomposition of the two +bm-forms of degree 2, ω0 and ω1). +Definition 2.15. The numbers [αi] = +� +γ αi are called modular weights for the +connected components γ ⊂ Z. +A relative version of Moser’s path method is proved in [GMW17]. As a corol- +lary we obtain the following local description of a bm-symplectic manifold: +Theorem 2.16 (bm-Darboux theorem, [GMW17]). Let ω be a bm-symplectic +form on (M, Z) and p ∈ Z. Then we can find a coordinate chart (U, x1, y1, . . . , xn, yn) +centered at p such that on U the hypersurface Z is locally defined by x1 = 0 and +ω = dx1 +xm +1 +∧ dy1 + +n +� +i=2 +dxi ∧ dyi. +Remark 2.17. For the sake of simplicity sometimes we will omit any explicit +reference to the critical set Z and we will talk directly about bm-symplectic struc- +tures on manifolds M implicitly assuming that Z is the vanishing locus of Πn where +Π is the Poisson vector field dual to the bm-symplectic form. +Next, we present two lemmas that allow us to talk about bm-symplectic struc- +tures and bm-Poisson as two different presentations of the same geometrical struc- +ture on a b-manifold. The lemma below shows that they are dual to each other +and, thus, in one-to-one correspondence. + +12 +2. A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS +Lemma 2.18. Let ω be a bm-symplectic and Π its dual vector field, then Π is a +bm-Poisson structure. +Proof. The quickest way to do this is to take the inverse, which is a bivector +field, and observe that it is a Poisson structure (because dω = 0 implies [Π, Π] = 0). +To see that it is bm-Poisson it is enough to check it locally for any point along the +critical set. Take a point p on the critical set Z and apply the bm-Darboux theorem +to get ω = dx1/xm +1 ∧ dy1 + � +i>1 dxi ∧ dyi This means that in the new coordinate +system +Π = xm +1 +∂ +∂x1 +∧ +∂ +∂y1 ++ +� +i>1 +∂ +∂xi +∧ ∂ +∂yi +and thus Π is a bm-Poisson structure. +□ +Conversely, +Lemma 2.19. Let Π be bm-Poisson and ω its dual vector field, then ω is a +bm-symplectic structure. +Proof. If Π transverse `a la Thom on Z with singularity of order m then +because of Weinstein’s splitting theorem we can locally write +Π = xm +1 +∂ +∂x1 +∧ +∂ +∂y1 ++ +� +i>1 +∂ +∂xi +∧ ∂ +∂yi +now its inverse is ω = dx1/xm +1 ∧ dy1 + � +i>1 dxi ∧ dyi which is a bm-symplectic +form. +□ +Hence we have a correspondence from bm-symplectic structures to bm-Poisson +structures. +2.3. Desingularizing bm-Poisson manifolds +In [GMW17] Guillemin, Miranda and Weitsman presented a desingularization +procedure for bm-symplectic manifolds proving that we may associate a family of +folded symplectic or symplectic forms to a given bm-symplectic structure depending +on the parity of m. Namely, +Theorem 2.20 (Guillemin-Miranda-Weitsman, [GMW17]). Let ω be a +bm-symplectic structure on a closed orientable manifold M and let Z be its critical +hypersurface. +• If m = 2k, there exists a family of symplectic forms ωǫ which coincide with +the bm-symplectic form ω outside an ǫ-neighborhood of Z and for which +the family of bivector fields (ωǫ)−1 converges in the C2k−1-topology to the +Poisson structure ω−1 as ǫ → 0 . +• If m = 2k + 1, there exists a family of folded symplectic forms ωǫ which +coincide with the bm-symplectic form ω outside an ǫ-neighborhood of Z. +As a consequence of Theorem 2.20, any closed orientable manifold that supports +a b2k-symplectic structure necessarily supports a symplectic structure. +In [GMW17] explicit formulae are given for even and odd cases. Let us refer +here to the even-dimensional case as these formulae will be used later on. + +2.3. DESINGULARIZING BM-POISSON MANIFOLDS +13 +Let us briefly recall how the desingularization is defined and the main result in +[GMW17]. Recall that we can express the b2k-form as: +(2.8) +ω = dx +x2k ∧ +�2k−1 +� +i=0 +xiαi +� ++ β. +This expression holds on a ǫ-tubular neighborhood of a given connected com- +ponent of Z. This expression comes directly from equation 2.6, to see a proof of +this result we refer to [Sco16]. +Definition 2.21. Let (S, Z, x), be a b2k-manifold, where S is a closed orientable +manifold and let ω be a b2k-symplectic form. Consider the decomposition given by +the expression (2.8) on an ǫ-tubular neighborhood Uǫ of a connected component of +Z. +Let f ∈ C∞(R) be an odd smooth function satisfying f ′(x) > 0 for all x ∈ [−1, 1] +and satisfying outside that +(2.9) +f(x) = +� +−1 +(2k−1)x2k−1 − 2 +for +x < −1, +−1 +(2k−1)x2k−1 + 2 +for +x > 1. +Let fǫ(x) be defined as ǫ−(2k−1)f(x/ǫ). +The fǫ-desingularization ωǫ is a form that is defined on Uǫ by the following +expression: +ωǫ = dfǫ ∧ +�2k−1 +� +i=0 +xiαi +� ++ β. +This desingularization procedure is also known as deblogging in the literature. +Remark 2.22. Though there are infinitely many choices for f, we will assume +that we choose one, and assume it fixed through the rest of the discussion. +It +would be interesting to discuss the existence of an isotopy of forms under a change +of function f. +Remark 2.23. Because ωǫ can be trivially extended to the whole S in such a +way that it agrees with ω (see [GMW17]) outside a neighborhood of Z, we can +talk about the fǫ-desingularization of ω as a form on S. + + +CHAPTER 3 +A crash course on KAM theory +The last part of this monograph is entirely dedicated to prove a KAM theorem +for bm-symplectic structures and to find applications. So the aim of this section is +to give a quick overview of the traditional KAM theorem. The setting of the KAM +theorem is a symplectic manifold with action-angle coordinates and an integrable +system in it. The theorem says that under small perturbations of the Hamiltonian +”most” of the Liouville tori survive. +Consider Tn×G ⊂ Tn×Rn with action-angle coordinates in it (φ1, . . . , φn, I1, . . . , In) +and the standard symplectic form ω in it. And assume the Hamiltonian function +of the system is given by h(I) a function only depending on the action coordinates. +Then the Hamilton equations of the system are given by +ιXhω = dh +where Xh is the vector field generating the trajectories. Because h does not de- +pend on φ the angular variables the system is really easy to solve, and the equations +are given by +x(t) = (φ(t), I(t)) = (φ0 + ut, I0), +where u = ∂h/∂I is called the frequency vector. These motions for a fixed +initial condition are inside a Liouville torus, and are called quasi-periodic. +The KAM theorem studies what happens to such systems when a small per- +turbation is applied to the Hamiltonian function, i.e. we consider the evolution of +the system given by the Hamiltonian h(I) + R(I, φ), where we think of the term +R(I, φ) as the small perturbation in the system. With this in mind, the Hamilton +equations can be written as +˙φ = u(I) + ∂ +∂I R(I, φ), ˙I = − ∂ +∂φR(I, φ), +Another important concept to have in mind is the concept of rational depen- +dency. A frequency u is rationally dependent if ⟨u, k⟩ = 0 for some k ∈ Zn, if there +exists no k satisfying the condition then the vector u is called rationally indepen- +dent. There is a stronger concept of being rationally independent and that is the +concept of being Diophantine. A vector u is γ,τ-diophantine if ⟨u, k⟩ ≥ +γ +|k|τ +1 for all +k ∈ Zn \ {0}. γ > 0 and τ > n − 1. +The KAM theorem states that the Liouville tori with frequency vector satisfying +the diophantine condition survive under the small perturbation R(I, φ). There are +conditions relating the size of the perturbation with γ and τ. Also, the set of tori +satisfying the Diophantine condition has measure 1 − Cγ for some constant C. +Now we give a proper statement of the theorem as was given in [DG96]. +15 + +16 +3. A CRASH COURSE ON KAM THEORY +Theorem 3.1 (Isoenergetic KAM theorem). Let G ⊂ Rn, n > 2, a compact, +and let H(φ, I) = h(I) + f(φ, I) real analytic on Dρ(G). +Let ω = ∂h/∂I, and +assume the bounds: +���� +∂2h +∂I2 +���� +G,ρ2 +≤ M, +|ω|G ≤ L +and +|ωn(I)| ≥ l∀I ∈ G. +Assume also that ω is µ-isoenergetically non-degenerate on G. For a = 16M/l2, +assume that the map Ω = Ωω,h,a is one-to-one on G, and that its range F = Ω(G) +is a D-set. Let τ > n − 1, γ > 0 and 0 < ν < 1 given, and assume: +ε := ∥f∥G,ρ ≤ ν2l6µ2ˆρ2τ+2 +24τ+32L6M 3 · γ2, +γ ≤ min +�8LMρ2 +νlˆρτ+1 , l +� +, +where we write ρ := min +� +νρ1 +12(τ+2), 1 +� +. Define the set +ˆG = ˆGγ := +� +I ∈ G − 2γ +µ : ω(I)isτ, γ − Diophantine +� +. +Then, there exists a real continuous map T : W ρ1 +4 (Tn) × ˆG → Dρ(G), analytic +with respect to the angular variables, such that: +(1) For every I ∈ ˆG, the set T (Tn × {I}) is an invariant torus of H, its +frequency vector is colinear to ω(I) and its energy is h(I). +(2) Writing +T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)), +one has the estimates +|Tφ| ˆ +G,( ρ1 +4 ,0),∞ ≤ 22τ+15L2M +ν2l2ˆρ2τ+1 +ε +γ2 , +|TI| ˆ +G,( ρ1 +4 ,0) ≤ 2τ+16L3M +νl3µˆρτ+1 +ε +γ +(3) meas[(Tn ×G)\T (Tn × ˆG)] ≤ Cγ, where C is a very complicated constant +depending on n, τ, diamF, D, ˆρ, M, L, l, µ. +Remark 3.2. This version of the KAM theorem is the isoenergetic one, this +version ensures that the energy of the Liouville Tori identified by the diffeomorphism +after the perturbation remains the same as before the perturbation. Our version of +the bm-KAM is not isoenergetic for the sake of simplifying the computations. +Also, we should outline that the KAM theorem has already been explored in +singular symplectic manifolds before. In [KMS16a] the authors proved a KAM +theorem for b-symplectic manifolds, for a particular kind of perturbations. +Theorem 3.3 (KAM Theorem for b-Poisson manifolds). Let Tn × Bn +r be en- +dowed with standard coordinates (ϕ, y) and the b-symplectic structure. Consider a +b-function +H = k log |y1| + h(y) +on this manifold, where h is analytic. Let y0 be a point in Bn +r with first component +equal to zero, so that the corresponding level set Tn × {y0} lies inside the critical +hypersurface Z. +Assume that the frequency map +˜ω : Bn +r → Rn−1, +˜ω(y) := ∂h +∂˜y (y) + +3. A CRASH COURSE ON KAM THEORY +17 +has a Diophantine value ˜ω := ˜ω(y0) at y0 ∈ Bn and that it is non-degenerate at y0 +in the sense that the Jacobian ∂˜ω +∂˜y (y0) is regular. +Then the torus Tn × {y0} persists under sufficiently small perturbations of H +which have the form mentioned above, i.e. they are given by ǫP, where ǫ ∈ R and +P ∈b C∞(Tn × Bn +r ) has the form +P(ϕ, y) = k′ log |y1| + f(ϕ, y) +f(ϕ, y) = f1( ˜ϕ, y) + y1f2(ϕ, y) + f3(ϕ1, y1). +More precisely, if |ǫ| is sufficiently small, then the perturbed system +Hǫ = H + ǫP +admits an invariant torus T . +Moreover, there exists a diffeomorphism Tn → T close1 to the identity taking +the flow γt of the perturbed system on T to the linear flow on Tn with frequency +vector +�k + ǫk′ +c +, ˜ω +� +. +1By saying that the diffeomorphism is “ǫ-close to the identity” we mean that, for given H, P +and r, there is a constant C such that ∥ψ − Id∥ < Cǫ. + + +Part 2 +Action-angle coordinates and +cotangent models + +In this part, we consider the semilocal classification for any bm-Poisson manifold +in a neighbourhood of an invariant compact submanifold. The compact subman- +ifolds under consideration are the compact invariant leaves of the distribution D +generated by the Hamiltonian vector fields Xfi of an integrable system. An in- +tegrable system is given by a set of n functions on a 2n-dimensional symplectic +manifold which we can order in a map F = (f1, . . . , fn). Historically, integrable +systems were introduced to actually integrate Hamiltonian systems XH using the +first-integrals fi and, classically, we identify H = f1. It turns out that in the sym- +plectic context the compact regular orbits of the distribution D coincide with the +fibers F −1(F(p)) for any point p on these orbits/fibers. The fact that the orbit +coincides with the connected fiber is part of the magic of symplectic duality. +The same picture is reproduced for singular symplectic manifolds of bm-type +or bm-Poisson manifolds as we will see in this chapter. +The study of action-angle coordinates has interest from this geometrical point +of view of the classification of geometric structures in a neighbourhood of a compact +submanifold of a bm-Poisson manifold. It also has interest from a dynamical point +of view as these compact submanifolds now coincide with invariant subsets of the +Hamiltonian system under consideration. +From a geometric point of view, the existence of action-angle coordinates deter- +mines a unique geometrical model for the bm-Poisson (or bm-symplectic) structure +in a neighbourhood of the invariant set. From a dynamical point of view, the exis- +tence of action-angle coordinates provides a normal form theorem that can be used +to study stability and perturbation problems of the Hamiltonian systems (as we +will see in the last chapter of this monograph). +An important ingredient that makes our action-angle coordinate theorem brand- +new from the symplectic perspective is that the system under consideration is more +general than Hamiltonian, it is bm-Hamiltonian as the first-integrals of the system +can be bm-functions which are not necessarily smooth functions. Dynamically, this +means that we are adding to the set of Hamiltonian invariant vector fields, the +modular vector field of the integrable system. +In contrast to the standard action-angle coordinates for symplectic manifolds, +our action-angle theorem comes with m additional invariants associated with the +modular vector field which can be interpreted in cohomological terms as the pro- +jection of the bm-cohomology class determined by the modular vector field on the +first cohomology group of the critical hypersurface under the Mazzeo-Melrose cor- +respondence. +The strategy of the proof of the action-angle coordinate systems is the search +of a toric action (so this takes us back to the motivation of the use of symmetries +in this monograph). In contrast to the symplectic case, it is not enough that this +action is Hamiltonian as then a direction of the Liouville torus would be missing. +We need the toric action to be bm-Hamiltonian. The structure of this proof looks +like the one in [KMS16a] but encounters serious technical difficulties as in order +to check that the natural action to be considered is bm-Hamiltonian we need to go +deeper inspired by [Sco16] in the relation between the geometry of the modular +vector field and the coefficients of the Taylor series ci of one of the first-integrals. +This allows us to understand new connections between the geometry and analysis +of bm-Poisson structures not explored before. + +21 +Once we prove the existence of this bm-Hamiltonian action the proof looks very +close to the one in [KMS16a]. +In the second chapter of this part we re-state the action-angle theorem as a +cotangent lift theorem with the following mantra: +Every integrable system on a bm-Poisson manifold looks like a bm-cotangent lift +in a neighborhood of a Liouville torus. + + +CHAPTER 4 +An action-angle theorem for bm-symplectic +manifolds +4.1. Basic definitions +4.1.1. On bm-functions. The definition of the analogue of b-functions in the +bm-setting is somewhat delicate. The set of bmC∞(M) needs to be such that for all +the functions f ∈bm C∞(M), its differential df is a b-form, where d is the bm-exterior +differential. Recall that a form in bmΩk(M) can be locally written as +α ∧ dx +xm + β +where α ∈ Ωk−1(M) and β ∈ Ωk(M). Recall also that +d +� +α ∧ dx +xm + β +� += dα ∧ dx +xm + dβ. +We need df to be a well-defined bm-form of degree 1. Let f = g +1 +xk−1 , then +df = dg +1 +xk−1 − g k−1 +xk dx. This from can only be a bm-form if and only if g only +depends on x. If f = g log(x), then dg log(x) + g 1 +xdx, which imposes dg = 0 and +hence g to be constant. +With all this in mind, we make the following definition. +Definition 4.1. The set of bm-functions is defined recursively according to the +formula +bmC∞(M) = x−(m−1)C∞(x) + bm−1C∞(M) +with C∞(x) the set of smooth functions in the defining function x and +bC∞(M) = {g log |x| + h, g ∈ R, h ∈ C∞(M)}. +Remark 4.2. A bmC∞(M)-function can be written as +f = a0 log x + a1 +1 +x + . . . + am−1 +1 +xm−1 + h +where ai, h ∈ C∞(M). +Remark 4.3. From this chapter on we are only considering bm-manifolds +(M, x, Z) with x defined up to order m. +I.e. +we can think of x as a jet of a +function that coincides up to order m to some defining function. This is the orig- +inal viewpoint of Scott in [Sco16] which we adopt from now on. The difference +with respect to the other chapters is that we do not fix an specific function. +Definition 4.4. We say that two bm-integrable systems F1, F2 are equivalent +if there exists ϕ, a bm-symplectomorphism, i.e. a diffeomorphism preserving both +ω and the critical set Z (“up to order m”1), such that ϕ ◦ F1 = F2. +1I.e. it preserves the jet x +23 + +24 +4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS +Remark 4.5. The Hamiltonian vector field associated to a bm-function f is a +smooth vector field. Let us compute it locally using the bm-Darboux theorem: +Π = xm +1 +∂ +∂x1 +∧ +∂ +∂y1 ++ +m +� +i=2 +∂ +∂xi +∧ ∂ +∂yi +and f = a0 log x1 + +m−1 +� +i=1 +ai +1 +xi +1 ++ h. +Then if we compute +df += +c1 +���� +a0 +1 +x1 ++ +m−1 +� +i=1 +ci +� +�� +� +(a′ +i − (i − 1)ai−1) 1 +xi +1 +dx1 +− +cm +� +�� +� +(m − 1)am−1 +1 +xm +1 dx1 + dh += +m +� +i=1 +ci +xi +1 +dx1 + dh. +Then, +(4.1) +Xf = Π(df, ·) = +m +� +i=1 +cixm−i +1 +∂ +∂y1 ++ Π(dh, ·), +we obtain a smooth vector field. +4.2. On bm-integrable systems +In this section we present the definition of a bm-integrable system as well as +some observations about these objects. +Definition 4.6. Let (M 2n, Z, x) be a bm-manifold, and let Π be a bm-Poisson +structure on it. F = (f1, . . . , fn)2 is a bm-integrable system3 if: +(1) df1, . . . , dfn are independent on a dense subset of M and in all the points +of Z where independent means that the form df1 ∧ . . . ∧ dfn is non-zero as +a section of Λn(bmT ∗(M)), +(2) the functions f1, . . . , fn Poisson commute pairwise. +Definition 4.7. The points of M where df1, . . . , dfn are independent are called +regular points. +The next remarks will lead us to a normal form for the first function f1. +Remark 4.8. Note that df1, . . . , dfn are independent on a point if and only if +Xf1, . . . , Xfn are independent at that point. This is because the map +bmT M →bm T ∗M : u �→ ωp(u, ·) +is an isomorphism. +Remark 4.9. The condition of df1, . . . , dfn being independent must be under- +stood as df1 ∧ . . . ∧ dfn being a non-zero section of �n( bmT ∗M). +2fi are bm-functions. +3In this monograph we only consider integrable systems of maximal rank n. + +4.2. ON bm-INTEGRABLE SYSTEMS +25 +Remark 4.10. By remark 4.8 the vector fields Xf1, . . . , Xfn have to be in- +dependent. +This implies that one of the f1, . . . , fn has to be a singular (non- +smooth) bm-function with a singularity of maximal degree. +If we write fi = +c0,i log(x1) + �m−1 +j=1 +cj,i +xj +1 + ˜f1 +Xfi = +m +� +j=1 +xm−j +1 +ˆcj,i +∂ +∂y1 ++ X ˜ +fi +where ˆcj,i(x) = d(cj,i) +dx +− (j − 1)cj−1,i. If there is no bm-function with a singularity +of maximum degree all the terms in the ∂/∂y1 direction become 0 at Z. And hence +Xf1, . . . , Xfn cannot have maximal rank at Z. +Lemma 4.11. Let F = (f1, . . . , fn) a bm-integrable system. If f1 has a singular- +ity of maximal degree, there exists an equivalent integrable system F ′ = (f ′ +1, . . . , f ′ +n) +where f ′ +1 has a singularity of maximal degree and no other f ′ +i has singularity of +any degree. +Proof. Let fi = c0,i log(x1) + +m−1 +� +j=1 +cj,1 +xj +1 +� +�� +� +ζi(x1) ++ ˜fi = ζi(x1) + ˜fi. By remark 4.104, +Xfi = +m +� +i=1 +xm−j +1 +ˆcj,i +� +�� +� +gi(x1) +∂ +∂y1 ++ X ˜ +fi = gi(x1) ∂ +∂y1 ++ X ˜ +fi. +Note that gi(x1) = gi(0) = ˆcm,i at Z. Let us look at the distribution given by the +Hamiltonian vector fields Xfi = gi(x1) ∂ +∂y1 +X ˜ +fi. This distribution is the same that +the one given by: +(4.2) +{Xf1, Xf2 − g2(x1) +g1(x1)Xf1, . . . , Xfn − gn(x1) +g1(x1) Xf1}. +Observe that for i > 1, Xfi − gi(x1) +g1(x1)Xf1 = X ˜ +fi + g2(x1) +g1(x1)X ˜ +f1. Also g1(x1) is different +from 0 close to Z because at Z g1(x1) = ˆcm,1. Since the distribution given by these +vector fields is the same, an integrable system that has Hamiltonian vector fields 4.2 +would be equivalent to F. From the expression 4.2 it is clear that the new vector +fields commute. And it is also true that this new vector fields are Hamiltonian. Let +us take F ′ the set of functions that have as Hamiltonian vector fields 4.2. +□ +From now on we will assume the integrable system to have only one singular +function and this function to be f1. +Remark 4.12. Because we asked Xf1, . . . , Xfn to be linearly independent at +all the points of Z and using the previous remarks cm := cm,1 ̸= 0 at all the points +of Z. +4Here have used the bm-Darboux theorem to do the computations. + +26 +4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS +Furthermore, we can assume f1 to have a smooth part equal to zero as sub- +tracting the smooth part of f1 to all the functions gives an equivalent system. Also, +we can assume that cm is 1 because dividing all the functions of the bm-integrable +system by cm also gives us an equivalent system. +As a summary, we can assume f1 = a0 log(x)+a11/x+. . .+am−21/xm−2+ +1/xm−1 and f2, . . . , fn to be smooth, a0 ∈ R and a1, . . . , am−2 ∈ C∞(x). +Also we are going to state lemma 3.2 in [GMPS17], because we are going to +use it later in this section. The result states that if we have a toric action on a bm- +symplectic manifold (which we will prove in a neighbourhood of a Liouville torus), +then we can assume the coefficients a2, . . . , am−2 to be constants. More precisely +Lemma 4.13. There exists a neighborhood of the critical set U = Z × (−ε, ε) +where the moment map µ : M → t∗ is given by +µ = a1 log |x| + +m +� +i=2 +ai +x−(i−1) +i − 1 ++ µ0 +with ai ∈ t0 +L and µ0 is the moment map for the TL-action on the symplectic leaves +of the foliation. +4.3. Examples of bm-integrable systems +The following example illustrates why it is necessary to use the definition of bm- +function as considered above. There are natural examples of changes of coordinates +in standard integrable systems on symplectic manifolds that yield bm-symplectic +manifolds but do not give well-defined bm-integrable systems. +Example 4.14. Consider a time change in the two body problem, to obtain +a b2-integrable system. In the classical approach to solve the 2-body problem the +following two conserved quantities are obtained: +f1 += +µy2 +2 + +l2 +2µr2 − k +r , +f2 += +l, +with symplectic form ω = dr ∧ dy + dl ∧ dα, where r is the distance between the +two masses and l is the angular momentum. We also know that l is constant along +the trajectories. Because l is a constant of the movement, we can do a symplectic +reduction on its level sets. The form on the symplectic reduction becomes dr ∧ dy. +To simplify the notation, we will use x instead of r. Then ω = dx ∧ dy. With +hamiltonian function given by f = µ +2 y2 + +l +2µ +1 +x2 − k 1 +x. Hence, the equations are: +˙x += +∂f +∂y , +˙y += +− ∂f +∂x. +Doing a time change τ = x3t then dx +dτ = +1 +x3 dx +dt . With this time coordinate, the +equations become: +˙x += +1 +x3 +∂f +∂y , +˙y += +− 1 +x3 +∂f +∂x. +These equations can be viewed as the motion equations given by a b3-symplectic +form ω = +1 +x3 dx ∧ dy. +Let us check that this is actually a bm-integrable system. + +4.3. EXAMPLES OF bm-INTEGRABLE SYSTEMS +27 +• All the functions Poisson commute is immediate because we only have +one. +• df = µydy +( k +x2 − l +µ +1 +x3 )dx is a b3-form because the term with dx does not +depend on y. +• All the functions are independent, this is true because df does not vanish +as a b3-form. +Example 4.15. In the paper [Mar19] the author builds an action of SL(2, R) +over (P, ωP ) where P = {ξ ∈ C|i(¯ξ − ξ) > 0} is the complex semi-plane, with +moment map JP (ξ) = +R +ξim ((|ξ|2 + 1), 2ξr, ±(|ξ|2 + 1)), where the ± sign depends +on the choice of the hemisphere projected by the stereographic projection. From +now on we will take the sign +. Also the symplectic form ωP has the following +expression: +ωP = ± R +ξ2 +im +dξr ∧ dξim +In order to simplify the notation we identify P with the real half-plane P = +{x, y ∈ R2|y > 0}. With this identification, the moment map becomes Jp(x, y) = +R +y (x2 + y2 + 1, 2x, x2 + y2 + 1). Obviously, this moment map does not give an +integrable system. The symplectic form writes as: +ωP = R +y2 dy ∧ dx. +This form can be viewed as a b2-form if we extend P including the line {y = 0} +as its singular set. +Let us consider only one of the components of JP as bm- +function and let us see if it gives a bm-integrable system. First we will try with +f1 = R +y (x2 + y2 + 1) and then f2 = R +y (2x). +(1) f1 = R +y (x2 + y2 + 1) We have to check three things to see if this gives a +b2-integrable system. +(a) All the functions Poisson commute is immediate because we only +have one. +(b) All the functions are bm-functions. This point does not hold because +df1 = R +y2 (2xydx + (y2 − x2 − 1)dy) and the first component makes no +sense as a section of Λ1(b2T ∗M). +(c) All the functions are independent. In this case, we need to check that +df1 does not vanish, but since it is not a bm-form it makes no sense +to be a non-zero section of Λ1(b2T ∗M). +(2) f2 = R +y (2x) +(a) Same as before. +(b) All the functions are bm-functions. This point does not hold because +df2 = 2R +y dx − 2Rx +y2 dy and the first component makes no sense as a +section of Λ1(b2T ∗M). +(c) Same as before. +Example 4.16. Toric actions give natural examples of integrable systems where +the component functions are given by the moment map. In the case of surfaces: +S1-actions on surfaces give natural examples of bm-integrable systems. Only torus +and spheres admit circle actions. + +28 +4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS +In the picture below two integrable systems on the 2-sphere depending on the +degree m. On the right the image of the moment map that defines the integrable +system. The action is by rotations along the central axis. +Namely consider the sphere S2 as a bm-symplectic manifold having as critical +set the equator: +(S2, Z = {h = 0}, ω = dh +hm ∧ dθ), +with h ∈ [−1, 1] and θ ∈ [0, 2π). +For m = 1: The computation ι ∂ +∂θ ω = − dh +h = −d(log |h|), tells us that the +function µ(h, θ) = log |h| is the moment map and defines a b-integrable system. +For higher values of m: ι ∂ +∂θ ω = − dh +hm = −d(− +1 +(m−1)hm−1 ), and the moment +map is µ(h, θ) = − +1 +(m−1)hm−1 which defines a bm-integrable system. +µ, m = 1 +µ, m = 2 +Figure 1. Integrable systems associated to the moment map of +an S1-action by rotations on a bm-symplectic 2-sphere S2. +Example 4.17. Consider now as b2-symplectic manifold the 2-torus +(T2, Z = {θ1 ∈ {0, π}}, ω = +dθ1 +sin2 θ1 +∧ dθ2) +with standard coordinates: θ1, θ2 ∈ [0, 2π). Observe that the critical hypersurface +Z in this example is not connected. It is the union of two disjoint circles. Consider +the circle action of rotation on the θ2-coordinate with fundamental vector field +∂ +∂θ2 . +As the following computation holds, +ι +∂ +∂θ2 ω = − dθ1 +sin2 θ1 += d +�cos θ1 +sin θ1 +� +. +The fundamental vector field of the S1-action defines b2C∞-integrable system given +by the function − cos θ1 +sin θ1 . +Example 4.18. The former example can be made general to produce examples +of bm-integrable systems on a bm-symplectic manifold for any integer m +(T2, Z = {θ1 ∈ {0, π}}, ω = +dθ1 +sinm θ1 +∧ dθ2). +Then +ι +∂ +∂θ2 ω = − +dθ1 +sinm θ1 += d +� +| cos θ1| +cos θ1 +2F1 +� 1 +2, 1−m +2 ; 3−m +2 ; sin2(θ1) +� +(1 − m) sinm−1 θ1 +� +, +with 2F1 the hypergeometric function. + +4.4. LOOKING FOR A TORIC ACTION +29 +µ +Figure 2. Integrable system given by an S1-action on a b2-torus +T2 and its associated moment map. +Thus, the associated S1-action has as bmC∞-Hamiltonian the function +−| cos θ1| +cos θ1 +2F1 +� 1 +2, 1−m +2 ; 3−m +2 ; sin2(θ1) +� +(1 − m) sinm−1 θ1 +which defines a bm-integrable system. +Now we give a couple of examples of bm-integrable systems. +Example 4.19. This example uses the product of bm-integrable systems on a +bm-symplectic manifold with an integrable system on a symplectic manifold. Given +(M 2n1 +1 +, Z, x, ω1) a bm-symplectic manifold with f1, . . . , fn1 a bm-integrable system +and (M 2n2 +2 +, ω2) a symplectic manifold with g1, . . . , gn2 an integrable system. Then +(M1×M2, Z×M2, x, ω1+ω2) is a bm-symplectic manifold and (f1, . . . , fn1, g1, . . . , gn2) +is a bm-integrable system on the higher dimensional manifold. +In particular by combining the former examples of bm-integrable systems on +surfaces and arbitrary integrable systems on symplectic manifolds we obtain exam- +ples of bm-integrable systems in any dimension. +Example 4.20. (From integrable systems on cosymplectic manifolds +to bm-integrable systems:) +Using the extension theorem (Theorem 50) of [GMP14] we can extend any +integrable system (f2, . . . , fn) to an integrable system in a neighbourhood of a +cosymplectic manifold (Z, α, ω) by just adding a bm-function f1 to the integrable +system so that the new integrable system is (f1, f2, . . . , fn) and considering the +associated bm-symplectic form: +(4.3) +˜ω = p∗α ∧ dt +tm + p∗ω. +(t is the defining function of Z). +4.4. Looking for a toric action +In this section we pursue the proof of action-angle coordinates for bm-integrable +systems by recovering a torus group action. This action is associated to the Hamil- +tonian vector fields associated to Xfi. +This is the same strategy used for b-integrable systems in [KMS16a]- One of +the main difficulties is to prove that the coefficients a1, . . . , an can be considered + +30 +4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS +as constant functions. This makes it more difficult to prove the existence of a Tn- +action in the general bm-case than in the b-case, but once we have it we can use +the results in [GMW17] to assume that the coefficients a1, . . . , an are constant +functions. +In this section we provide some preliminary material that will be needed later: +Proposition 4.21. Let (M, Z, x, ω) be a bm-symplectic manifold such that Z +is connected with modular period k. Let π : Z → S1 ≃ R/kZ be the projection +to the base of the corresponding mapping torus. Let γ : S1 = R/kZ → Z be any +loop such that π ◦ γ is positively oriented and has constant velocity 1. Then the +following are equal: +(1) The modular period of Z, +(2) +� +γ ιLω, +(3) The value am−1 for any bmC∞(M)-function +f = a0 log(x) + +m−1 +� +j=1 +aj +1 +xj + h +such that the hamiltonian vector field Xf has 1-periodic orbits homotopic +in Z to some γ. +Proof. Let us first prove that (1)=(2) and then that (2)=(3). +(1)=(2) Let us denote by Vmod the modular vector field. Recall from [GMW17] +that ιL(Vmod) is the constant function 1. Let s : [0, k] → Z be the trajec- +tory of the modular vector field. Because the modular period is k, s(0) +and s(k) are in the same leaf L. Let ˆs : [0, k + 1] → Z a smooth extension +of s such that s|[k,k+1] is a path in L joining ˆs(k) = s(k) to ˆs(k+1) = s(0). +This way ˆs becomes a loop. Then, +k = +� k +0 +1dt = +� +S +ιLω = +� +ˆs +ιLω = +� +γ +ιLω +(2)=(3) Let r : [0, 1] �→ Z be the trajectory of Xf the hamiltonian vector field of +f. Recall that Xf satisfies +ιXf ω = +m +� +j=1 +cj +dx +xi + dh. +Let xm ∂ +∂x be a generator of the linear normal bundle L. We know that +Xf is 1-periodic and its trajectory is homotopic to γ. Hence, +k = +� +r ιLω += +� 1 +0 +ιxm ∂ +∂x ω(Xf|r(t))dt += +� 1 +0 +−( +m +� +j=1 +ci +dx +xi + dh) · (xm ∂ +∂x)|r(t)dt += +−cm = −am−1 +□ + +4.4. LOOKING FOR A TORIC ACTION +31 +We will also need a Darboux-Carath´eodory theorem for bm-symplectic mani- +folds: +Theorem 4.22 (Darboux-Carath´eodory (bm-version)). Let +(M 2n, x, Z, ω) +be a bm-symplectic manifold and m be a point on Z. Let f1, . . . , fn be a bm-integrable +system. Then there exist bm-functions (q1, . . . , qn) around m such that +ω = +n +� +i=1 +dfi ∧ dqi +and the vector fields {Xfi, Xqj}i,j commute. If f1 is not smooth (recall that f1 = +a0 log(x) + �m−1 +j=1 aj 1 +xi with an ̸= 0 on Z and a0 ∈ R) the qi can be chosen to be +smooth functions, and (x, f2, . . . , fn, q1, . . . , qn) is a system of local coordinates. +Proof. The first part of this proof is exactly as in [KMS16a]. Assume now +f1 = a0 log(x) + +m−1 +� +j=1 +aj +1 +xi . We modify the induction requiring also that µi (in +addition to be in Ki) is also in T ∗M ⊆b T ∗M. We can also ask this extra condition +while asking µi(Xfi) = 1, we only have to check that Xfi does not vanish in T M. +This is clear because Xfi does not vanish at bT M and +0 = {fn, fi} = +� m +� +i=1 +˜ai +dx +xi +� +(Xfi) = +� +dx +xm +m +� +i=1 +aixi +� +(Xfi). +All the terms in the last expression vanish except for the one of degree m. +Then dx/xm is in the kernel of Xfi, hence Xfi does not vanish on T M and the +qi can be chosen to be smooth. +{Xx, Xf2, . . . , Xfn, Xq1, . . . Xqn} commute because {Xfi, Xqi}i,j commute. Then +dx ∧ df2 . . . ∧ dfn ∧ dq1 ∧ . . . ∧ dqn +is a non-zero section of �n(bT M). And hence +(x, f2, . . . , fn−1, q1, . . . , qn) +are local coordinates. +□ +Before proceeding with the proof of the action-angle coordinates, we need to +prove that in a neighbourhood of a Liouville torus the fibration is semilocally trivial: +Lemma 4.23 (Topological Lemma). Let m ∈ Z be a regular point of a bm- +integrable system (M, x, Z, ω, F). Assume that the integral manifold Fm through m +is compact. Then there exists a neighborhood U of Fm and a diffeomorphism +φ : U ≃ Tn × Bn +which takes the foliation F to the trivial foliation {Tn × {b}}b∈Bn. +Proof. We follow the steps of [LGMV08]. In this case, the only extra step +that must be checked is that the foliation given by the bm-hamiltonian vector fields +of F = (f1, f2, . . . , fn) is the same as the one given by the level sets of ˜F := +(x, f2, . . . , fn). In our case f1 = a0 log(x) + �m−1 +u=1 ai 1 +xi , where a0 ∈ R, ai ∈ C∞(x), +am−1 = 1. Hence the foliations are the same. Then as in [LGMV08], we take an + +32 +4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS +Figure 4. Fibration by Liouville tori: The middle fiber of the +point p ∈ Z in magenta, the neighbouring Liouville tori in blue. +arbitrary Riemannian metric on M and this defines a canonical projection ψ : U → +Fm. Let us define φ := ψ × ˜F. We obtain the commutative diagram (Figure 3). +U +Tn × Bn +Bn +φ +˜ +F +p +Figure 3. Commutative diagram of the construction of the iso- +morphism of bm-integrable systems. +which provides the necessary equivalence of bm-integrable systems. +□ +4.5. Action-angle coordinates on bm-symplectic +manifolds +In a neighbourhood of one of our Liouville tori all we can assume about the +form of our bm-symplectic structure is that is given by the Laurent series defined +in [Sco16]. +That is to say, we can assume that in a tubular neighborhood U of Z +ω = +m−1 +� +j=1 +dx +xi ∧ π∗(αi) + β, +where π : U → Z is the projection of the tubular neighborhood onto Z, αi are +closed smooth de Rham forms on Z and β a de Rham form on M of degree 2. +In [RBM, MM22] normal forms are given for group actions in a neighbour- +hood of the orbit. Below we provide a normal for the integrable system in a neigh- +bourhood of an orbit of the torus action associated to the integrable system. This +theorem is finer than the bm-symplectic slice theorem provided in [MM22] as it +also gives information about the first integrals. +One of the non-trivial steps of the proof is to associate a toric action to the +integrable system. The connection to normal forms of group actions will become +even more evident when we discuss the associated cotangent models. + +4.5. ACTION-ANGLE COORDINATES ON bm-SYMPLECTIC +MANIFOLDS +33 +Theorem A (Action-angle coordinates for bm-symplectic manifolds). Let (M, x, ω, F) +be a bm-integrable system, where F = (f1 = a0 log(x) + �m−1 +j=1 aj 1 +xj , . . . , fn) with +aj for j > 1 functions in x. Let m ∈ Z be a regular point and let us assume that the +integral manifold of the distribution generated by the Xfi through m is compact. +Let Fm be the Liouville torus through m. Then, there exists a neighborhood U of +Fm and coordinates (θ1, . . . , θn, σ1, . . . , σn) : U → Tn × Bn such that: +(1) We can find an equivalent integrable system F = (f1 = a′ +0 log(x) + +�m−1 +j=1 a′ +j +1 +xj , . . . , fn) such that the coefficients a′ +0, . . . , a′ +m−1 of f1 are con- +stants ∈ R, +(2) +ω|U = + + +m +� +j=1 +c′ +j +c +σj +1 +dσ1 ∧ dθ1 + + + +n +� +i=2 +dσi ∧ dθi +where c is the modular period and c′ +j = −(j − 1)a′ +j−1, also +(3) the coordinates σ1, . . . , σn depend only on f1, . . . fn. +Proof. The idea of this proof is to construct an equivalent bm-integrable +system whose fundamental vector fields define a Tn-action on a neighborhood of +Tn × {0}. It is clear that all the vector fields Xf1, . . . , Xfn define a torus action on +each Liouville tori Tn × {b} where b ∈ Bn, but this does not guarantee that their +flow defines a toric action on all Tn × Bn. The proof is structured in three steps. +The first one is the uniformization of the periods, i.e. we define an Rn-action on a +neighborhood of Tn × {0} such that the lattice defined by its kernel at every point +is constant. This allows to induce an actual action of a torus (as the periods are +constant) of rank n: A Tn action by taking quotients. The second step consists +in checking that this action is actually bm-Hamiltonian. And in the final step we +apply theorem 4.22 to obtain the expression of ω. +(1) Uniformization of periods. +Let Φs +XF be defined as the joint flow by the Hamiltonian vector fields +of the action: +(4.4) +Φ : Rn × (Tn × Bn) +→ +(Tn × Bn) +((s1, . . . , sn), (x, b)) +�→ +Φs1 +Xf1 ◦ · · · ◦ Φsn +Xfn ((x, b)) +this defines an Rn-action on Tn×Bn. For each b ∈ Bn at a single orbit +Tn × {b} the kernel of this action is a discrete subgroup of Rn. We will +denote the lattice given by this kernel Λb. Because the orbit is compact, +the rank of Λb is maximal i.e. n. This lattice is known as the period lattice +of Tn × {b} as we know by standard arguments in group theory that the +lattice has to be of maximal rank so as to have a torus as a quotient. In +general we can not assume that Λb does not depend on b. The process of +uniformization of the periods modifies the action 4.4 in such a way that +Λb = Zn for all b. Let us consider the following Hamiltonian vector field +�n +i=1 kiXfi. The bm-function that generates this Hamiltonian vector field +is: +k1 + +a0 log(x) + +m−1 +� +j=1 +aj +1 +xj + + + +n +� +i=2 +kifi + +34 +4. AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS +where recall that am−1 is constant equal 1. Observe that the coefficient +multiplying 1/xm−1 is k1. By proposition 4.21 k1 = c the modular period. +In this case c = [αm]. +Hence, for b ∈ Bn−1 × {0} the lattice Λb is contained in Rn−1 × cZ ⊆ +Rn. Pick (λ1, . . . , λn) : Bn → Rn such that: +• (λ1(b), . . . , λn(b)) is a basis of Λb for all b ∈ Bn, +• λn +i vanishes along Bn−1 × {0} at order m for i < n and λi is equal +to c along Bn−1 × {0}. +In the previous points, λj +i denotes the j-th component of λi. The first +condition can be satisfied by using the implicit function theorem. That +is because Φ(λ, m) = m is regular with respect to the s coordinates. The +second condition is automatically true because Λb ⊆ Rn−1×cZ. We define +the uniformed flow as: +(4.5) +˜Φ : Rn × (Tn × Bn) +→ +(Tn × Bn) +((s1, . . . , sn), (x, b)) +�→ +Φ(�n +i=1 siλi, (x, b)) +(2) The Tn-action is bm-Hamiltonian. The objective of this step is to find +bm-functions σ1, . . . , σn such that Xσi are the fundamental vector fields +of the Tn-action Yi = �n +j=1 λj +iXfj. +By using the Cartan formula for a bm-symplectic form, we obtain: +LYiLYiω += +LYi(d(ιYiω) + ιYidω) += +LYi(d(− �n +j=1 λj +idfi)) += +−LYi(�n +j=1 dλj +i ∧ dfj) = 0 +Note that λj +i are constant on the level sets of F as Φ(λ, m) = m and +the level sets of F are invariant by Φ. +Recall that if Y is a complete periodic vector field and P is a bivector +such that LY LY P = 0, then LY P = 0. So, the vector fields Yi are Poisson +vector fields. To show that each ιYiω has a bmC∞ primitive we will see +that [ιYiω] = 0 in the bm-cohomology. +One one hand, if i > 1, ιYiω vanishes at Z. This holds because Yi has +not any component ∂/∂Y . +Recall Proposition 6 from [GMP14]: +Proposition 4.24. If ω ∈b Ω(M) with ω|Z = 0, then ω ∈ Ω(M). +In a similar way for bm-forms we have, +Proposition 4.25. If ω ∈bm Ω(M) with ω|Z vanishing up to order +m, then ω ∈ Ω(M). +Thus as ιYiω vanishes at Z, the bm-forms ιYiω are indeed smooth. +Thus we can now apply the standard Poincar´e lemma and as these forms +are closed they are locally exact. This proves that all the vector fields Yi +with i > 1 are indeed Hamiltonian. +On the other hand, the fact that ιY1ω = cdf1 is obvious. +Then, because we have a toric action that is Hamiltonian, we can use +lemma 3.2 in [GMPS17], and we get an equivalent system such that ai +are all constant and moreover ⟨a′ +i, X⟩ = αi(Xω). Note that by dividing by +a′ +m−1, we can still assume a′ +m−1 = 1 to be consistent with our notation, +but we then have to multiply f1 · c in the next step. + +4.5. ACTION-ANGLE COORDINATES ON bm-SYMPLECTIC +MANIFOLDS +35 +(3) Apply Darboux-Carath´eodory theorem. +The construction above gives us some candidates σ1 = cf1, σ2, . . . , σn +for the action coordinates. +We now apply the Darboux-Carath´eodory theorem and express the +form in terms of x: +ω = + + +m +� +j=1 +c cj +xj dx ∧ dq1 + + + +n +� +i=2 +dσi ∧ dqi. +Since the vector fields Xσi = +∂ +∂qi are fundamental fields of the Tn- +action the flow 4.5 gives a linear action on the qi coordinates. +Observe that the coordinate system is only defined in U. It may not +be valid at points outside U that may be in the orbit of points in U. Let +us see that the charts can be extended to these points. +Define U′ the union of all tori that intersect U. We will see that the +coordinates are valid at U′. +Let {pi, θj} be the extension of {σi, qj}. It is clear that {pi, θj} = δij +by its construction in the Darboux-Carath´eodory theorem. +To see that {θi, θj} = 0 we take the flows by Xpk and extend the +expression to the whole U′: +Xpk({θi, θj}) = {{θi, θj}, pk} = {θi, δij} − {θj, δjk} = 0. +The fact that ω is preserved is obvious because Xpk are hamiltonian +vector fields and thus they preserve the bm-symplectic forms. Moreover, +t, θ1, p2, θ2, . . . , pn, θn are independent on U′ and hence are a coordinate +system in a neighbourhood of the torus. +□ +Remark 4.26. In the proof we have seen that there exists an equivalent inte- +grable system where the coefficients of the singular function are indeed constant. +From now on, when considering a bm-integrable system we are going to make this +assumption. +Remark 4.27. By means of the desingularization transformation we may ob- +tain an action-angle coordinate theorem for folded manifolds as we do in Part 3 +for the KAM theorem for folded symplectic manifolds. This folded action-angle +theorem is a particular case of the one obtained in [CM22]. + + +CHAPTER 5 +Reformulating the action-angle coordinate via +cotangent lifts +The action-angle theorem for symplectic manifolds (also known as action-angle +coordinate theorem) can be reformulated in terms of a cotangent lift. +Recall that given a Lie group action on any manifold its cotangent lifted action +is automatically Hamiltonian. By considering the action of a torus on itself by +translations this action can be lifted to its cotangent bundle and give a semilocal +normal form theorem as the Arnold-Liouville-Mineur theorem for symplectic man- +ifolds. If we now replace this cotangent lift to the cotangent bundle to a lift to the +bm-cotangent bundle we obtain the semilocal normal form of the main theorem of +this chapter. +Let start recalling the symplectic and b-symplectic case following [KM17]. +5.1. Cotangent lifts and Arnold-Liouville-Mineur in Symplectic +Geometry +Let G be a Lie group and let M be any smooth manifold. Given a group action +ρ : G × M −→ M, we define its cotangent lift as the action on T ∗M given by +ˆρg := ρ∗ +g−1 where g ∈ G. We then have a commuting diagram +T ∗M +T ∗M +M +M +ˆ +ρg +ˆπ +π +ρg +Figure 1. Commutiative diagram of the construction of the iso- +morphism of bm-integrable systems. +where π is the canonical projection from T ∗M to M. +The cotangent bundle T ∗M is a symplectic manifold endowed with the exact +symplectic form given by the differential of the Liouville one-form ω = −dλ. The +Lioville one-form can be defined intrinsically: +(5.1) +⟨λp, v⟩ := ⟨p, (πp)∗(v)⟩ +with v ∈ T (T ∗M), p ∈ T ∗M. +A standard argument (see for instance [GS90]) shows that the cotangent lift ˆρ +is Hamiltonian with moment map µ : T ∗M → g∗ given by +⟨µ(p), X⟩ := ⟨λp, X#|p⟩ = ⟨p, X#|π(p)⟩, +37 + +38 +5. ACTION-ANGLE COORDINATES AND COTANGENT LIFTS +where p ∈ T ∗M, X is an element of the Lie algebra g and we use the same symbol +X# to denote the fundamental vector field of X generated by the action on T ∗M +or M. This construction is known as the cotangent lift. +In the special case where the manifold M is a torus Tn and the group is Tn +acting by translations, we obtain the following explicit structure: Let θ1, . . . , θn be +the standard (S1-valued) coordinates on Tn and let +(5.2) +θ1, . . . , θn +� +�� +� +=:θ +, t1, . . . , tn +� +�� +� +=:t +be the corresponding chart on T ∗Tn, i.e. +we associate to the coordinates (5.2) +the cotangent vector � +i tidθi ∈ T ∗ +θ Tn. The Liouville one-form is given in these +coordinates by +λ = +n +� +i=1 +tidθi +and its negative differential is the standard symplectic form on T ∗Tn: +(5.3) +ωcan = +n +� +i=1 +dθi ∧ dti. +Denoting by τβ the translation by β ∈ Tn on Tn, its lift to T ∗Tn is given by +ˆτβ : (θ, t) �→ (θ + β, t). +The moment map µcan : T ∗Tn → t∗ of the lifted action with respect to the canonical +symplectic form is +(5.4) +µcan(θ, t) = +� +i +tidθi, +where the θi on the right hand side are understood as elements of t∗ in the obvious +way. Even simpler, if we identify t∗ with Rn by choosing the standard basis +∂ +∂θi +of t then the moment map is just the projection onto the second component of +T ∗Tn ∼= Tn × Rn. Note that the components of µ naturally define an integrable +system on T ∗Tn. +We can rephrase the Arnold-Liouville-Mineur theorem in terms of the symplec- +tic cotangent model: +Theorem 5.1. Let F = (f1, . . . , fn) be an integrable system on the symplectic +manifold (M, ω). Then semilocally around a regular Liouville torus the system is +equivalent to the cotangent model (T ∗Tn)can restricted to a neighbourhood of the +zero section (T ∗Tn)0 of T ∗Tn. +5.2. The case of bm-symplectic manifolds +Let us start by introducing the twisted bm-cotangent model for torus actions. +This model has additional invariants: the modular vector field of the connected +component of the critical set and the modular weights of the associated toric action. +Consider T ∗Tn be endowed with the standard coordinates (θ, t), θ ∈ Tn, t ∈ Rn and +consider again the action on T ∗Tn induced by lifting translations of the torus Tn. +We will now view this action as a bm-Hamiltonian action with respect to a suitable +bm-symplectic form. In analogy to the classical Liouville one-form we define the +following non-smooth one-form away from the hypersurface Z = {t1 = 0} : + +5.2. THE CASE OF bm-SYMPLECTIC MANIFOLDS +39 +� +cc1 log |t1| + +m +� +i=2 +cci +t−(i−1) +1 +−(i − 1) +� +dθ1 + +n +� +i=2 +tidθi. +When differentiating this form we obtain a bm-symplectic form on T ∗Tn which +we call (after a sign change) the twisted bm-symplectic form on T ∗Tn with +invariants (cc1, . . . , ccm): +(5.5) +ωtw,c := + + +m +� +j=1 +cj +c +tj +1 +dt1 ∧ dθ1 + + + +n +� +i=2 +dti ∧ dθi, +where c is the modular period. The moment map of the lifted action is then given +by +(5.6) +µtw,q0,...,qm−1) := (q0 log |t1| + +m +� +i=2 +qit−(i−1) +1 +, t2, . . . , tn), +where we are identifying t∗ with Rn and cj = −(j − 1)qj−1. +We call this lift together with the bm-symplectic form 5.5 the twisted bm- +cotangent lift with modular period c and invariants (c1, . . . , cm). Note that the +components of the moment map define a bm-integrable system on (T ∗Tn, ωtw,(cc1,...,ccm)). +The model of twisted bm-cotangent lift allows us to express the action-angle +coordinate theorem for bm-integrable systems in the following way: +Theorem 5.2. Let F = (f1, . . . , fn) be a bm-integrable system on the bm- +symplectic manifold (M, ω). +Then semilocally around a regular Liouville torus +T, which lies inside the critical hypersurface Z of M, the system is equivalent to +the cotangent model (T ∗Tn)tw,(cc1,...,ccm) restricted to a neighbourhood of (T ∗Tn)0. +Here c is the modular period of the connected component of Z containing T and the +constants (c1, . . . , cm) are the invariants associated to the integrable system and its +associated toric action. + + +Part 3 +A KAM theorem for bm-symplectic +manifolds + +The KAM theorem explains how integrable systems behave under small per- +turbations. +More precisely, it studies how an integrable system in action-angle +coordinates responds to a small perturbation on its Hamiltonian. The trajectories +of an integrable system in action-angle coordinates can be seen as linear trajectories +over a torus. The KAM theorem finds a way to transform these original trajectories +to other linear trajectories over some transformed torus. The KAM theorem states +that most of these tori, and the linear solutions of the system on these tori, survive +if the perturbation is small enough. +In this part, we give a new KAM theorem for bm-symplectic manifolds with +detailed proof. This is contained in the first chapter of this part. Moreover, we +devote three more chapters to applications: +(1) Desingularization of bm-integrable systems. We present a way to use +the desingularization of bm-symplectic manifolds presented in [GMW17] +to construct standard smooth integrable systems from bm-integrable sys- +tems. This desingularized integrable system is uniquely defined. +(2) Desingularization of the KAM theorem on bm-symplectic man- +ifolds. In this section we use the desingularization of bm-integrable sys- +tems in conjunction with the KAM theorem for bm-symplectic manifolds +to deduce the original KAM theorem as well as a completely new KAM +theorem for folded symplectic forms. +(3) Potential applications to Celestial mechanics. We overview a list of +motivating examples from Celestial mechanics where regularization trans- +formations give rise to bm-symplectic forms. We discuss some potential +applications of perturbation theory in this set-up. + +CHAPTER 6 +A new KAM theorem +The objective of this chapter is to give a construction of KAM theory in the +setting of bm-symplectic manifolds and with bm-integrable systems. The core of the +chapter is the construction of the proper statement and the proof of the equivalent +of the KAM theorem on bm-symplectic manifolds. +This chapter is divided different sections: +(1) On the structure of the proof. On this section we are going to present +the main ideas that are going to appear in the proper statement and proof +of the main theorem. The idea of the theorem is to build a sequence of +bm-symplectomorphisms such that its limit transforms the hamiltonian to +only depend on the action coordinates. +(2) Technical results and definitions. On this section we present some +technical results and definitions that are key for the proof of the main +theorem. +(3) KAM theorem on bm-symplectic manifolds. +On this section we +present the statement and the proof of the main result of this chap- +ter. +The proof is structured in 6 parts. +In the first part we define +the parameters that are going to be used to define the sequence of bm- +symplectomorphisms. In the second part we build precisely this sequence +of bm-symplectomorphisms. In the third part we see that the sequence of +frequency maps of the transformed Hamiltonian functions at every step +converges. In the fourth part we see that the sequence of bm-symplectomorphisms +converges. In the fifth part we obtain results on the stability of the trajec- +tories under the original perturbation. In the sixth part, we find bounds +to explain how close the invariant tori are from the unperturbed. Finally, +we obtain a bound for the measure of the set of invariant tori. +6.1. On the structure of the proof +The first thing we do is to reduce our study to the case the perturbation is not +a bm-function but an analytic one. This is because any purely singular perturbation +only affects the component in the direction of the modular vector field and can be +easily controlled. +The idea of the proof is really similar to the classical KAM case. We want +to build a diffeomorphism such that its transformed hamiltonian only depends on +the action coordinates. But it is not possible to build this diffeomorphism in one +step. What we do, as it is done in the classical case, it is to build a sequence of +diffeomorphisms such that the part of the hamiltonian depending on the angular +variables decreases at every step. The idea is to remove the first K terms of its +Fourier expression at every step while making K rapidly increase. This is done by +43 + +44 +6. A NEW KAM THEOREM +assuming the diffeomorphism comes as the flow at time 1 generated by a Hamil- +tonian function. In this way one can use the Lie Series in conjunction with the +Fourier series to find the expression for the hamiltonian function that generates +our diffeomorphism. The final diffeomorphism will be the composition of all the +diffeomorphisms obtained at each step. One of the main difficulties of the proof, as +in the classical case, is to prove that these diffeomorphisms converge and to prove +some bounds of its norm. +We also note that for our bm-symplectic setting, the diffeomorphisms we con- +sider leave the defining function of the critical set invariant up to order m, this will +have an important role later. Also observe in particular that the critical set can +not be transformed by any perturbation given by a bm-function. +Next we give some technical definitions and results. We define the norms we are +going to use to do all the estimates. We set the notation for the proof and the state- +ment of the theorem. We define the notion of non-resonance for a neighborhood +of the critical set of the bm-symplectic manifold. We study the set of all possible +non-resonant vectors. And we state the inductive lemma, which gives us estimates +and constructions for every step of our sequence of diffeomorphisms. +After all this discussion we are in conditions to properly state the bm-version of +the KAM theorem. One important difference to the classical KAM theorem is that +we have to guarantee that at Z the set of non-resonant vectors does not become +the whole set of frequencies. This condition can be understood as the perturbation +being smaller than some constant multiplied by the inverse of the modular period. +The proof of the theorem is done in six different steps by following the structure +on [DG96]. Since we are going to use the inductive lemma at every step, first we +define the parameters and sets to which we are going apply such lemma. Then we +check that we can actually apply the lemma and obtain some extra estimates for +the results of the lemma. After this we see that the sequence of frequency vectors +converges. We do the same with the sequence of canonical transformations. Then +we get some bounds for the size of the components of the final diffeomorphism. +Next we characterize the tori that survive by the perturbation. Finally we give +some estimates for the measure of the set of these tori. +Note that our version of the bm-KAM theorem improves the one in [KMS16a] +in several ways. Firstly it is applicable to bm-symplectic structures not only for +b-symplectic. Also we give several estimates that are not obtained in [KMS16a], +this estimates have sense in a neighborhood of the critical set Z, while [KMS16a] +only studied the behavior at Z. Finally the type of perturbation we consider is far +more general, since we do not have any condition of the form of the perturbation +but only on its size. +6.1.1. Reducing the problem to an analytical perturbation. In the +standard KAM, we assume to have an analytic Hamiltonian h(I) depending only +on the action coordinates and we add to it a small analytical perturbation R(φ, I). +This perturbed system receives the name of nearly integrable system. And then find +a new coordinate system such that h(I) + R(φ, I) = ˜h(˜I) where most of the quasi- +periodic orbits are preserved and can be mapped to the unperturbed quasi-periodic +orbits by means of the coordinate change. +In our setting we may assume h(I) to not be analytical and be a bm-function. +Also the perturbation R(φ, I) may as well be considered a bm-function. +In the + +6.1. ON THE STRUCTURE OF THE PROOF +45 +following lines we justify without loss of generality that actually we can assume the +perturbation to be analytical. +Let us state this more precisely. Let (M, x, Z, ω, F) be a bm-manifold with a +bm integrable system F on it. Consider action angle coordinates on a neighborhood +of Z. Then we can assume the expressions: +ω = + + +m +� +j=1 +cj +Ij +1 + + dI1 ∧ dφ1 + +n +� +i=2 +dIi ∧ dφi, and +F = (q′ +0 log I1 + +m−1 +� +i=1 +q′ +i +1 +Ii +1 ++ h(I), f2, . . . , fn) +where h, f2, . . . , fn are analytical. +Let the Hamiltonian function of our system be the first component of the +moment map ˆh′ = q′ +0 log I1 + �m−1 +i=1 q′ +i +1 +Ii +1 + h = ζ′ + h, where ζ′ := q′ +0 log I1 + +�m−1 +i=1 q′ +i +1 +Ii +1 . Note that dζ′ = �m +i=1 ˆq′ +i +1 +I′ +1 , where ˆq′ +i = −(i − 1)q′ +i−1. Note that by +the result of the previous chapter cj/ˆq′ +j = K the modular period. In particular +cm/ˆq′ +m = K. +The hamiltonian system given by ˆh′ can be easily solved by φ = φ0 +u′t, I = I0 +where u′ is going to be defined in the following sections. Consider now a pertur- +bation of this system: +ˆH′ = ˆh′(I) = ˆR(I, φ), where ˆR is a bm-function ˆR(I, φ) = +Rζ(I1)+R(I, φ) where Rζ(I1) = (r0 log I1 +�m−1 +i=1 ri 1 +Ii +1 ) is the singular part. Then +we can consider the perturbations Rζ(I1) and R(I, φ) separately. This way, we may +consider Rζ(I) as part of ˆh′(I). Then we have a new hamiltonian +ˆh(I) = (q′ +0 + r0) log I1 + +m−1 +� +i=1 +(q′ +i + ri) 1 +Ii +1 ++ h = q0 log I1 + +m−1 +� +i=1 +qi +1 +Ii +1 ++ h. +Now, instead of the identity Kˆq′ +j = cj we will have K(ˆqj − ˆrj) = cj, which +implies K +� +1 − +ˆrj +ˆq′ +j+ˆrj +� += cj +ˆqj . In particular +K +� +1 − +ˆrm +ˆq′m + ˆrm +� += cm +ˆqm +Let us define K′ = K +� +1 − +ˆrm +ˆq′m+ˆrm +� +. So from now on we assume ˆh = q0 log I1 + +�m−1 +i=1 qi 1 +Ii +1 + h, that the perturbation R(φ, I) is analytical, and we have the condi- +tion cm +ˆqm = K′. Observe that this system with only the singular perturbation is still +easy to solve in the same way that the system previous to this perturbation was. +6.1.2. Looking for a bm-symplectomorphism. Assume we have a Hamil- +tonian function H = ˆh(I) + R(φ, I) in action-angle coordinates. Where ˆh(I) is the +singular component of the bm-integrable system, i.e. +(6.1) +ˆh(I) = h(I) + q0 log(I1) + +m−1 +� +i=1 +qi +1 +Ii +1 +, + +46 +6. A NEW KAM THEOREM +where h(I) is analytical1. Assume also that the bm-symplectic form ω2 in these +coordinates is expressed as: +(6.2) +ω = + + +m +� +j=1 +cj +Ij +1 + + dI1 ∧ dφ1 + +n +� +i=2 +dIi ∧ dφi. +And finally, the expression for the frequency vector is: +ˆu = ∂ˆh +∂I = +∂(h(I) + q0 log(I1) + �m−1 +i=1 qi 1 +Ii +1 ) +∂I += +� +u1 + +m +� +i=1 +ˆqi +Ii +1 +, u2, . . . , un +� +, +where ˆq1 = q0 and ˆqi−1 = −iqi if i ̸= 0. +The objective is to follow the steps of the usual KAM construction (the steps +followed are highly inspired in [DG96]) replacing the standard symplectic form for +ω and taking as hamiltonian the bm-function ˆh. +Remark 6.1. The objective of the construction is to find a diffeomorphism +(actually a bm-symplectomorphism) ψ such that H ◦ ψ = h(˜I). This is done induc- +tively, by taking H ◦ ψ = H ◦ φ1 ◦ . . .◦ φq ◦ . . ., while trying to make R(φ, I) smaller +at every step. +Let us focus in one single step +Recall the classical formula: +Lemma 6.2. See [DG96]. +f ◦ φt = +∞ +� +j=0 +tj +j!Lj +W f, +Lj +W f = {Lj−1 +W f, W} +Where W is the Hamiltonian that generates the flow φt, and {·, ·} is the correspond- +ing Poisson bracket. +We will denote rk(H, W, t) = �∞ +j=k +tj +j! Lj +W H. +1If another component of the moment map is chosen to be the hamiltonian of the system, +the result still holds: the computations can be replicated assuming ˆh(I) = h(I). +2In classical KAM, ω is used to denote the frequency vector ∂h +∂I . We need ω to denote the +bm-symplectic form so we are going to use u to denote the frequency vector. + +6.1. ON THE STRUCTURE OF THE PROOF +47 +(6.3) +H ◦ φ = H ◦ φ|t=1 += +∞ +� +j=0 +tj +j! Lj +W H +���� +ˆh+R +������ +t=1 += +ˆh + R{ˆh + R, W} + r2(H, W, 1) += +ˆh + R + {ˆh, W} + {R, W} + r2(ˆh, W, 1) ++r2(R, W, 1) += +ˆh + +R + {ˆh, W} +� +�� +� +We want to cancel +this term as +fast as we can ++r2(ˆh, W, 1) + r2(R, W, 1) +We want {ˆh, W} +R≤k = 0, equivalently {W, ˆh} = R≤k, where R≤k means the +Fourier expression of R up to order K: +R≤k = +� +k∈Rn +|k|1≤K +Rk(I)eik·φ +Let us impose the condition {W, ˆh} = R≤K. Let us write the expression of the +Poisson bracket associated to the bm-symplectic form. +{W, ˆh} += +� +1 +�m +j=1 +cj +Ij +1 +� � +∂W +∂φ1 +∂ˆh +∂I1 +− ∂W +∂I1 +∂ˆh +∂φ1 +� ++ +n +� +i=2 +� +∂W +∂φi +∂ˆh +∂Ii +− ∂W +∂Ii +∂ˆh +∂φi +� +Because ˆh depends only on I, +∂ˆh +∂φi = 0 for all i. Moreover, the singular part of +the bm-function only depends on I1 and hence its derivatives with respect to the +other variables are also 0. Using that ∂ˆh +∂I = u + �m +i=1 +ˆqi +Ii +1 the previous expression +can be simplified: +{W, ˆh} += + +u1 + �m +i=1 +ˆqi +Ii +1 +�m +j=1 +cj +Ij +1 + + ∂W +∂φ1 ++ +n +� +i=2 +∂W +∂φi +ui +To expand the expression further we develop W in its Fourier expression: W = +� +k∈Rn +|k|1≤K +Wk(I)eikφ. The Fourier expansion is added up to order K, because it is +only necessary for the expressions to agree up to order K. With this notations the +condition becomes: +{W, ˆh}≤K += + +u1 + �m +i=1 +ˆqi +Ii +1 +�m +j=1 +cj +Ij +1 + + +∂ +∂φ1 + + + + +� +k∈Rn +|k|1≤K +Wk(I)eikφ + + + + + +48 +6. A NEW KAM THEOREM ++ +n +� +j=2 +uj +∂ +∂φj + + + + +� +k∈Rn +|k|1≤K +Wk(I)eikφ + + + + += + +u1 + �m +i=1 +ˆqi +Ii +1 +�m +j=1 +cj +Ij +1 + + + + + + +� +k∈Rn +|k|1≤K +Wk(I)eikφik1 + + + + ++ +n +� +j=2 +uj + + + + +� +k∈Rn +|k|1≤K +Wk(I)eikφikj + + + + += +� +k∈Rn +|k|1≤K +Wk(I)eikφ · + +ik1 + +u1 + �m +i=1 +ˆqi +Ii +1 +�m +j=1 +cj +Ij +1 + + + +n +� +j=2 +ikjuj + + += R≤K +Then, it is possible to make the two sides of the equation equal by imposing +the condition term by term: +(6.4) +Wk(I) += +Rk(I) +1 +i +� +k1 +� +u1+�m +i=1 +ˆ +qi +Ii +1 +�m +j=1 +cj +Ij +1 +� ++ �n +j=2 kjuj +� += +Rk(I) +1 +i +� +k1 +� +u1+�m +i=1 +ˆ +qi +Ii +1 +�m +j=1 +cj +Ij +1 +� ++ ¯k¯u +�, +where we adopted the notation �n +j=2 kjuj = ¯k¯u. +Remark 6.3. Observe that the expression 6.4 has no sense when k = ⃗0 and +hence {W, h}0 = R03 can not be solved. Let W0(I) = 0, then {h, W}≤K = R≤K − +R0. +Plugging the results above into the equation 6.3, one obtains: +H ◦ φ = ˆh + R0 + R≥K + r2(ˆh, W, 1) + r1(R, W, 1) +With this construction the diffeomorphism φ is found. But this is only the +first of many steps. If q denotes the number of the iteration of this procedure, in +general, we obtain: +3The zero term of the Fourier series can be seen as the angular average of the function + +6.1. ON THE STRUCTURE OF THE PROOF +49 +(6.5) +H(q) = H(q−1) ◦ φ(q) += +ˆh(q−1) + R(q−1) +0 ++ R(q−1) +≥K ++r2(h(q−1), W (q), 1) + r1(R(q−1), W (q), 1), +and at every step: +(6.6) +�ˆh(q) = ˆh(q−1) + R(q−1) +0 +R(q) = R(q−1) +>K ++ r2(ˆh(q−1), W (q), 1) + r1(R(q−1), W (q), 1) +6.1.3. On the change of the defining function under +bm-symplectomorphisms. Note that since we are considering bm-manifolds it +only makes sense to consider I1 up to order m, see [Sco16]. When talking about +defining functions we are interested in [I1], its jet up to order m. By definition bm- +maps preserve I1 up to order m and bm-vector fields X are such that LX(I1) = g·Im +1 +for g ∈ C∞(M). +Lemma 6.4. Let φt be the integral flow of X a bm-vector field, then φt is a +bm-map. +Proof. We want +I1 ◦ φt = I1 + Im +1 · g +for some g ∈ C∞(M). We will use 6.2. +I1 ◦ φt = +∞ +� +j=0 +tj +j!Lj +XI1 = I1 + LX(I1) + +∞ +� +j=2 +tj +j!Lj +XI1 += I1 + Im +1 + +∞ +� +j=2 +tj +j!Lj +XI1. +On the other hand, let us prove by induction Lk +XI1 = g(k)Im +1 . The first case is +obvious, assume the case k holds and let us prove the case k + 1. +Lk+1 +X +I1 += +{Lk +XI1, X} += +{g(k)Im +1 , X} += +(LXg(q))Im +1 + g(k) · mIm−1 +1 +LXI1 += +(LXg(k) + g(k) · m · Im−1 +1 +· g)Im +1 += +g(k+1)Im +1 +where g(k+1) = LXg(k) + g(k) · m · Im−1 +1 +· g. +□ +Lemma 6.5. The Hamiltonian vector flow of some smooth hamiltonian function +h is a bm-vector field. +Proof. At each point of Z the following identity holds LXhI1 = Im +1 +∂f +∂φ1 . The +result can be extended at a neighborhood of Z. +□ +Observe that combining the two previous results we get that the hamiltonian +flow of a function preserves I1 up to order m. + +50 +6. A NEW KAM THEOREM +6.2. Technical results +As the non-singular part of our functions we will be considering analytic func- +tions on T × G, G ⊂ Rn. +The easiest way to work with these functions is to +consider them as holomorphic functions on some complex neighborhood. Let us +define formally this neighborhood. +Wρ1(Tn) := {φ : ℜφ ∈ Tn, |ℑφ|∞ ≤ ρ1}, +Vρ2(G) := {I ∈ Cn : |I − I′| ≤ ρ2 for some I′ ∈ G}, +Dρ(G) := Wρ1(Tn) × Vρ2(G), +where | · |∞ denotes the maximum norm and | · |2 denotes de Euclidean norm. +Now it is necessary to clarify the norms that are going to be used on these sets. +Definition 6.6. Let f be an action function (only depending on the I-coordinates), +and F an action vector field. +|f|G,η := supI∈Vη(G) |f(I)|, +|f|G := |f|G,0 +|F|G,η,p := supI∈Vη(G) |F(I)|p, +|F|G,η := |F|G,η,2 +Now, assume f(I, φ) to be an action-angle function written using its Fourier ex- +pansion as � +k∈Zn fk(I)eik·φ, and F to be an action-angle vector field. +|f|G,ρ := sup(φ,I)∈Dρ(G) |f(I)|, +∥f∥G,ρ := � +k∈Zn |fk|G,ρ2e|k|1ρ1 +|F|G,ρ,p := � +k∈Zn |Fk|G,ρ2,pe|k|1ρ1, +∥F∥G,ρ = ∥F∥G,ρ,2 +Lemma 6.7 (Cauchy Inequality). +���� +∂f +∂φ +���� +G,(ρ1−δ1,ρ2),1 +≤ +1 +eδ1 +∥f∥G,ρ +���� +∂f +∂I +���� +G,(ρ1,ρ2−δ2),∞ +≤ 1 +δ2 +∥f∥G,ρ +Definition 6.8. If Df = ( ∂f +∂φ, ∂f +∂I ), +∥Df∥G,ρ,c := max +� +∥∂f +∂φ∥G,ρ,1, c∥∂f +∂I ∥G,ρ,∞ +� +Definition 6.9. To simplify our notation, let us define: +A(I1) = +�m +j=1 +ˆqj +Ij +1 +�m +j=1 +cj +Ij +1 +and +B(I1) = +1 +�m +j=1 +cj +Ij +1 +. +Remark 6.10. With this notation, equation 6.4 can be written as: +Wk(I) = +Rk(I) +i(k1B(I1)u1 + ¯k¯u + k1A(I1)) +Observe that A(I1) and B(I1) are analytic (holomorphic on the complex ex- +tended domain) where the denominator does not vanish. We can assume that this +does not happen by shrinking the domain G in the direction of I1. Observe, in +particular, that when I1 → 0, A(I1) → ˆqm/cm = 1/K′ the inverse of the modular +period and B(I1) → 0. In this way, the norms of A(I1) and B(I1) are bounded and +well defined. We will denote these norms by KA and KB respectively. Also, since + +6.2. TECHNICAL RESULTS +51 +A(I1) and B(I1) are analytic, their derivatives will also be bounded, and we will +denote the norms of these derivatives by KA′ and KB′. +To further simplify the notation in the following computations we introduce +the definition: +Definition 6.11. +¯ +A = +� +A +0 +� +and +¯B = +� +B +0 +0 +Idn−1,n−1 +� +Remark 6.12. With this notation, equation 6.4 can be written as: +(6.7) +Wk(I) = +Rk(I) +i(k ¯B(I1)u + k ¯ +A(I1)) +Definition 6.13. Having fixed ω, a bm-symplectic form (as in equation 6.2) +and ˆh a bm-function (as in equation 6.1) as a hamiltonian. Given an integer K and +α > 0, F ⊂ Rn (or Cn) the space of frequencies is said to be α, K-non-resonant +with respect to (c1, . . . , cm) and (ˆq1, . . . , ˆqm) if +|k ¯B(I1)u + k ¯ +A(I1)| ≥ α, ∀k ∈ Z \ {0}, |k|1 ≤ K, ∀u ∈ F. +We are going to use the notation α, K, c, ˆq-non-resonant. +Remark 6.14. The non-resonance condition is established on u = ∂h/∂I, not +on ˆu = ∂ˆh/∂I, because our non-resonance condition already takes into account the +singularities. In this way we can use the analytic character of u. +Remark 6.15. If +�� ∂u +∂I +�� +G,ρ2 is bounded by M ′, then +�� ∂ +∂I +� ¯Bu + ¯ +A +��� +G,ρ2 is also +bounded: +(6.8) +�� ∂ +∂I +� ¯Bu + ¯ +A +��� +G,ρ2 +≤ +��� ∂ ¯ +B +∂I u + ¯B ∂u +∂I + ∂ ¯ +A +∂I +��� +G,ρ2 +≤ +KB′|u|G,ρ2 + KBM ′ + KA =: M. +Remark 6.16. When we consider the standard KAM theorem, the frequency +vector u is relevant because the solution to the Hamilton equations of the unper- +turbed problem has the form: +I = I0, +φ = φ0 + ut. +Let us see what plays the role of u in our bm-KAM theorem. Let us find the +coordinate expression of the solution to ιXˆhω = dˆh, where ω is a bm-symplectic +form in action-angle coordinates. +Xˆh = ˙I1 +∂ +∂I1 ++ . . . + ˙In +∂ +∂In +, +where ˙I1, . . . , ˙In are the functions we want to find. +dˆh = + + +m +� +j=1 +ˆqi +1 +Ij +1 + + dI1 + dh, +and hence, + +52 +6. A NEW KAM THEOREM +Xˆh = Π(dˆh, ·) = +�m +i=1 +ˆqi +Ii +1 +�m +i=1 +cj +Ij +1 +∂ +∂φi ++ Xh. +Hence φ = φ0 + ( ¯Bu + ¯ +A +� �� � +u′ +)t. So the frequency vector that we are going to be +concerned about is going to be u′ instead of ˆu = +∂ +∂I ˆh. +Lemma 6.17. If u is one-to-one from G to its image then u′ = ¯Bu + ¯ +A is +also one-to-one from G′ to its image in a neighborhood of Z, while at Z it is the +projection of u such that the first coordinate is sent to ˆqm +cm = 1/K′ the inverse of the +modular period, were G′ ⊆ G. +Proof. Because +u′ = + + +1 +�m +j=1 +cj +Ij +1 +u1 + +�m +j=1 +ˆqj +Ij +1 +�m +j=1 +cj +I1 +, u2, . . . , un + + , +and B is invertible outside I1 = 0, shrinking G if necessary in the first dimension +the map is one-to-one. But at the critical set {I1 = 0}, u′ is a projection of u where +the first component is sent to the constant value ˆqm +cm = +1 +K′ . +□ +Lemma 6.18. If u(G) is α, K, c, ˆq-non-resonant, then u(Vρ2(G)) is α +2 , K, c, ˆq- +non-resonant, assuming that ρ2 ≤ +α +2MK and +�� ∂u +∂I +�� +G,ρ2 ≤ M ′ +Proof. Fix k ∈ Z \ {0}, we want to bound |k ¯B(I1)v + k ¯ +A(I1)| where v ∈ +u(Vρ2(G)) as a function on v. Given v ∈ u(Vρ2(G)) we ask whether there is any +bound for the distance to some v′ ∈ u(G). +v ∈ u(Vρ2(G)) ⇒ v = u(x), x ∈ Vρ2(G) ⇒ ∃y ∈ G such that |x − y| ≤ ρ2. +Take v′ = u(y). +|v − v′| ≤ |x − y| +���� +∂u +∂I +���� +G,ρ2 +≤ ρ2M ′ ≤ ρ2M/KB ≤ +α +2MK M/KB = +α +2KKB +. +Where we used equation 6.8 in the third inequality. +|k1B(I1)v1 + ¯k¯v + k1A(I1)| +≥ +|k1B(I1)v′ +1 + ¯k ¯v′ + k1A(I1)| +� +�� +� +≥α +−|k1B(I1)(v1 − v′ +1) + ¯k(¯v − ¯v′)| +≥ +α − KB |k · (v − v′)| +� +�� +� +≤Kα/(2KKB) +≥ +α − α/2 = α/2 +□ +Proposition 6.19. Let ˆh(I) be a bm-function as in equation 6.1. Assume h(I) +and R(φ, I) be real analytic on Dρ(G), u(G) = ∂h +∂I (G) is α, K, c, ˆq-non-resonant. +Assume also that | ∂ +∂I u|G,ρ2 ≤ M ′ and ρ2 ≤ +α +2MK . Let c > 0 given. Then R0(φ, I), + +6.2. TECHNICAL RESULTS +53 +W≤K(φ, I) given by the previous construction are both real analytic on Dρ(G) and +the following bounds hold +(1) ||DR0||G,ρ,c ≤ ||DR||G,ρ,c +(2) ||D(R − R0)||G,ρ,c ≤ ||DR0||G,ρ,c +(3) ||DW||G,ρ,c ≤ 2A +α ||DR0||G,ρ,c +Where A = 1 + 2Mc +α +Proof. Inequalities 1 and 2 are obvious because of the Fourier expression. +Let us prove inequality 3. +Let us expand R(φ, I) and W(φ, I) in their Fourier +expression: +R = +� +k∈Rn +Rk(I)eik·φ, +W = +� +k∈Rn +Wk(I)eik·φ. +We will bound this expression finding term-by-term bounds. +∂R +∂φ = +� +k∈Rn +Rk(I)eik·φik. +Hence, if we denote [ ∂R +∂φ ]k the k-th term of the Fourier expansion of ∂R +∂φ , we +have: +�∂R +∂φ +� +k += Rkik. +Let us compute the derivative of Wk with respect to the I variables: +∂Wk +∂I += +∂ +∂I +� +Rk +i(k ¯B(I1)u + k ¯ +A(I1)) +� += +∂Rk/∂I +i(k ¯B(I1)u + k ¯ +A(I1))) − Rki ∂ +∂I (k ¯B(I1)u + k ¯ +A(I1))) +[i(k ¯B(I1)u + k ¯ +A(I1)))]2 += +∂Rk/∂I +i(k ¯B(I1)u + k ¯ +A(I1))) + Rkik ∂ +∂I ( ¯B(I1)u + ¯ +A(I1))) +[(k ¯B(I1)u + k ¯ +A(I1)))]2 += +∂Rk/∂I +i(k ¯B(I1)u + k ¯ +A(I1))) + +[ ∂Rk +∂φ ]k ∂ +∂I ( ¯B(I1)u + ¯ +A(I1))) +[(k ¯B(I1)u + k ¯ +A(I1)))]2 +. +Then, we take norms (| · |G,ρ2,∞) on each side of the equation. +���� +∂Wk +∂I +���� +G,ρ2,∞ +≤ +2 +α +���� +∂Rk +∂I +���� +G,ρ2,∞ ++ 4M +α2 +���� +�∂Rk +∂φ +� +k +���� +G,ρ2,∞ +≤ +2 +α +���� +∂Rk +∂I +���� +G,ρ2,∞ ++ 4M +α2 +���� +�∂Rk +∂φ +� +k +���� +G,ρ2,1 +. +Taking the supremum at the whole domain: +���� +∂Wk +∂I +���� +G,ρ2,∞ +≤ +2 +α +���� +∂Rk +∂I +���� +G,ρ2,∞ ++ 4M +α2 +���� +�∂Rk +∂φ +� +k +���� +G,ρ2,1 +. +Moreover, + +54 +6. A NEW KAM THEOREM +∂W(I) +∂φ += +∂ +∂φ +� � +k∈Rn +Wk(I)eik·φ +� += +∂ +∂φ +� � +k∈Rn +ikWk(I)eik·φ +� +. +Hence, the k-th term of the Fourier series of ∂W +∂φ is +�∂W +∂φ +� +k += Wkik = +Rk +i(k ¯B(I1)u + k ¯ +A(I1)))ik += +1 +i(k ¯B(I1)u + k ¯ +A(I1))) +�∂R +∂φ +� +k +. +Taking norms (∥ · ∥G,ρ,1) at each side: +���� +∂W +∂φ +���� +G,ρ,1 +≤ 2 +α +���� +∂W +∂φ +���� +G,ρ,1 +. +Then, +∥DW∥G,ρ,c += +max +����� +∂W +∂φ +���� +G,ρ,1 +, c +���� +∂W +∂I +���� +G,ρ,∞ +� +≤ +max +� +2 +α +���� +∂R +∂φ +���� +G,ρ,1 +, c 2 +α +���� +∂R +∂I +���� +G,ρ2,∞ ++ c4M +α2 +���� +∂R +∂φ +���� +G,ρ2,1 +� +≤ +max +� +2 +α +���� +∂R +∂φ +���� +G,ρ,1 +, 2 +α ∥DR∥G,ρ2,c + c4M +α2 ∥DR∥G,ρ2,c +� += +max +� +2 +α +���� +∂R +∂φ +���� +G,ρ,1 +, 2 +α +� +1 + 2M +α c +� +∥DR∥G,ρ2,c +� +≤ +2 +α +� +1 + 2M +α c +� +∥DR∥G,ρ2,c +≤ +2 +αA ∥DR∥G,ρ2,c, +where A is as desired. +□ +Recall the Cauchy inequalities, see [P¨os93]: +(6.9) +���� +∂f +∂φ +���� +G,(ρ1,ρ2),1 +≤ +1 +eδ1 +∥f∥G,ρ +���� +∂f +∂I +���� +G,(ρ1,ρ2−δ2),∞ +≤ +1 +δ2 +∥f∥G,ρ + +6.2. TECHNICAL RESULTS +55 +Lemma 6.20. Let f, g be analytic functions on Dρ(G), where 0 < δ = (δ1, δ2) < +ρ = (ρ1, ρ2) and c > 0. Define ˆδc := min(cδ1, δ2). The following inequalities hold: +(1) ∥Df∥G,ρ−δ,c ≤ +c +ˆδc ∥f∥G,ρ +(2) ∥{f, g}∥G,ρ ≤ 2 +c∥Df∥G,ρ,c · ∥Dg∥G,ρ,c +(3) ∥D(f>K)∥G,(ρ−δ1,ρ2),c ≤ e−Kδ1∥Df∥G,ρ,c +Proof. Let us prove each point separately. +(1) Using the Cauchy inequalities one obtains the following: +���� +∂f +∂φ +���� +G,ρ−δ,1 += +���� +∂f +∂φ +���� +G,(ρ1−δ1,ρ2−δ2),1 +≤ +���� +∂f +∂φ +���� +G,(ρ1−δ1,ρ2),1 +≤ +1 +eδ1 +∥f∥G,ρ, +���� +∂f +∂I +���� +G,ρ−δ,∞ += +���� +∂f +∂I +���� +G,(ρ1−δ1,ρ2−δ2),∞ +≤ +���� +∂f +∂I +���� +G,(ρ1,ρ2−δ2),∞ +≤ 1 +δ1 +∥f∥G,ρ. +Putting the two inequalities inside the definition of the norm: +∥Df∥G,ρ−δ,c += +max +����� +∂f +∂φ +���� +G,ρ−δ,1 +, c +���� +∂f +∂I +���� +G,ρ−δ,∞ +� +≤ +max +� 1 +eδ1 +∥f∥G,ρ, c +δ2 +∥f∥G,ρ +� +≤ +max +� 1 +eδ1 +c +c, c +δ2 +� +∥f∥G,ρ +≤ +max +� c +eˆδc +, c +ˆδc +� +∥f∥G,ρ, +where the last inequality holds because ˆδc = min(cδ1, δ2). +(2) Let us find the expression of {f, g} for a bm-symplectic structure. {f, g} = +ω(Xf, Xg) where Xf and Xg are such that ιXf ω = df and ιXgω = dg. +Let restrict the computations only to f. +df = +n +� +i=1 +∂f +∂φ1 +dφ1, +Xf = +n +� +i=1 +ai +∂ +∂φi ++ +n +� +i=1 +bi +∂ +∂φi +. +Where ai and bi are coefficients to be determined by imposing the +following condition: +ιXf ω = + + +m +� +j=1 +cj +Ij +1 + + (a1dI1 − b1dφ1) + +n +� +i=2 +(aidIi − bidφi) = df. +Then, solving for the coefficients the following expressions are ob- +tained: + +56 +6. A NEW KAM THEOREM +a1 = +1 +��m +j=1 +cj +Ij +1 +� ∂f +∂φ1 +and +ai = ∂f +∂φi +for i ̸= 1, +b1 = − +1 +��m +j=1 +cj +Ij +1 +� ∂f +∂φ1 +and +bi = − ∂f +∂φi +for i ̸= 1. +Hence, the expression for the hamiltonian vector fields becomes: +Xf = +1 +��m +j=1 +cj +Ij +1 +� +� ∂f +∂φ1 +∂ +∂φ1 +− ∂f +∂I1 +∂ +∂I1 +� ++ +n +� +i=1 +� ∂f +∂φi +∂ +∂φi +− ∂f +∂Ii +∂ +∂Ii +� +, +Xg = +1 +��m +j=1 +cj +Ij +1 +� +� ∂g +∂φ1 +∂ +∂φ1 +− ∂g +∂I1 +∂ +∂I1 +� ++ +n +� +i=1 +� ∂g +∂φi +∂ +∂φi +− ∂g +∂Ii +∂ +∂Ii +� +. +Then the Poisson bracket applied to the two functions: +{f, g} = ω(Xf, Xg) += +1 +��m +j=1 +cj +Ij +1 +� +� ∂f +∂I1 +∂g +∂φ1 +− ∂f +∂φ1 +∂g +∂I1 +� ++ +n +� +i=2 +� ∂f +∂Ii +∂g +∂φi +− ∂f +∂φi +∂g +∂Ii +� +. +And hence the norm of the Poisson bracket becomes: +∥{f, g}∥G,ρ += +������� +1 +��m +j=1 +cj +Ij +1 +� +� ∂f +∂I1 +∂g +∂φ1 +− ∂f +∂φ1 +∂g +∂I1 +� ++ +n +� +i=2 +� ∂f +∂Ii +∂g +∂φi +− ∂f +∂φi +∂g +∂Ii +������ +G,ρ +≤ +����� +n +� +i=1 +� ∂f +∂Ii +∂g +∂φi +− ∂f +∂φi +∂g +∂Ii +������ +G,ρ +Where we assumed +����m +j=1 +cj +Ij +1 +��� ≥ 1. This assumption makes sense, +because we are interested in the behaviour close the critical set Z. Close +enough to the critical set this expression holds. Then, +∥{f, g}∥G,ρ +≤ +n +� +i=1 +���� +∂f +∂Ii +���� +G,ρ +���� +∂g +∂φi +���� +G,ρ ++ +n +� +i=1 +���� +∂f +∂φi +���� +G,ρ +���� +∂g +∂Ii +���� +G,ρ +≤ +���� +∂f +∂I +���� +G,ρ,∞ +���� +∂g +∂I +���� +G,ρ,1 ++ +���� +∂f +∂I +���� +G,ρ,1 +���� +∂g +∂I +���� +G,ρ,∞ +≤ +1 +c |Df∥G,ρ,c∥Dg∥G,ρ,c + 1 +c|Df∥G,ρ,c∥Dg∥G,ρ,c + +6.2. TECHNICAL RESULTS +57 +≤ +2 +c ∥Df∥G,ρ,c∥Dg∥G,ρ,c. +(3) Lastly, +∥D(f>K)∥G,(ρ1−δ1,ρ2),1 += max +����� +∂f>K +∂φ +���� +G,(ρ1−δ1,ρ1),1 +, c +���� +∂f>K +∂I +���� +G,(ρ1−δ1,ρ1),∞ +� +. +We will proceed by bounding each term separately. On one hand: +���� +∂f +∂φ +���� +G,(ρ1,ρ2),1 += +����� +� +k∈Zn +ikfk(I)eikφ +����� +G,(ρ1,ρ2),1 +≥ +� +k∈Zn +k ∥fk(I)∥G,ρ2,1 e|k|1ρ1 +≥ +� +k∈Zn +|k|1>K +k ∥fk(I)∥G,ρ2,1 e|k|1(ρ1+δ1−δ1) +≥ +eKδ1 +� +k∈Zn +|k|1>K +k ∥fk(I)∥G,ρ2,1 e|k|1(ρ1−δ1) += +eKδ1 +���� +∂f>K +∂φ +���� +G,(ρ1−δ1,ρ2),1 +. +On the other hand: +���� +∂f +∂I +���� +G,(ρ1,ρ2),∞ += +����� +� +k∈Zn +∂fk(I) +∂I +eikφ +����� +G,(ρ1,ρ2),∞ +≥ +� +k∈Zn +���� +∂fk(I) +∂I +���� +G,ρ2,∞ +e|k|1ρ1 +≥ +� +k∈Zn +|k|1>K +���� +∂fk(I) +∂I +���� +G,ρ2,∞ +e|k|1(ρ1+δ1−δ1) +≥ +eKδ1 +� +k∈Zn +|k|1>K +���� +∂fk(I) +∂I +���� +G,ρ2,∞ +e|k|1(ρ1−δ1) +≥ +eKδ1 +���� +∂f>K +∂I +���� +G,(ρ1−δ1,ρ2),∞ +. +Hence ∥D(f>k)∥G,(ρ1−δ1,ρ2),c ≤ e−Kδ1∥Df∥G,ρ,c. + +58 +6. A NEW KAM THEOREM +□ +Now we define a norm that indicates how close a map Φ is to the identity. +Definition 6.21. Let x = (φ, I) ∈ C2n, then +|x|c := max(|φ|1, c|I|∞) +Definition 6.22. For a map Υ : Dρ(G) → C2n its norm and the norm of its +derivative its defined as: +|Υ|G,ρ,c := +sup +x∈Dρ(G) +|Υ(x)|c, +|DΥ|G,ρ,c := +sup +x∈Dρ(G) +|DΥ(x)|c, +where |DΥ(x)|c = sup +y∈R2n +|y|c=1 +|DΥ(x) · y|c +Lemma 6.23. If Υ is analytic on Dρ(G), then |DΥ|G,ρ−δ,C ≤ |Υ|G,ρ,c +ˆδc +Proof. Observe that if we consider ∥.∥ any norm on Cn and a matrix A of +size n × n, and ∥A∥ defines the induced norm of matrices i.e. +∥A∥ = sup +y∈C2n +∥y∥=1 +∥A · y∥ +then one has that ∥(∥a1∥′, . . . , ∥an∥′)∥ ≤ ∥A∥ where aj denotes the j-th row of A. +Also note that ∥ · ∥′ can be a any norm consider the infinity norm. This can be +easily proven in the following way: +∥A · y∥ = +������� + + + +a1 · y +... +an · y + + + +������� +≤ +������� + + + +∥a1∥′∥y∥′ +... +∥an∥′∥y∥′ + + + +������� +Where ∀y ∈ Cn such that ∥y∥ = 1. Let aj be the rows of DΥ(x), +aj = +�∂Υj +∂φ , ∂Υj +∂I +� +, +and be ∥aj∥′ its norm. With this property in mind we proceed as follows: +|DΥ|G,ρ−δ,c += +sup +x∈Dρ−δ(G) +|DΥ(x)|c +≤ +sup +x∈Dρ−δ(G) +|(|a1|∞, . . . , |an|∞)|c +≤ +��� +� +supx∈Dρ−δ ∥DΥ1∥∞ , . . . , supx∈Dρ−δ ∥DΥ2n∥∞ +���� +c += +��� +� +∥DΥ1∥G,ρ−δ,∞ , . . . , ∥DΥ2n∥G,ρ−δ,∞ +���� +c +≤ +��� +� +1 +δ1 ∥Υ1∥G,ρ , . . . , 1 +δ1 ∥Υ2n∥G,ρ +���� +c + +6.2. TECHNICAL RESULTS +59 +≤ +1 +ˆδc +���∥Υ1∥G,ρ , . . . , ∥Υ2n∥G,ρ +��� +c += +1 +ˆδc supx∈Dρ(G) |Υ1, . . . , Υ2n|c = +1 +ˆδc supx∈Dρ(G) |Υ|c += +1 +ˆδc |Υ|G,ρ,c +□ +Lemma 6.24. Let W be an analytic function on Dρ(G), ρ > 0 and let Φt be its +Hamiltonian flow at time t (t > 0). Let δ = (δ1, δ2) > 0 and c > 0 given. Assume +that ∥DW∥G,ρ,c ≤ ˆδc. Then, Φt maps Dρ−tδ(G) into Dρ(G) and one has: +(1) |Φt − Id|G,ρ−tδ,c ≤ t∥DW∥G,ρ,c, +(2) Φ(Dρ(G)) ⊃ Dρ−tδ(G) for ρ′ ≤ ρ − tδ, +(3) Assuming that ∥DW∥G,ρ,c < ˆδc/2e, for any given function f analytic on +Dρ(G), and for any integer m ≥ 0, the following bound holds: +∥rm(f, W, t)∥G,ρ−tδ +≤ +∞ +� +l=0 +� +1 +�l+m +m +� · +�2e∥DW∥G,ρ,c +ˆδc +�l� +tm +m!∥Lm +Wf∥G,ρ += γm +�2e∥DW∥G,ρ,c +ˆδc +� +· tm∥Lm +Wf∥G,ρ, +where for 0 ≤ x ≤ 1 we define +γm(x) := +∞ +� +l=0 +l! +(l + m)!xl +Proof. During the proof we are going to denote Φs(φ0, I0) by (φ(s), I(s)). +Let us find the coordinate expression of the hamiltonian flow for the expression +6.2 of a bm-symplectic form. Recall that the equation for the hamiltonian flow is +d +dsφi(s) = {φi, W} and +d +dsIi(s) = {Ii, W}. +{φi, W} = +1 +��m +j=1 +cj +Ij +1 +� +�∂φi +∂I1 +· ∂W +∂φ1 +− ∂φi +∂φ1 +· ∂W +∂I1 +� ++ +n +� +j=2 +�∂φi +∂Ij +· ∂W +∂φj +− ∂φi +∂φj +· ∂W +∂Ij +� +. +Hence, +d +dsφi(s) = − +1 +��m +j=1 +cj +Ij +1 +� ∂W +∂I1 +if i = 1 and d +dsφi(s) = −∂W +∂Ii +if i ̸= 1. +On the other side, +{Ii, W} = +1 +��m +j=1 +cj +Ij +1 +� +� ∂Ii +∂I1 +· ∂W +∂φ1 +− ∂Ii +∂φ1 +· ∂W +∂I1 +� ++ +n +� +j=2 +� ∂Ii +∂Ij +· ∂W +∂φj +− ∂Ii +∂φj +· ∂W +∂Ij +� +. + +60 +6. A NEW KAM THEOREM +Hence, +d +dsIi(s) = +1 +��m +j=1 +cj +Ij +1 +� ∂W +∂φ1 +if i = 1 and d +dsIi(s) = ∂W +∂φi +if i ̸= 1. +(1) Assume now that 0 < s0 ≤ t. Then, +|φ(s0) − φ0|∞ +≤ +s0 sup0 0, A = 1 + 2Mc +α . +Assume that ρ2 ≤ +α +2MK , ∥DR∥G,ρ,c ≤ αˆδc +74A. Then, there exists a real analytic map +Φ : Dρ− δ +2 (G) → Dρ(G), such that H ◦ Φ = ˆh + ˜R,with: +(1) ∥D ˜R∥G,ρ−δ,c ≤ e−Kδ1∥DR∥G,ρ,c + 14A +αˆδc ∥DR∥2 +G,ρ,c, +(2) |Φ − Id|G,ρ− δ +2 ,c ≤ 2A +α ∥DR∥G,ρ,c, +(3) Φ(Dρ′(G)) ⊃ Dρ′− δ +2 (G) for ρ′ ≤ ρ − δ +2 +Proof. Recall that +�� ∂ +∂I u +�� +G,ρ2 ≤ M ′ implies +�� ∂ +∂I ( ¯Bu + ¯ +A) +�� +G,ρ2 ≤ M by equa- +tion 6.8. By equation 6.6 +R(q) = R(q−1) +>K ++ r2(ˆh(q−1), W (q), 1) + r1(R(q−1), W (q), 1). +To simplify the notation we are going to omit the index of the iteration: +(6.11) +˜R = R>K + r2(ˆh,W, 1) + r1(R, W, 1). +Where W is defined in terms of its Fourier expressions by equation 6.7: +Wk(I) = +Rk(I) +i(k ¯B(I1)u + k ¯ +A(I1)) +By proposition 6.19: ∥DW∥G,ρ,c ≤ 2A +α ∥DR∥G,ρ,c ≤ 2A +α +αˆδc +74A = +ˆδc +37. And Φ is +defined as in lemma 6.20: Φ : Dρ− δ +2 (G) → Dρ(G). +(1) Differentiating equation 6.11 we obtain: +D ˜R = DR>K + Dr2(ˆh, W, 1) + Dr1(R, W, 1). +Taking norms at every side of the expression: +∥D ˜R∥G,ρ−δ,c += +∥DR>K + Dr2(ˆh, W, 1) + Dr1(R, W, 1)∥G,ρ−δ,c +≤ +∥DR>K∥G,ρ−δ,c + ∥Dr2(ˆh, W, 1)∥G,ρ−δ,c ++∥Dr1(R, W, 1)∥G,ρ−δ,c +≤ +e−Kδ1∥DR∥G,ρ,c ++ 2c +ˆδc +� +∥r2(ˆh, W, 1)∥G,ρ− δ +2 ,c + ∥r1(R, W, 1)∥G,ρ− δ +2 ,c +� +Let us further develop the two last terms of the previous expression, +by using lemma 6.24: +∥r2(ˆh, W, 1)∥G,ρ− δ +2 ,c +≤ +γ2 +� +2e∥DW∥G,ρ,c +ˆδc/2 +� +∥L2 +W h∥G,ρ +≤ +γ2 +� +4e∥DW∥G,ρ,c +ˆδc +� +∥{{h, W}, W}∥G,ρ, +∥r1(ˆh, W, 1)∥G,ρ− δ +2 ,c +≤ +γ1 +� +2e∥DW∥G,ρ,c +ˆδc/2 +� +∥L1 +W R∥G,ρ +≤ +γ1 +� +4e∥DW∥G,ρ,c +ˆδc +� +∥{R, W}∥G,ρ. + +62 +6. A NEW KAM THEOREM +Then, using the second statement of lemma 6.20 and that {W, h} = +R≤K: +∥{R, W}∥G,ρ ≤ 2 +c∥DR∥G,ρ,c∥DW∥G,ρ,c, and +|{{h, W}, W}∥G,ρ += +∥{R≤K, W}∥G,ρ +≤ +2 +c∥DR≤K∥G,ρ,c∥DW∥G,ρ,c +≤ +2 +c∥DR∥G,ρ,c∥DW∥G,ρ,c. +Moreover, it is easy to see that γ1(x) = +− log(1−x) +x +and γ2(x) = +x+(1−x) log(1−x) +x2 +. Observe that these functions are monotonously increas- +ing in x. Recall that ∥DW∥G,ρ,c ≤ 2A +α ∥DR∥G,ρ,c. Then, +∥r1(ˆh, W, 1)∥G,ρ− δ +2 ,c ++∥r2(ˆh, W, 1)∥G,ρ− δ +2 ,c +≤ +γ1 +� +4e∥DW∥G,ρ,c +ˆδc +� +∥{R, W}∥G,ρ ++γ2 +� +4e∥DW∥G,ρ,c +ˆδc +� +∥{{h, W}, W}∥G,ρ +≤ +γ1 +� +4e∥DW∥G,ρ,c +ˆδc +� +2 +c∥DR∥G,ρ,c∥DW∥G,ρ,c ++γ2 +� +4e∥DW∥G,ρ,c +ˆδc +� +2 +c∥DR∥G,ρ,c∥DW∥G,ρ,c +≤ +γ1 +� +4e∥DW∥G,ρ,c +ˆδc +� +2 +c +2A +α ∥DR∥2 +G,ρ,c ++γ2 +� +4e∥DW∥G,ρ,c +ˆδc +� +2 +c +2A +α ∥DR∥2 +G,ρ,c +≤ +2 +c[γ1( 4e +37) + γ2( 4e +37)] 2A +α ∥DR∥2 +G,ρ,c += +4A +αc [γ1( 4e +37) + γ2( 4e +37)]∥DR∥2 +G,ρ,c. +Moreover γ1( 4e +37) + γ2( 4e +37) ≈ 1.741 . . . < 7 +4. +Then, +∥D ˜R∥G,ρ−δ,c +≤ +e−Kδ1∥DR∥G,ρ,c + 2c +ˆδc +4A +αc +7 +4∥∥2 +G,ρ,c +≤ +e−Kδ1∥DR∥G,ρ,c + 14A +ˆδcα ∥DR∥2 +G,ρ,c, +as we wanted to prove. +(2) Direct from lemma 6.24: +|Φ − Id|G,ρ. δ +2 ,c ≤ ∥DW∥G,ρ,c ≤ 2A +α ∥DR∥G,ρ,c +(3) Also direct from lemma 6.24: +Φ(Dρ(G)) ⊃ Dρ′− δ +2 (G), for ρ′ ≤ ρ − δ/2 +□ +Definition 6.26. ∆c,ˆq(k, α) = {J ∈ Rn such that |k ¯B(I1)J + k ¯ +A(I1)| < α} +Lemma 6.27. With the previous definitions we have the following bounds. +Outside of Z: +meas (F ∩ ∆c,ˆq(k, α)) ≤ (diamF)n−1 2α +|k|2,ω +. + +6.2. TECHNICAL RESULTS +63 +At Z: +meas (F ∩ ∆c,ˆq(k, α)) +� += 0 +if α ≤ |k1| +K′ +≤ (diamF)n +if α > |k1| +K′ +Proof. It is important to understand the geometry of the set ∆c,ˆq(k, α). Re- +call that k ¯B(I1)J = k1B(I1)J1 + ¯k ¯J, hence this part of the expression can be inter- +preted as the scalar product of the vector J with the vector (k1B(I1), k2, . . . , kn). +Then the set {J ∈ Rn such that |k ¯B(I1)J| < α} is the space between two hyper- +planes orthogonal to (k1B(I1), k2, . . . , kn). Adding the term k ¯ +A(I1) only applies +a transition to the previous set. Let us find what is the separation between the +hyperplanes. Assume J is parallel to (k1B(I1), k2, . . . , kn) with lengths a: +J = a(k1B(I1), k2, . . . , kn) +|k|2,ω +, +where |k|2,ω = +� +B(I1)2k2 +1 + k2 +2 + . . . k2n. Then, +J · (B(I1), k1, . . . , kn) += +c(B(I1)k2 +1 + k2 +2 + . . . k2 +n) +1 +|k|2,ω += +a|k|2,ω ≤ α ⇔ a ≤ +α +|k|2,ω . +And finally, +meas (F ∩ ∆c,ˆq(k, α)) ≤ (diamF)n−1 2α +|k|2,ω +. +The previous formula can not be applied if when we are at Z and k = (k1, 0, . . . , 0). +(B(I1)k1, k2, . . . , kn) +|k ¯B(I1)J + k ¯ +A(I1)| < α +|k ¯B(I1)J| < α +−k ¯ +A(I1) +Figure 1. Graphical representation of the set ∆c,ˆq(α) +At Z, +∆c,ˆq(K, α) = {J ∈ Rn such that | ¯K ¯J + k1 +ˆqm +cm +| < α}. +And if k = (k1, 0, . . . , 0) then +∆c,ˆq(K, α) = {J ∈ Rn such that |k1 +ˆqm +cm +| < α}. + +64 +6. A NEW KAM THEOREM +Then +∆c,ˆq(k, α) = +� Rn +if |k1| < α cm +ˆqm = αK′, +{∅} +if |k1| ≥ α cm +ˆqm = αK′. +Using this last identity, the statement we wanted to prove is immediate. +□ +Definition 6.28. G − b := {I ∈ G such that Ub(I) ⊂ G}, where Ub(I) is the +ball of radius b centered at I. +Definition 6.29. F is a D-set if meas[(F − b1) \ (F − b2)] ≤ D(b2 − b1). +Lemma 6.30. Let F ⊂ Rn be a D-set for d ≥ 0, τ > 0, β ≥ 0 and k ≥ 0 an +integer. Consider the set +F(d, β, K) := (F − d) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, β +|k|τ +1 +� +. +Then, outside of Z: +(1) If d′ ≥ d, β′ ≥ β, k′ ≥ k, then +meas[F(d, β, k) \ F(d′, β′, k′)] ≤ +D(d′ − d) + 2(diamF)n−1 + + + + +� +k∈Zn\{0} +|k|1≤K +β′ − β +|k|τ +1|k|2,ω ++ +� +k∈Zn\{0} +0<|k|1≤K +β′ +|k|τ +1|k|2,ω + + + + +(2) For every b ≥ 0 +meas[F(d, β, K) \ (F(d, β, K) − b)] ≤ (D + 2n+1(dim F)n−1Kn)b +And inside of Z, if we assume β ≤ +1 +K′ , the equation 1 holds adding only the terms +¯k ̸= 0 and 2 holds without any change. +Proof. Recall that +∆c,ˆq +� +k, β +|k|τ +1 +� += +� +J ∈ Rn such that +��k ¯B(I1)J + k ¯ +A(I1) +�� < +β +|k|τ +1 +� +. +First we will prove the results outside of Z and then +(1) Let us expand the expression of meas[F(d, β, k) \ F(d′, β′, k′)]: + +(F − d) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, β +|k|τ +1 +� + + \ + +(F − d) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, β +|k|τ +1 +� + + . +Now we use the following property on the previous expression: +(A \ B) \ (C \ D) += +[(A \ B) \ C] ∪ [(A \ B) ∩ D] +⊂ +(A \ C) ∪ [(A \ B) ∩ D] += +(A \ C) ∪ (A ∩ (D \ B)), +where the last equality holds true because D ⊃ B. Using this property +we have that meas[F(d, β, k) \ F(d′, β′, k′)] is included in + +6.2. TECHNICAL RESULTS +65 +[(F − d) \ (F − d′)] +∪ + +(F − d) ∩ + + + + + + +� +k∈Zn\{0} +|k|1≤K′ +∆c,ˆq +� +k, β′ +|k|1 +� + + + + +\ + + + + +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, β +|k|1 +� + + + + + + + + . +And this expression is equivalent to: +[(F − d) \ (F − d′)] +∪ +� +k∈Zn\{0} +|k|1≤K +� +(F − d) ∩ +� +∆c,ˆq +� +k, β′ +|k|τ +1 +� +\∆c,ˆq +� +k, β +|k|τ +1 +��� +∪ +� +k∈Zn\{0} +K<|k|1≤K′ +� +(F − d) ∩ ∆c,ˆq +� +k, β′ +|k|τ +1 +�� +. +Now, using lemma 6.27 we obtain: +meas(F(d, β, K) \ F(d′, β′, K′)) ≤ +≤ D(d′ − d) + (diamF)n−1 + + + + +� +k∈Zn\{0} +|k|1≤K +2(β′ − β) +|k|τ +1|k|2,ω ++ +� +k∈Zn\{0} +K<|k|1≤K′ +2β′ +|k|τ +1|k|2,ω + + + + +(2) Observe that: +F(d, β, K) − b += + +(F − d) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, β +|k|τ +1 +� + + − b +⊃ +(F − (d + b)) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, β +|k|τ +1 ++ b|k|2,ω +� +. +Then, +meas[(F(d, β, K)) \ (F(d, β, K) − b)] +≤ +meas +�� +(F − d) \ � +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, +β +|k|τ +1 +�� +\ +� +(F − (d + b)) \ � +k∈Zn\{0} +|k|1≤K +∆c,ˆq +� +k, +β +|k|τ +1 +��� +≤ +meas [(F − d) \ (F − (d + b))∪ + +66 +6. A NEW KAM THEOREM +� +k∈Zn\{0} +|k|1≤K +� +(F − d) ∩ +� +∆c,ˆq +� +k, +β +|k|τ +1 + b|k|2,ω +��� +\ +� +∆c,ˆq +� +k, +β +|k|τ +1 +��� +≤ +Db + � +k∈Zn\{0} +|k|1≤K +(diamF)n−1 2b|k|2,ω +|k|2,ω +≤ +Db + 2nKn(diamF)n−1 · 2 = Db + 2n+1Kn(diamF)n−1, +where in the last inequality we used that the number of vectors k such +that |k|1 ≤ K is less or equal than 2nKn. +The previous identities worked outside of Z. Let us understand the set F(d, β, K) +when we are ate Z. +F(d, β, K) +:= +(F − d) \ � +k∈Zn\{0} +|k|1≤K +∆cˆq(k, +β +|k|τ +1 ) += +(F − d) \ + + + + + +� +k∈Zn\{0} +|k|1≤K +¯k̸=0 +∆cˆq(k, +β +|k|τ +1 ) + + + +∪ + + + +� +k∈Zn\{0} +|k|1≤K +¯k=0 +∆cˆq(k, +β +|k|τ +1 ) + + + + + += +(F − d) \ + + + + + +� +k∈Zn\{0} +|k|1≤K +¯k̸=0 +∆cˆq(k, +β +|k|τ +1 ) + + + +∪ + +� +k1∈Z\{0} +|k|1≤ +β +|k1|τ K′ +Rn + + + +. +Note that if for some k1 ∈ Z \ {0}, |k|1 ≥ +β +|k|τ +1 K′, we take out all the possible +frequencies. Then seems natural to ask |k|1 ≥ +β +|k|τ +1 K′ for all k1 ∈ Z \ {0}, which +holds if and only if |k1|1+τ ≥ βK′ for all k1 ∈ Z \ {0} or simply β ≤ +1 +K′ which we +assumed. Then +F(d, β, K) := (F − d) \ +� +k∈Zn\{0} +|k|1≤K +¯k̸=0 +∆cˆq(k, β +|k|τ +1 +). +Hence we can replicate the proof of 1 only with the terms ¯k ̸= 0. And the bound +of 2 can be slightly improved by using that the number of vectors k ∈ Zn \ {0} +such that |k|1 ≤ K and |¯k| ̸= 0 is bounded by 2nKn − K, but since it is not a big +improve, for the sake of simplicity we assume the bound 2 at Z. +□ +Lemma 6.31. Let G ⊂ Rn be compact. +u, ˜u : G → Rn maps of class C2. +|˜u − u| ≤ ε. +Assume that u is one-to-one on G, let F = u(G). +Consider the +following bounds: +���� +∂u +∂I +���� +G +≤ M, +���� +∂u +∂I (I) · v +���� ≥ µ|v| +∀v ∈ Rn, ∀I ∈ G, + +6.2. TECHNICAL RESULTS +67 +���� +∂˜u +∂I +���� +G +≤ ˜ +M, +���� +∂˜u +∂I2 +���� +G +≤ ˜ +M2, +���� +∂˜u +∂I (I)v +���� ≥ ˜µ|v| +∀v ∈ Rn, ∀I ∈ G, +˜µ < µ and ˜ +M < M. Assume ε ≤ ˜mu2/(4 ˜ +M2). Then, given a subset ˜F ⊂ F − 4Mε +˜µ +and writing ˜G = (˜u)−1( ˜F ), the map ˜u is one-to-one from ˜G to ˜F and +˜G ⊂ G − 2ǫ +˜µ , +u( ˜G) ⊃ ˜F − ε. +Moreover, +|(˜u)−1 − u−1| ˜ +F ≤ ε +µ +Proof. The statement is not any different than the classical one, so we are +not going to prove it in here. A proof can be found in [DG96]. +□ +Lemma 6.32 (Inductive lemma). Let G ⊂ Rn be a compact. +H(φ, I) = ˆh(I) + R(φ, I) +where ˆh is defined as in 6.1 in the domain Dρ(G),and R(φ, I) analytic on the same +domain. Let ˆu = ∂ˆh +∂I and u = ∂h +∂I . Assume that | ∂ +∂I u|G,ρ2 ≤ M ′ and |u|G ≤ L. Also, +assume that u is non-degenerate: +���� +∂u +∂I v +���� ≥ µ|v| +∀I ∈ G. +Let ˜ +M > M, ˜L > L and ˜µ < µ. Assume u is one-to-one on G and denote F = u(G). +Assume τ > 0, 0 < β ≤ 1 and K given. Assume also that +F ∩ ∆c,ˆq +� +K, β +|k|τ +1 +� += ∅, +∀k ∈ Zn, |k|1 ≤ K, k ̸= 0. +Denote ǫ := ∥DR∥G,ρ,c, η := |R0|G,ρ2 and ξ := +�� ∂R0 +∂I +�� +G,ρ2. +(1) ρ2 ≤ +β +2MKτ+1 +(2) ǫ ≤ min +� +βˆδc +74AKτ , ˜µ2(ρ2−δ2) +4 ˜ +M +� +(3) ξ ≤ min +� +( ˜ +M − M)δ2/R, (µ − ˜µ)ρ2 +� +Then there exists a real canonical transformation +Φ : Dρ− δ +2 (G) → Dρ(G) +and a decomposition H ◦ Φ = ˜ˆh(I) + ˜R(φ, I). Writing ˜u = +∂ +∂I ˜h one has. +(1) |˜u − u|G,ρ2 = ξ, +|˜h − h|G,ρ2 = η, +(2) ˜ǫ := ∥D ˜R∥G,ρ−δ,c ≤ e−Kδ1ǫ + 14AKτ +βˆδc +ǫ2, +(3) ˜η := | ˜R0|G,ρ2− δ2 +2 ≤ 7AKτ +cβ +ǫ2, +(4) |Φ − Id|G,ρ− δ +2 ,c ≤ 2AKτ +β +ǫ, +(5) +�� ∂ +∂I ˜u +�� +G,ρ2 ≤ ˜ +M ′, |˜u|G ≤ ˜L, +(6) | ∂˜u +∂I v| ≥ ˜µ|v| +∀I ∈ G, +(7) Given a subset ˜F ⊂ F − 4Mǫ +˜µ , ˜G(˜u)−1( ˜F) the map ˜u is one-to-one from +˜G to ˜F, ˜G ⊂ G − 2ǫ +˜µ , u( ˜G) ⊃ ˜F − ǫ. Moreover |˜u−1 − u−1| ˜ +F ≤ ǫ/µ. + +68 +6. A NEW KAM THEOREM +Proof. The set u(I) is β/Kτ, K-non-resonant with respect to ω. This implies +that +(6.12) +|k1B(I1)u1 + ¯k¯u + A(I1)u1| ≥ β/Kτ. ≥ +β +|k|τ +1 +≥ β +Kτ . +Then ρ2 ≤ β/Kτ +2MK = +β +2MKτ+1 , ∥DR∥G,ρ,c ≤ β/Kτ ˆδc +74A += +βˆδc +74AKτ . We apply the iterative +lemma (Theorem 6.25) to obtain Φ : Dρ− δ +2 (G) → Dρ(G), such that H ◦ Φ = ˜h + ˜R +where ˜h = h + R0. +We have taken out the points that are not β/Kτ, K-non-resonant with respect +to ω. Because of conditions 1 and 2 we can apply the Iterative lemma. Now let us +prove each of the points in the statement. +(1) We know by definition that ˜u = ∂˜h +∂I = ∂(h+R0) +∂I += ∂h +∂I + R0 +∂I , hence: +|˜u − u|G,ρ2 = |∂h +∂I + ∂R0 +∂I − ∂h +∂I |G,ρ2 = |∂R0 +∂I |G,ρ2 = ξ +˜h = h + R0 ⇒ |˜h − h|G,ρ2 = |h + R0 − h|G,ρ2 = |R0|G,ρ2 = η +(2) By the iterative lemma: +∥D ˜R∥G,ρ−δ,c +≤ +e−Kδ1∥DR∥G,ρ,c + 14A +αˆδc ∥DR∥G,ρ,c +≤ +e−Kδ1ε + 14A +αˆδc ε2 += +e−Kδ1ε + 14AKτ +βˆδc +ε2, +where we have used that α = +β +Kτ . +(3) At this point we use an inequality used in the proof of the iterative Lemma +(theorem 6.25). +| ˜R0|G,ρ2−δ2/2 +≤ +|r2(h, W, 1) + r1(R, W, 1)|G,ρ2−δ2/2 +≤ +7A +αc ∥DR∥2 +G,ρ,c = 7AKτ +β +ε2. +(4) Also using the the iterative Lemma: +|Φ − id|G,ρ−δ/2,c ≤ 2A +α ∥DR∥G,ρ,c = 2AKτ +β +∥DR∥G,ρ,c. +(5) Recall that | ∂ +∂I Aω˜u|G,ρ2−δ2 ≤ ˜ +M, |˜u|G ≤ ˜L, ˜h = h + R0, | ∂ +∂I Aωu|G,ρ2 ≤ +M, |u|G ≤ L. Note that A(I1) ≤ m · maxj(qj)/ minj(cj) and B(I1) ≤ +1/ minj(cj). Hence A(I1) + B(I1) ≤ maxj(qj)/ minj(cj) + 1/ minj(cj) := +R, and we have that |Aω| ≤ R. +| ∂ +∂I Aω˜u|G,ρ2−δ2 += +| ∂ +∂I Aω˜u + ∂ +∂I Aωu − ∂ +∂I Aωu|G,ρ2−δ2 +≤ +| ∂ +∂I Aω(˜u − u)|G,ρ2−δ2 + | ∂ +∂I Aωu|G,ρ2−δ2 +≤ +| ∂ +∂I AωR0|G,ρ2−δ2 + M +≤ +|Aω|G,ρ2|R0|G,ρ +δ2 ++ M +≤ +|Aω|G,ρ2·ξ +δ2 ++ M +≤ +Rξ +δ2 + M +≤ +( ( ˜ +M−M)δ2 +R +)R +δ2 ++ M ≤ ˜ +M − M + M = ˜ +M, +where ξ ≤ ( ˜ +M − M)δ2/R. + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +69 +(6) We know | ∂u +∂I (I)v| ≥ µ|v| for all I ∈ G, then | ∂u +∂I (I)v|G ≥ µ|v|. We want +to find | ∂˜u +∂I (I)v|G ≥ µ′|v| if µ′ < µ. +| ∂˜u +∂I v|G += +|( ∂˜u +∂I + ∂u +∂I − ∂u +∂I )v|G += +|( ∂2R0 +∂I2 + ∂u +∂I )v|G +≥ +−| ∂2R0 +∂I2 v|G + | ∂u +∂I v|G +≥ +µ|v| − | ∂2R0 +∂I2 |G|v| +≥ +µ|v| − | ∂R0 +∂I |G 1 +δ2 |v| +≥ +µ|v| − ξ +ρ2 |v| = (µ − ξ/ρ2)|v| ≥ µ′|v|, +where we have used that | ∂2R0 +∂I2 |G ≤ | ∂R0 +∂I | 1 +ρ2 , and also that µ′ < µ − ξ/ρ2, +hence ξ ≤ (µ − µ′)ρ2. +(7) To apply lemma 6.31 we only need to check that ε ≤ +˜µ2 +˜ +M2 . +˜ +M2 can be +chosen such that | ∂2u +∂I2 |G ≤ ˜ +M2. Note that | ∂2u +∂I2 |G ≤ | ∂2u +∂I2 |G,ρ2−δ2. +| ∂u +∂I |G,ρ2−δ2 ≤ ˜ +M +⇒ +| ∂2u +∂I2 |G,ρ2−δ2(ρ2 − δ2) ≤ | ∂u +∂I |G,ρ2−δ2 ≤ ˜ +M +⇒ +| ∂2u +∂I2 |G,ρ2−δ2 ≤ +˜ +M +ρ2−δ2 = ˜ +M2 +⇒ +| ∂2u +∂I2 |G ≤ ˜ +M2 +Then ε ≤ +˜ +M +4 ˜ +M2 if and only if ε ≤ µ2/(4 +˜ +M +(ρ2−δ2)) if and only if ε ≤ µ2(ρ2−δ2) +4 ˜ +M +which it is assumed in the statement. +□ +6.3. +A KAM theorem on bm-symplectic manifolds +Theorem B ( A bm-KAM theorem). Let G ⊂ Rn, n ≥ 2 be a compact set. Let +H(φ, I) = ˆh(I)+f(φ, I), where ˆh is a bm-function ˆh(I) = h(I)+q0 log(I1)+�m−1 +i=1 +qi +Ii +1 +defined on Dρ(G), with h(I) and f(φ, I) analytic. Let ˆu = ∂ˆh +∂I and u = ∂h +∂I . Assume +| ∂u +∂I |G,ρ2 ≤ M, |u|G ≤ L. Assume that u is µ non-degenerate (| ∂u +∂I v| ≥ µ|v| for some +µ ∈ R+ and I ∈ G. Take a = 16M. Assume that u is one-to-one on G and its range +F = u(G) is a D-set. Let τ > n − 1, γ > 0 and 0 < ν < 1. Let +(1) +(6.13) +ε := ∥f∥G,ρ ≤ +ν2µ2ˆρ2τ+2 +24τ+32L6M 3 γ2, +(2) +(6.14) +γ ≤ min(8LMρ2 +ν ˆρτ+1 , L +K′ ) +(3) +(6.15) +µ ≤ min(2τ+5L2M, 27ρ1L4Kτ+1, βντ+122τ+1ρτ +1), +where ˆρ := min +� +νρ1 +12(τ+2), 1 +� +. Define the set ˆG = ˆGγ := {I ∈ G− 2γ +µ |u(I) is τ, γ, c, ˆq− +Dioph.}. Then, there exists a real continuous map T : W ρ1 +4 (Tn) × ˆG → Dρ(G) an- +alytic with respect the angular variables such that +(1) For all I ∈ ˆG the set T (Tn ×{I}) is an invariant torus of H, its frequency +vector is equal to u(I). + +70 +6. A NEW KAM THEOREM +(2) Writing T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)) with estimates +|Tφ(φ, I)| ≤ 22τ+15ML2 +ν2ˆρ2τ+1 +ε +γ2 +|TI(φ, I))| ≤ 210+τL(1 + M) +ν ˆρτ+1 +ε +γ +(3) meas[(Tn ×G)\T (Tn× ˆG)] ≤ Cγ where C is a really complicated constant +depending on n, µ, D, diamF, M, τ, ρ1, ρ2, K and L. +Proof. This proof, as the one in [DG96] is going to be structured in six +sections. First we define the parameters used in each iteration while building the +diffeomorphism. After that, we prove that we can apply the inductive lemma 6.32 +and we exhibit some bound that hold using the results of the inductive lemma. Next, +we find that the sequence of frequency vectors and the sequence of diffeomorphisms +that we built actually converges. Then we find estimates of the components of the +canonical transformation that we have built. Then we find a way to identify the +invariant tori and finally we give a bound for the measure of the set of invariant +tori. +(1) Choice of parameters +We are going to make iterative use of proposition 6.32. So we need +to properly define all the parameters in the statement for every iteration. +Let: + + + +Mq += +(2 − 1 +2q )M, +Lq += +(2 − 1 +2q )L, +µq += +(1 + 1 +2q ) µ +2 . +Note that Mq, Lq monotonically increase from M to 2M and L to 2L +when q → ∞. On the other hand µq monotonically decreases from µ to +µ/2. Also, let: +� K0 += +0, +Kq += +K · qq−1, q ≥ 1, +where K is the minimum natural number greater or equal than 1/ˆρ +and greater or equal than ( +νβ +µ22τ+12 )1/τ. Moreover, β := γ/L ≤ 1, and + + + + + +ρ(q) += +(ρ(q) +1 , ρ(q) +2 ), +ρ(q) +1 += +(1 + +1 +2νq ) ρ1 +4 , +ρ(q) +2 += +νβ +32MKτ+1 +q+1 . +Notice that ρ(q) +1 +decreases monotonically from ρ1/2 to ρ1/4. Also, ρ(q) +2 +decreases to 0. We also denote: + + + + + + + +δ(q) +1 += +ρ(q−1) +1 +− ρ(q) +1 , +δ(q) +2 += +ρ(q−1) +2 +− ρ(q) +2 , +cq += +δ(q) +2 +δ(q) +1 . + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +71 +Note that +δ(q) +1 += +� +1 + +1 +2ν(q−1 +� ρ1 +4 − +� +1 + +1 +2νq +� ρ1 +4 += +� +1 +2ν(q−1) − +1 +2νq +� ρ1 +4 += +1−1/2ν +2ν(q−1) +ρ1 +4 . +Also, since 0 < ν < 1 then ν/2 ≤ 1 − 1/2ν ≤ ν. Plugging this in the +previous equation we obtain: +(6.16) +νρ1 +2ν(q−1)8 ≤ δ(q) +1 +≤ +νρ1 +2ν(q−1)4. +Also, +δ(q) +2 += +νβ +32MKτ+1 +q +− +νβ +32MKτ+1 +q+1 += +νβ +32M(K2q−1)τ+1 − +νβ +32M(K2q)τ+1 += +νβ +32M(K2q−1)τ+1 +� +1 − +1 +2τ+1 +� +. +Also, since τ > 0 then 1/2 ≤ (1 − 1/2τ+1) ≤ 1. Using this in the +previous equation: +(6.17) +νβ +64MKτ+1 +q +≤ δ(q) +2 +≤ +νβ +32MKτ+1 +q +. +Using equations 6.16 and 6.17 we find bounds for cq + + + + + + + + + + + +cq +≤ +� +νβ +32MKτ+1 +q +� +� +νρ1 +2ν(q−1) +� += +β2ν(q−1) +4MKτ+1 +q +ρ1 , +cq +≥ +� +νβ +64MKτ+1 +q +� +� +νρ1 +2ν(q−1)4 +� = +β2ν(q−1) +16MKτ+1 +q +ρ1 . +Then, we also define +� βq += +(1 − +1 +2νq )β, +β′ +q += +βq+βq+1 +2 +. +Observe that both βq and β′ +q tend to β. Also observe that β′ +q ≥ ν +4β, +because: +β′ +q += +βq+βq+1 +2 += +(1− +1 +2νq )+� +1− +1 +2ν(q+1) +� +2 +β += +� +1 − +� 1+ 1 +2ν +2νq +� +1 +2 +� +β +≥ +� +1 − (1 − 1/2ν) 1 +2 +� +β ≥ ν +4β. +As K is the minimal natural number such that K ≥ 1/ˆρ then K ≤ +2/ˆρ. Hence ˆρ ≤ 2 +K . Also +1 +ˆρτ+1 ≥ +�K +2 +�τ+1 +. +Recall that ˆρ = min( +νρ1 +12(τ+2), 1) and, in particular, ˆρ ≤ νρ1 and ˆρ ≤ 1. + +72 +6. A NEW KAM THEOREM +By definition γ ≤ 8LMρ2 +ν ˆρτ+1 . And because β = γ/L: +βL ≤ 8LMρ2 +ν ˆρτ+1 ≤ 8LMρ2Kτ+1 +ν +. +Because we assumed ε ≤ +ν2µ2 ˆρ2τ+2 +24τ+32L6M3 γ2 then, using that γ = Lβ and +ˆρ ≤ 2/K: +(6.18) +ε ≤ ν2µ2 � 2 +K +�2τ+2 +24τ+32L6M 3 ≤ +ν2µ2β2 +24τ+30L4M 3K2τ+2 . +Also using again the assumption that ε ≤ +ν2µ2 ˆρ2τ+2 +24τ+32L6M3 γ2 we want to +prove that +(6.19) +ε ≤ +ν3ρ1β2 +22τ+22MK2τ+1 . +It is enough to check that: +ν2µ2ˆρ2τ+2L2β2 +24τ+32L6M 3 +≤ +ν3ρ1β2 +22τ+22MK2τ+1 +where we used γ = Lβ. Now observing that ˆρ ≤ νρ1 it suffices to see +ν2µ2ν2τ+2ρ2τ+2 +1 +L2β2 +24τ+32L6M 3 +≤ +ν3ρ1β2 +22τ+22MK2τ+1 , +which simplifies to +µ2ρ2τ+2 +1 +24τ+10L4M 2 ≤ +1 +K2τ+1 . +Using that K ≥ 1/(νρ1) is enough to check that +µ2ρ2τ+1 +1 +ν2τ+2 +22τ+12L4M 2 ≤ (νρ1)2τ+1, +which holds if and only if µ ≤ 2τ+5L2M as we assumed. +(2) Induction +Let us take G0 = G. Now the goal is to construct a decreasing se- +quence of compact sets Gq ⊂ G and a sequence of real analytic canonical +transformations +Φ(q) : Dρ(q)(Gq) → Dρ(q−1)(Gq−1), +q ≥ 1. +Denoting Ψ(q) = Φ1 ◦· · ·◦Φ(q) the transformed Hamiltonian functions +will be noted by H(q) = H ◦ Ψ(q) = ˆh(q)(I) + R(q)(φ, I). Moreover, u(q) = +∂h(q) +∂I +and ˆu(q) = ∂ˆh(q) +∂I . +We are going to show that the following bounds hold for all q ≥ 0: +(a) εq := ∥DR(q)∥Gq,ρ(q),cq+1 ≤ +8ε +νρ12(2τ+2)q , +(b) ηq := |R(q) +0 |Gq,ρ(q) +2 +≤ +ε +2(2τ+3)q and ξq := | ∂R(q) +0 +∂I |Gq,ρ(q) +2 +≤ 4MKτ+1ε +νβ2(τ+2)q , +(c) | ∂2h(q) +∂I2 |Gq,ρ(q) +2 +≤ Mq, +|u(q)| ≤ Lq +∀I ∈ Gq, +(d) u(q) is µq-non-degenerate on Gq, + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +73 +(e) u(q) is one-to-one on Gq, and u(q)(Gq) = Fq where we define: +Fq := (F − βq) \ +� +k∈Zn\{0} +|k|1≤K +∆cq,ˆq(K, βq +|k|τ +1 +) +To prove this we proceed by induction. For q = 0: + + + +G0 = G, +h(0) = h, ˆh(0) = ˆh, +R(0) = f. +Using the definitions from the previous point: +� +ρ(0) +1 += +(1 + 1) ρ1 +4 = ρ1/2, +ρ(0) +2 += +νβ +32MKτ+1 ≤ ρ2 +2 , +where in the last inequality we have used that β ≤ +8Mρ2Kτ+1 +ν +and +hence ρ2 ≥ +βν +8MKτ+1 . +Then, +ε0 = ∥Df∥G,ρ(0),c1 = ∥Df∥G,ρ(1)+δ(1). +Now, let us use that |DΥ|G,ρ−δ,c ≤ 2|Υ|G,ρ,c +ˆδc +while having in mind that +ˆδ(1) +c1 = min(c1δ(1) +1 , δ(1) +2 ). Then, +∥Df∥G,ρ(1),c1 ≤ c1|f|G,ρ(0) +ˆδc1 +≤ |f|G,ρ(0) +δ(1) +1 +≤ |f|G,ρ(0)8 +νρ1 += 8ε +νρ1 +, +where we have used δ(1) +1 +≥ +νρ1 +8·2ν(1−1) = νρ1 +8 . +This proves the first step of +the induction for 2a). +Let us prove now the base case for 2b). On one side η0 = |R(0) +0 |G0,ρ2(0) ≤ +ε +2(2τ+3)0 = ε, which holds because |R(0) +0 |G0,ρ(0) +2 +≤ |R(0)|G,ρ(0) = |f|G,ρ(0) = +ǫ. On the other hand ξ0 = | ∂R(0) +0 +∂I |G0,ρ(0) +2 +≤ | ∂R(0) +0 +∂I |G0,ρ2−ρ2/2 ≤ +1 +ρ2/2∥R0∥G,ρ ≤ +2ε +ρ2 ≤ +ε +ρ(0) +2 , where we used that ρ2(0) ≤ ρ2/2 = ρ2 − ρ2/2. +The base case of 2c) is immediate because | ∂2h(0) +∂I2 |G0,ρ(0) +2 +≤ | ∂2h +∂I2 |G,ρ2 = +M = M0 and also |u(0)|G0 = |u|G ≤ L = L0. +The base case of 2d holds because u(0) = u is µ non-degenerate in +G = G0. +The base case of 2e holds because u(0) = u is one-to-one in G0 = G +by hypothesis. u(0)(G0) = F0 where F0 = (F − β0) \ {∅} = F because +K0 = 0 and β0 = 0. +For q ≥ 1, we assume the statements hold for q − 1 and we prove +it for q. Let us apply proposition 6.32 (Inductive Lemma) to H(q−1) = +hq−1 + Rq−1 with Kq instead of K. + +74 +6. A NEW KAM THEOREM +We have to be careful with the condition F ∩ ∆c,ˆq(k, +β +|k|τ +1 ) = ∅ ∀k ∈ +Zn, |k|1 ≤ Kq, k ̸= 0 and with the definition +Fq−1 := (F − βq−1) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, βq−1 +|k|τ +1 +), +because the resonances have to be removed up to order Kq, not Kq−1. +Let us define +F ′ +q−1 := (F − βq−1) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, β′ +q−1 +|k|τ +1 +) +where we simply replaced βq−1 for β′ +q−1 because ∆c,ˆq(k, βq−1 +|k|τ +1 ) makes +no sense when q = 1, βq−1 = 0. +Accordingly let us define G′ +q−1 := (u(q−1))−1(F ′ +q−1). The conditions in +proposition 6.32 are going to be satisfied with F ′ +q−1, β′ +q−1, Kq, Mq−1, Lq−1, +µq−1,ρ(q−1), δ(q), cq, Mq, Lq, µq replacing F, β, K, M, L, µ, ρ, δ, c, ˜ +M, ˜L, ˜µ. +And also a = 16M ≥ 8Mq. +We are now going to check that 1, 2 and 3 are satisfied so we can +apply proposition 6.32. +– 1 We want to see that ρ(q−1) +2 +≤ +β′ +q−1 +2MqKτ+1 +q +. +By definition ρq−1 +2 += +νβ +32MKτ+1 +q +≤ +4β′ +q−1 +32MKτ+1 +q +≤ +β′ +q−1 +8MqKτ+1 +q +≤ +βq−1 +2MqKτ+1 +q +, where we used that +Mq ≥ M. +– 2 We want to see that εq−1 ≤ min +� +βq−1 ˆρ(q) +c +74AqKτ +q−1 , +µτ +q (ρ(q−1) +2 +−δ(q−1) +c +) +4Mq +� +, +where Aq := 1 + +2Mq−1cqKτ +q +β′ +q−1 +. +Notice that: +Aq +:= +1 + +2Mq−1cqKτ +q +β′ +q−1 +≤ +1 + +8Mq−1cqKτ +q +νβ +≤ +1 + +8Mq−1β2ν(q−1)Kτ +q +4MKτ+1 +q +ρ1νβ += +1 + 2Mq−12ν(q−1) +MKqρ1ν += +1 + 2Mq−12ν(q−1) +MK2q−1ρ1ν +≤ +1 + +4M2q−1 +MK2q−1ρ1ν += +1 + +4 +Kρ1ν ≤ 1 + 4 = 5 +First, we check that εq−1 ≤ +βq−1 ˆρ(q) +c +74AqKτ +q−1 . +By induction hypothesis we know that εq−1 ≤ +8ε +νρ12(2τ+2)(q−1) . Hence +it is enough to see +8ε +νρ12(2τ+2)(q−1) ≤ β′ +q−1δ(q) +2 +75 · 5Kτq +. + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +75 +Notice that +β′ +q−1δ(q) +2 +379Kτq +≥ +νβ +4 +νβ +64MKτ+1 +q +1 +37Kτq += +ν2β2 +4·64·370 +1 +MK2τ+1 +q += +ν2β2 +4·64·370·M2(q−1)(2τ+1)K2τ+1 . +And this holds if the following is true: +8ε +νρ1 +≤ +ν2β2 +4 · 64 · 379 · K2τ+1M ⇔ ε ≤ +ν3β2ρ1 +212135MK2τ+1. +This is true because in the previous section we have seen that ε ≤ +ν3ρ1β2 +22τ+30MK2τ+1 +Let us now prove that ε ≤ +µ2 +q(ρ(q−1) +2 +−δ(q−1) +2 +) +2Mq +. First of all observe that +(ρ(q−1) +2 +− δ(q) +2 ) = ρ(q) +2 . So, what we want to prove is equivalent to +proving εq−1 ≤ +µ2 +qρ(q) +2 +2Mq . +On the other hand, we know that εq−1 ≤ +8ε +νρ12(2τ+2)(q−1) . And observe +also that +µ2 +qρ(q) +2 +2Mq +≥ +(µ/2)2 +νβ +32MKτ+1 +2M += +µ2νβ +28M2Kτ+1 . +If are able to check that +8ε +νρ12(2τ+2)(q−1) ≤ +µνβ +28M2Kτ+1 we would be fine. +The previous equation holds if and only if the following holds, +ε ≤ µν2βρ12(2τ+2)(q−1) +211M 2Kτ+1 +. +If we knew beforehand that ε ≤ µν2βρ12−(2τ+2) +211M2Kτ+1 += +µν2βρ1 +22τ+13M2Kτ+1 we +would be done. +But we also know that +ǫ ≤ +ν2µ2β2 +22τ+30L4M 3K2τ+2 . +Then it is enough to check that +ν2µ2β2 +22τ+30L4M 3K2τ+2 ≤ +µν2βρ1 +22τ+13M 2Kτ+1 . +And this holds because µ ≤ 27ρ1L4Kτ+1 +– 3 Lastly we want to see that +ξq−1 ≤ min((Mq − Mq−1)δ(q) +2 +R , (µq−1 − µq)ρ(q−1) +2 +). +Observe that R does not depend on q because at each iteration ˆh(q) +singular part is not modified. ˆh(q) = ˆh(q) + R(q) +0 +and R0 is analytic +depending only on the action coordinates. By induction hypothesis, +we know that +ξq−1 = |∂R(q) +0 +∂I +|Gq,ρ(q) +2 +≤ 4MKτ+1ε +νβ2(τ+2)q . +We are going to check the two different inequalities separately + +76 +6. A NEW KAM THEOREM +(a) ξq−1 ≤ (Mq − Mq−1) δ(q) +2 +R . Note that Mq = (2 − +1 +2q )M, then +Mq − Mq−1 = M +2q . +δ(q) +2 +≥ +νβ +64M(K2q−1)τ+1 ≥ +νβ +64M(K2q)τ+1 += +νβ +64MKτ+1 +1 +2qτ+q . +We deduce +(Mq − Mq−1)δ(q) +2 +≥ +νβ +64Kτ+1 +1 +2τq+2q . +Hence we only need to check that +4MKτ+1ε +νβ2(τ+2)q ≤ +νβ +64Kτ+1 +1 +2τq+2q . +The previous condition holds if and only if +4MKτ+1ε +νβ +≤ +νβ +26Kτ+1 ⇔ ε ≤ +ν2β2 +2K2τ+2M . +On the other hand, let us use again that ε ≤ +ν2µ2β2 +22τ+30L4M3K2τ+2 . +If we apply the condition µ ≤ 2τ+6L2M in the last expression +we obtain: +ε ≤ ν2β222τ+12L4M 2 +22τ+30L4M 3K2τ+2 = +ν2β2 +28K2τ+2M . +(b) ξq−1 ≤ (µq−1 − µq)ρ(q−1) +2 +. +Observe that +µq = (1 + 1 +2q )µ +2 , +(µq−1 − µq) = ((1 + +1 +2q−1 ) − (1 + 1 +2q ))µ +2 = ( +1 +2q−1 − 1 +2q )µ +2 += +�2 − 1 +2q +� µ +2 = 1 +2q +µ +2 = +µ +2q+1 +Also, +ρ(q−1) +2 += +νβ +32MKτ+1 +q += +νβ +32M(K2q−1)τ+1 +≥ +νβ +32MKτ+12q(τ+1) . +Then, +(µq−1 − µq)ρ(q−1) +2 +≥ +µ +2q+1 +νβ +32MKτ+12q(τ+1) . +Then we only have to check that +4MKτ+1ε +νβ2τq+2q−2 +≤ +µ +2q+1 +νβ +32MKτ+12q(τ+1) += +µ +2τq+2q+1 +νβ +32MKτ+1 . +Which holds if and only if + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +77 +MKτ+1ε +νβ2−2 +≤ µ +2 +νβ +32MKτ+1. +Then, +ε ≤ µ2−2 +2 +ν2β2 +32M 2K2τ+2 = +µν2β2 +28M 2K2τ+2 . +But we know +ε +≤ +ν2µ2β2 +2τ+30L4M3K2τ+2 +≤ +ν2µ2τ+5L4Mβ2 +2τ+30L4M3K2τ+2 += +ν2µβ2 +225M2K2τ+2 +≤ +µν2β2 +28M2K2τ+2 +as we wanted. +In the second inequality we used that µ ≤ +2τ+5L4M. +So, finally, we can apply the inductive lemma 6.32 with the pa- +rameters mentioned previously in this section. Hence we obtain a +canonical transformation Φ(q) and a transformed hamiltonian H(q) = +h(q) + R(q). The new domains Gq ⊂ G′ +q−1 are going to be specified +in the following lines. So now we are going to prove 2a,2b,2c,2d,2e. +– 2a. We want to see εq := ∥DR(q)∥Gq,ρ(q),cq+1 ≤ +8ε +νρ12(2τ+2)q. +By the second result of proposition 6.32 we have: +(6.20) +εq ≤ e−Kqδ(q) +1 εq−1 + 14AqKτ +q +β′ +q−1δ(q) +2 +ε2 +q−1. +Now we are going to bound each term of the right hand of the +expression at a time. +Recall that δ(q) +1 +≥ +νρ1 +82ν(q−1) . +Kqδ(q) +1 +≥ +K2q−1 νρ1 +8 2−ν(q−1) += +νρ1 +8 K2(1−ν)(q−1) +≥ +12(τ+2) +8 +2(1−ν)(q−1) += +(3/2τ + 3)2(1 − ν)(q − 1) +≥ +(2τ + 3) 3 +4 ≥ (2τ + 3) ln 2, +where we used that K ˆρ ≥ 1 and hence K ≥ 12(τ+2) +νρ1 +. So we +conclude that e−Kqδ(q) +1 +≤ +1 +22τ+3 , and we have bounded the first +term of 6.20. Let us bind the second one. +On one hand, we have that +14AqKτ +q +β′ +q−1 +≤ 14 · 5Kτ +q +νβ +4 +≤ 29K2 +q +νβ +where we have used that β′ +q ≥ νβ +4 and Aq ≤ 5. + +78 +6. A NEW KAM THEOREM +Now we are going to apply that εq−1 ≤ +8ε +νρ12(2τ+2)(q−1) , δ(q) +2 +≥ +νβ +64MKτ+1 +q +and ǫ ≤ +ν3ρ1β2 +22τ+22MK2τ+1 to obtain +14AqKτ +q +β′ +q−1δ(q) +2 εq−1 += +14AqKτ +q +β′ +q−1 +1 +δ(q) +2 εq−1 +≤ +29Kτ +q +νβ +64MKτ+1 +q +νβ +8ε +νρ12(2τ+2)(q−1) +≤ +218MK2τ+1 +q +ν3β2ρ12(2τ+2)(q−1) +ν3ρ1β2 +22τ+22MK2τ+1 +≤ +2182(q−1)(2τ+1)−(2τ+2)(q−1)−(2τ+22) += +2(1−q)2−2τ−4 = +1 +22τ+32q−1 . +This gives us the bound of the second term of 6.20. Now we +put both bounds together: +εq ≤ +1 +22τ+3 εq−1 + +1 +22τ+3 +1 +2q−1 εq−1 ≤ +1 +22τ+2 εq−1. +That implies εq ≤ +ǫ +2(2τ+2)(q−1) as we wanted. Because we can +assume νρ1 ≤ 1. +– 2b +Let us write σ(q) +2 += ρ(q−1) +2 +− δ(q) +2 /2 = ρ(q) +2 ++ δ(q) +2 /2 ≥ ρ(q) +2 , then +ηq = |R(q) +0 |Gq,ρ(q) +2 +≤ |R(q) +0 |Gq,σ(q) +2 . +By the inductive lemma 6.32: +ηq +≤ +7AqKτ +q +cqβ′ +q−1 ε2 +q−1 +≤ +7AqKτ +q +β′ +q−1 ε2 +q−1 +δ(q) +1 +δ(q) +2 += +14AqKτ +q +β′ +q−1δ(q) +2 ε2 +q−1 +δ(q) +1 +2 +≤ +1 +22τ+32q−1 εq−1 +δq) +1 +2 +≤ +1 +2 +δ(q) +1 +22τ+32q−1 +ε +νρ12(2τ+2)(q−1) +≤ +1 +2 +νρ1 +4·2ν(q−1) +1 +22τ+32q−1 +8ε +νρ12(2τ+2)(q−1) +≤ +ε +2(2τ+3)q . +For the second part we only need to apply Cauchy inequalities: +ξq ≤ +2 +δ(q) +2 +|Rq +0|Gq,ρ(q) +2 +≤ +2 +δ(q) +2 +ε +2(2τ+3)q . +– 2c and 2d are direct from lemma 6.32. +– 2e We need to consider again the results from lemma 6.32 +with Fq as F. We have to check the condition Fq ⊂ F ′ +q−1 − +4Mq−1εq−1 +µq +. Let us define dq := βq−βq−1 +2Kτ+1 +q +. Using that F ′ +q−1 := +(F − βq−1) \ � +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k. +β′ +q−1 +|k|τ +1 ) we have +F ′ +q−1 − dq ⊃ (F − (βq−1 + dq)) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, β′ +q−1 +|k|τ +1 ++ |k|dq). + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +79 +Moreover, + + + + + + + + + + + +βq−1 + dq +≤ +βq, and +β′ +q−1 +|k|τ +1 + |k|dq += +β′ +q−1+|k|τ +1 |k| +βq−βq−1 +2Kτ+1 +q +|k|τ +1 +≤ +β′ +q−1+Kτ+1 +q +βq−βq−1 +2Kτ+1 +q +|k|τ +1 += +β′ +q−1+ +βq +2 − +βq−1 +2 +|k|τ +1 += +βq +|k|τ +1 . +Now if we see that 4Mq−1εq−1 +µq +≤ dq we will have the inclusion +we want. Observe that 4Mq−1 +µq +≤ 4·2M +µ/2 = 16M +µ . So, it is enough +to check that 16M +µ εq−1 ≤ dq. +εq−1 +≤ +8ε +νρ12(2τ+2)q +≤ +8ν3ρ1β2 +νρ12(2τ+2)q2(2τ+22)MK2τ+1 +≤ +8ν2β2 +2(2τ+2)q+2τ+20MK2τ+1 +≤ +8ν2β +2(βq−βq−1) +ν +2(τ+1)q+(τ+1)q+2τ+20MK2τ+1 += +8νβ2(βq−βq−1) +2(τ+1)+(τ+1)q+2τ+20MKτ+1 +q +Kτ += +8νβ2 +2(τ+1)+(τ+1)q+2τ+19MKτ +(βq−βq−1) +2Kτ+1 +q += +νβ +2(τ+1)+(τ+1)q+2τ+15MKτ dq +≤ +νβ +23τ+16MKτ dq. +Hence, it is enough to prove the following: +16M +µ +νβ +23τ+16MKτ dq ≤ dq. +Wich holds if and only if +16M +µ +νβ +2τ+16MKτ ≤ 1 ⇔ Kτ ≥ +νβ +µ2τ+12 , +which we assumed when choosing K. +(3) Convergence of diffeomorphisms +Now we are going to prove the convergence of the successive maps +u(q) : Gq → Fq +i.e. we want to see that exist proper sets G∗, F ∗ and an analytical +map u∗ such that u(q) : Gq → Fq converge to u∗ : G∗ → F ∗. +Let us use lemma 6.32 as before. +For q ≥ 1 we obtain +|u(q) − u(q−1)|Gq ≤ ξq +and +|(u(q))−1 − (u(q−1))−1|Fq ≤ εq +µq +. +Now, because the following two inequalities hold +� +ξq +≤ +4MKτ+1ε +νβ2(τ+2)q +εq +µq +≤ +8ε +νρ122τ+2q +1 +(1+ 1 +2q ) µ +2 = +8ε2q−1 +νβ2(2τ+2)q(2q+1)µ + +80 +6. A NEW KAM THEOREM +the sequences uq and (u(q))−1 converge to maps u∗ and Υ respectively. +These maps are defined on the following sets: +G∗ +:= +� +q≥0 Gq, +F ∗ +:= +� +q≥0 Fq = (F − β) \ � +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, +β +|k|τ +1 ). +The second equality holds because F ∗ is a compact for being the +intersection of compact sets. We can now deduce that +|u∗ − u(q)|G∗ +≤ +� +s≥q |u(q) − u(q−1)|G∗ +≤ +� +s≥q |u(q) − u(q−1)|G +≤ +� +s≥q ξq. +with the same argument we see that |Υ−(u(q))−1|F ∗ ≤ . . . ≤ � +s≥q +εq +µq . +The next steps are going to be to prove that Gq ⊂ Gq−1 − 2εq−1 +µq−1 and +Fq ⊂ Fq−1 − 4Mq−1εq−1 +µq−1 +. If we check it and take the limit we would have: +G∗ ⊂ Gq − +� +s≥q +2εq +µq +and +F ∗ ⊂ Fq − +� +s≥q +4Mqεq +µq +. +Let us first check Fq ⊂ Fq−1 − 4Mq−1εq−1 +µq−1 +. Let us define x := 4Mq−1 +µq−1 . +Fq−1 − x +⊃ +(F − (βq−1 + x)) \ � +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, βq−1 +|k|τq + |k|x) +⊃ +(F − (βq−1 + x)) \ � +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, +βq−1+Kτ+1 +q +x +|k|τq +). +To have the inclusion we want, we have to check that: +(a) βq−1 + x ≤ βq. +(b) +βq−1+Kτ+1 +q +x +|k|τ +1 +≤ +βq +|k|τ +1 ⇔ βq−1 + Kτ+1 +q +x. +Since the second one implies the first we will only check the second +one. +βq−1 + Kτ+1 +q +x += +βq−1 + Kτ+1 +q +4Mq−1εq−1 +µq−1 +≤ +βq−1 + Kτ+1 +q +16Mεq−1 +µ +≤ +βq−1 + Kτ+1 +q +dq += +βq−1 + Kτ+1 +q +βq−βq−1 +2Kτ+1 +q += +βq−1 − βq−1/2 + βq/2 += +βq−1+βq +2 += +βq +Where we have used that 16Mεq/µ ≤ dq and that βq is monotonically +increasing with q. +The inclusion Gq ⊂ Gq−1 − 2εq−1 +µq−1 is given as a result of the lemma +6.32. +So we proved what we wanted. We are now going to see that u∗ is +one-to-one on G∗ and that u∗(G∗) = F ∗. + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +81 +Tale I ∈ G∗, we have that u(q)(I) ∈ Fq for every q. Hence u∗(I) ∈ F ∗, +and we deduce that u∗(G∗) ⊂ F ∗. +With the same argument, we see +Υ(F ∗) ⊂ G∗. Let us prove that Υ(u∗(I)) = I. +|Υ(u∗(I)) − I| +≤ +|Υ(u∗(I)) − (u(q))−1(u∗(I)) ++(u(q))−1(u∗(I)) − (u(q))−1(u(q)(I))| +≤ +|Υ(u∗(I)) − (u(q))−1(u∗(I))| ++|(u(q))−1(u∗(I)) − (u(q))−1(u(q)(I))| +≤ +|Υ − (u(q))−1|F ∗ + +1 +µq |u∗ − u(q)|G∗. +Where to bound the second term we used the mean value theorem, +i.e. |u(q)(x) − u(q)(y)|Gq ≤ | ∂ +∂I u(q)|Gq|x − y|, and the fact that because +of the µq-non-degeneracy, | ∂u(q) +∂I | ≥ µq|v|, ∀v ∈ Rn and ∀I′ ∈ Gq. Note +that we can use the mean value theorem because u∗(I) − u(q)(I) belongs +to Fq because 4Mqεq +µq +≥ ξq. Let us prove this inequality. If we want to see +4Mqεq +µq +≥ ξq, it is enough to see 4Mεq +µ +≥ ξq. +ξq +≤ +2 +δ(q) +2 |R(q) +0 |Gq,σ(q) +2 +≤ +2 +δ(q) +2 +δ(q) +1 +εq−1 +2 +1 +22τ+32q−1 += +1 +cq +1 +22τ+22q−1 εq−1 ≤ 4M +µ εq−1 +The last inequality holds true if and only if +µ +≤ +β2ν(q−1)22τ+32q−1 +Kτ+1 +q +ρ14 += +β2ν(q−1)22τ+32q−1 +Kτ+12(τ+1)(q−1)ρ14 +≤ +β22τ+3 +Kτ+1ρ14 += +β22τ+1 +Kτ+1ρ1 +≤ +β22τ+1 +( +1 +νρ1 )τ+1ρ1 = βντ+122τ+1ρt +1au +as we assumed in the statement of the theorem. +Since the bound +obtained tends to 0, we have Υ(u∗(I)) = I and hence u∗ is one-to-one. +Analogously we obtain u∗(Υ(J)) = J +∀J ∈ F ∗. Finally, u∗ is one-to-one +and u∗(G∗) = F ∗. Note also that from the inductive lemma we obtain +|h(q) − h(q−1)|Gq,ρ(q−1) +2 +≤ ηq−1. Also, observe the following bound that we +are going to use in the next sections. +|u∗ − u(q)|G∗ ≤ +� +s≥q +4MKτ+1ε +νβ2(τ+2)s . +(4) Convergence of the canonical transformations +Let σ(q) = ρ(q−1) − δ(q) +2 /2. Observe that this definition implies that +σ(q) − ρ(q) = δ(q) +2 +and σ(q) − δ(q) +2 += ρ(q). +Observe that applying the +inductive lemma 6.32: +|Φ(q) − id|Gq,σ(q),cq +≤ +2Aq−1Kτ +q +β′ +q−1 +εq−1 + +82 +6. A NEW KAM THEOREM +≤ +2·5·4 +νβ +8ε +νρ12(2τ+2)(q−1) +≤ +29Kτ ε +ν2ρ1β2(τ+2)(q−1) +≤ +29Kτν3ρ1β2 +ν2ρ1β2(τ+2)(q−1)22τ+22MK2τ+1 +≤ +29νβ +2(τ+2)(q−1)22τ+20MKτ+1 += +νβ +26M(K2q−1)τ+1 +29 +2(q−1)22τ+14 +≤ +δ(q) +2 +1 +2(q−1)22τ+5 +≤ +δ(q) +2 +2(q−1)32, +where we have used that δ(q) +2 +≥ +νβ +84MKqP τ+1, ε ≤ +ν3ρ1β2 +22τ+20MK2τ+1 , β ≤ +8MKτ+1ρ2 +ν +and β′ +q−1 ≥ νβ +4 . +Now, recall that ˆδc = min(cδ1, δ2), then ˆδcq = min(cqδ(q) +1 , δ(q) +2 ) = +min(δ(q) +2 , δ(q) +2 ) = δ(q) +2 . +Now using that |DΥ|G,ρ−δ,c ≤ |Υ|G,ρ,c +ˆδc +, we can obtain: +|DΦ(q) − Id|Gq,ρ(q),cq += +|D(Φ(q)) − id|Gq,ρ(q),cq +≤ +|D(Φ(q)) − id|Gq,σ(q)−δ(q) +2 +,cq +≤ +|Φ(q)−id|Gq,σ(q),cq +ˆδcq +≤ +|Φ(q)−id|Gq,σ(q),cq +δ(q) +2 +≤ +2|Φ(q)−id|Gq,σ(q),cq +δ(q) +2 +≤ +2 +δ(q) +2 +δ(q) +2 +2(q−1)·32 ≤ +1 +2q−116 ≤ +1 +2(q−1)4 +Let x, y be such that the segment joining them is contained in Dρ(q)(Gq). +Using the mean value theorem one can deduce the following bound: +|Φq(x) − Φq(y)|cq ≤ |DΦ(q)|Gq,ρ(q),cq · |x − y|cq. +By 22, in particular |Φ(q)(x)−x|cq ≤ δq +2 and |Φ(q)(y)−y|cq ≤ δq +2. Then +the segment that join Φ(q)(x) and Φ(q)(y) is contained in Dρ(q−1)(Gq−1) = +Dρ(q)+δ(q), because Gq ⊂ Gq−1 − 2εq−1 +µq−1 and because ρ(q) − ρ(q−1) ≤ δ(q) +2 +because ρ(q) − ρ(q−1) = δ(q) +2 . +Therefore we can apply the mean value theorem once again: +|Φ(q−1)(Φ(q)(x)) − Φ(q−1)(Φ(q)(y))|cq−1 +≤ |DΦ(q−1)|Gq−1,ρq−1,cq−1|Φ(q)(x) − Φ(q)(y)|cq−1 +≤ 2τ+1−ν|DΦ(q−1)|Gq−1,ρq−1,cq−1|Φ(q)(x) − Φ(q)(y)|cq, +where we have used that cq−1/cq = +δ(q−1) +2 +/δ(q−1) +1 +δ(q) +2 +/δ(q) +1 += +δ(q−1) +2 +δ(q) +2 +δ(q) +1 +δ(q−1) +1 += +2τ+1 1 +2ν = 2τ+1−ν. +Using the previous bounds and iterating by q, we obtain the following: +|Ψ(q)(x) − Ψ(q)(y)|c1 + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +83 +≤ 2(τ+1−ν)(q−1)|DΦ(1)|G1,ρ(1),c1 · . . . · |DΦ(q)|Gq,ρ(q),cq|x − y|cq +≤ 2(τ+1−ν)(q−1)(1 + 1 +4)(1 + +1 +4·2) · . . . · (1 + +1 +4·2q−1 )|x − y|cq +≤ 2(τ+1−ν)(q−1)e1/2|x − y|cq ≤ 2(τ+1−ν)(q−1) · 2|x − y|cq. +Which holds for q ≥ 1 and for every x, y such that the segment joining +them is contained in Dρ(q)(Gq). Now, given q ≥ 2 and x ∈ Dρ(q)(Gq) let +y = Φ(q)(x): +|Ψ(q)(x) − Ψ(q−1)(x)|c1 += +|Ψ(q−1)(Φ(q)(x)) − Ψ(q−1)(x)|c1 +≤ +2(τ+1+ν)(q−2)2|Φ(q)(x) − x|cq−1 +≤ +2(τ+1+ν)(q−1)2|Φ(q)(x) − x|cq +≤ +2(τ+1+ν)(q−1)2δ(q) +2 +≤ +2(τ+1+ν)(q−1)2 +28Kτε +ν2ρ1β2(τ+2)(q−1) += +29Kτε +ν2ρ1β2(1+ν)(q−1) . +Which holds even for q = 1 by setting Ψ(0) = id by 22. Hence 25 +implies that Ψ(q) converges to a map +Ψ∗ : D(ρ1/4,0)(G∗) = W ρ1 +4 (Tn) × G∗ → Dρ(G). +And we deduce for every q ≥ 0 that +|Ψ∗ − Ψ(q)|G∗,( ρ1 +4 ,0),c1 ≤ +210Kτε +ν2ρ1β2(1+ν)q . +Moreover by taking the limit to the equation +H ◦ Ψ(q) = h(q) + R(q) +we see that H ◦ Ψ∗ = h∗(I) on D( ρ1 +4 ,0)(G∗). +(5) Stability estimates +Next we see that for q → ∞, the motions associated to the trans- +formed hamiltonian ˆH(q) = ˆh(q) + R(q) and the quasi-periodic motions of +ˆh(q) get closer and closer. +Let us denote +� x(q)(t) = (φ(q)(t), I(q)(t)) +the trajectory of H(q), +ˆx(q)(t) = (ˆφ(q)(t), ˆI(q)(t)) +the trajectory of ˆH(q) +corresponding to a given initial condition x(q)(0) = x∗ +0 = (φ∗ +0, I∗ +0) ∈ +Tn × Gq. Let +� +˜x(q)(t) +:= +(˜φ(q)(t), I∗ +0) = (φ∗ +0 + u(q)(I∗ +0))t, I∗ +0 , +ˆ˜x(q)(t) +:= +(ˆ˜φ(q)(t), I∗ +0) = (φ∗ +0 + u′(q)(I∗ +0))t, I∗ +0 +the corresponding trajectories of the integrable parts of h(q) and ˜h(q) +respectively. Recall that ˆh(q)(I) = h(q)(I)+ζ(q)(I1) = h(q)(I)+q0 log(I1)+ +�m−1 +i=1 qi 1 +Ii +1 and u′(q) = ¯Bu(q) + ¯ +A(I1). It is clear that ˜x(q)(t) and ˆ˜x(q)(t) +are defined for all t ∈ R. +Let us denote: + +84 +6. A NEW KAM THEOREM +Tq = inf{t > 0 : |I(q)(t)−I∗ +0| > δ(q+1) +2 +or |φ(q)(t)−˜φ(q)(t)|∞ > δ(q+1) +1 +}. +ˆTq = inf{t > 0 : |ˆI(q)(t) − I∗ +0| > δ(q+1) +2 +or |ˆφ(q)(t) − ˆ˜φ(q)(t)|∞ > δ(q+1) +1 +}. +Observe that x(q)(t) and ˆx(q)(t) are defined and belong do Dρ(q)(Gq), +for 0 ≤ t ≤ Tq and 0 ≤ t ≤ ˆTq respectively, because δ(q) ≤ ρ(q). Also +recall the Hamiltonian equations. Let us first state the motion equations +for our Hamiltonian function ˆH(q): +ιX ˆ +H(q) ω = d ˆH(q), +or +X ˆ +H(q) = Π(d ˆH(q), ·). +Let us write +X ˆ +H(q) = ˙ˆI(q) +1 +∂ +∂I1 ++ . . . ˙ˆI(q) +n +∂ +∂In ++ ˙ˆφ(q) +1 +∂ +∂φ1 ++ . . . + ˙ˆφ(q) +n +∂ +∂φn +. +Moreover +d ˆH(q) += +dˆh(q) + dR(q) += +dζ(q) + dh(q) + dR(q) += +�n +i=1 +∂ζ(q) +∂Ii + +n +� +i=1 +∂ζ(q) +∂φi +� +�� +� +=0 ++ �n +i=1 +∂h(q) +∂Ii ++ +n +� +i=1 +∂h(q) +∂φi +� +�� +� +=0 ++ �n +i=1 +∂R(q) +∂Ii ++ �n +i=1 +∂R(q) +∂φi . +Recall +ω = + + +m +� +j=1 +cj +Ij +1 + + dI1 ∧ dφ1 + +n +� +i=2 +dIi ∧ dφi, +Π = +1 +��m +j=1 +cj +IJ +1 +� ∂ +∂I1 +∧ +∂ +∂φ1 ++ +n +� +i=2 +∂ +∂Ii +∧ +∂ +∂φi +. +Then: + + + + + +˙ˆI(q) +j += +− ∂R(q) +∂φj (ˆx(q)(t)), +if j ̸= 1 and +˙ˆI(q) +1 += +− +1 +��m +i=1 +ci +Ii +1 +� ∂R(q) +∂φj (ˆx(q)(t)) = −B(I1) ∂R(q) +∂φ1 (ˆx(q)(t)). +Observe that +(6.21) +| ˙ˆI(q) +1 (t)| ≤ +���� +∂R(q) +∂φ1 +(ˆx(q)(t)) +���� . +Moreover, + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +85 + + + + + + + + + + + + + +˙ˆφ(q) +j += +ˆu(q) +j (ˆI(q)(t)) + ∂R(q) +∂Ij (ˆx(q)(t)) += +u(q) +j (ˆI(q)) + ∂R(q) +∂Ij (ˆx(q)(t)) +if j ̸= 1 +˙ˆφ(q) +1 += +(B(I1)u(q) +1 ++ A(I1)) +� +�� +� +u(q) +1 +(ˆI(q)(t)) + B(I1) ∂R(q) +∂I1 (ˆx(q)(t)), +where we have used that ˆu(q) +j += u′(q) +j +if j ̸= 1. Using 6.21 we obtain +| ˙ˆI(q)(t)| ≤ +���� +∂R(q) +∂φ +���� +Gq,ρ(q) +≤ εq. +Hence, +| ˙ˆφ(q) − u′(q)(I∗ +0 )|∞ += +|u′(q)(ˆI(q)(t)) + ¯B ∂R(q) +∂I1 (ˆx(q)(t)) − u′(q)(I∗ +0)|∞ +≤ +|u′(q)(ˆI(q))(ˆI(q)(t)) − u′(q)(I∗ +0)|∞ + | ∂R(q) +∂I (ˆx(q)(t))|∞ +≤ +M ′ +q|ˆI(q)(t) − I∗ +0| + ∥ ∂R(q) +∂I ∥Fq,ρ(q),∞ +≤ +Mq|ˆI(q)(t) − I∗ +0| + +εq +cq+1 +≤ +2Mδ(q+1) +2 ++ +εq +cq+1 ≤ 3Mδ(q+1) +2 +. +Where in the last bound we used that +(6.22) +εq +cq+1 +≤ Mδ(q+1) +2 +, +that holds because: +εq +cq+1 +≤ +16MKτ+1 +q+1 ρ1 +β2νq +8ε +νρ12(2τ+2)q +≤ +16MKτ+1 +q+1 ρ1 +βeνq +8 +νρ12(2τ+2)q +ν2µ2β2 +2τ+30L4M2K2τ+2 +≤ +27Kτ+1 +q+1 νµ2β +2(2τ+2)q+νq+τ+30L4M2K2τ+2 +≤ +νβµ2 +Kτ+1 +q+1 2νq+τ+23L4M2 += +µ2 +2νq+τ+17L4M +νβ +26MKτ+1 +q+1 +≤ +µ2 +2τ+17+νqL4M δ(q+1) +2 +≤ +22τ+12L4M2 +2τ+17+νqL4M δ(q+1) +2 +≤ +2τ−5−νqδ(q+1) +2 +≤ +2τ +25+νq Mδ(q+1) +2 +≤ +Mδ(q+1) +2 +if q is large enough. +Thus, since one of the inequalities defining ˆTq has to be an equality +for t = Tq we obtain, +δ(q+1) +2 += +|ˆI(q)(Tq) − I∗ +0| ≤ Tqεq, +or +δ(q+1) +1 += +|ˆφ(q)(Tq) − ˆ˜φ(q)(T1)|∞ ≤ Tq3Mδ(q+1) +2 +. +Hence, ˆTq ≥ min( δ(q+1) +2 +εq +, +δ(q+1) +1 +3Mδ(q+1) +2 +) ≥ +1 +3Mcq+1 , where we used again +6.22. + +86 +6. A NEW KAM THEOREM +Let us denote T ′ +q := +1 +3Mcq+1 , then ˆTq ≥ T ′ +q. This implies +|ˆx(q)(t) − ˆ˜x(q)(t)|cq+1 ≤ δ(q+1) +2 +for |t| ≤ T ′ +q. +Since ˆH(q) = ˆH◦Ψ(q) and Ψ(q) is canonical it turns out that Ψ(q)(ˆx(q)(t)) +is a trajectory of ˆH defined for t ≤ T ′ +q. It is important to observe that +for q big enough this trajectory remains near the torus Ψ(q)(Tn × {I∗ +0}). +Moreover T ′ +q tends to infinity when q → ∞. +(6) Invariant tori +Assume now that x∗ +0 ∈ Tn × G∗ and let us write +� x∗(t) += +(φ∗ +0 + u∗(I∗ +0 )t, I∗ +0) +ˆx∗(t) += +(φ∗ +0 + u′∗(I∗ +0)t, I∗ +0) +for t ∈ R. +Note that +|ˆ˜x(q)(t) − ˆx∗(t)|cq+1 +≤ +cq+1|u′(q)(I∗ +0 ) − u′∗(I∗ +0 )|∞|t| +≤ +cq+1|u′(q) − u′∗|G∗,∞|t|. +And observe that if |t| ≤ +δ(q+1) +1 +|u′(q)−u′∗|G∗,∞ =: T ′′ +q then, +|ˆ˜x(q)(t) − ˆx∗(t)|cq+1 +≤ +cq+1|u′(q) − u′∗|G∗,∞ +δ(q+1) +1 +|u′(q)−u′∗|G∗,∞ +≤ +δq+1 +2 +δq+1 +1 +δq+1 +1 += δq+1 +2 +. +Observe that +|u′∗ − u′(q)|G∗ = | ¯Bu∗ + ¯ +A − ¯Bu(q) − ¯ +A|G∗ += |B(u∗ − u(q))|G∗ ≤ |u∗ − u(q)|G∗, +close enough to Z. +Hence the bound obtained for |u∗−u(q)|G∗ also holds for |u′∗−u′(q)|G∗. +|u′∗ − u′(q)|G∗ ≤ +� +s≥q +4MKτ+1ε +νβ2(τ+2)s ≤ 8MKτ+1ε +νβ2(τ+2)q . +Using this bound, we see that T ′′ +q tends to infinity because +T ′′ +q ≥ +� νρ1 +8 · 2νq +� � νβ2(τ+2)q +8MKτ+1ε +� += +ν2βρ1 +64MKτ+1ε2(τ+2−ν)q. +Then +|ˆx(q)(t) − ˆx∗(t)|cq+1 ≤ |ˆx(q)(t) − ˆ˜x(q)(t)|cq+1 + |ˆ˜x(q)(t) − ˆx∗(t)|cq+1 ≤ 2δ(q+1) +2 +. +when t ≤ T ′′′ +q := min(T ′ +q, T ′′ +q ). +Next, we see that the trajectory Ψ(q)(x(q)(t)) is very close to Ψ∗(x∗(t)) +for large values of q. This is true because when |t| ≤ T ′′′ +q . +|Ψ(q)(ˆx(q) − Ψ∗(ˆx∗(t)))|c1 +≤ |Ψ(q)(ˆx(q)(t)) − Ψ(q)(ˆx∗(t))|c1 + |Ψ(q)(ˆx∗(t)) − Ψ∗(ˆx∗(t))|c1 +≤ 2(τ+1−ν)(q−1) · 2|ˆx(q)(t) − ˆx∗(t)|cq + |Ψ(q) − Ψ∗|G∗,(ρ1/4,0),c1 + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +87 +≤ 2(τ+1−ν)(q−1) · 4δ(q+1) +2 ++ |Ψ(q) − Ψ∗|G∗,(ρ1/4,0),c1 +≤ 2(τ+1−ν)(q−1) · 4δ(q+1) +2 ++ +210Kτε +ν2ρ1β2(1+ν)q +≤ +c1 +cq+1 +4δ(q+1) +2 +2(τ+1−ν) + +210Kτε +ν2ρ1β2(1+ν)q +≤ +c14 +2(τ+1−ν) +δ(q+1) +1 +δ(q+1) +2 +δ(q+1) +2 ++ +210Kτ ε +ν2ρ1β2(1+ν)q +≤ +c14 +2(τ+1−ν) δ(q+1) +1 ++ +210Kτ ε +ν2ρ1β2(1+ν)q +where we used that cq−1/cq = 2τ+1−ν then c1/cq+1 = 2(τ+1−ν)q. +The bound 28 tends to zero. So we deduce, for every fixed t, Ψ(q)(ˆx(q)(t)) +exits or q large enough and its limit is Φ∗(ˆx∗(t)). This fact and the con- +tinuity of the flow of ˆH imply that Ψ∗(ˆx∗(t)) is also a trajectory of ˆH, +which is defined for all t ∈ R. +This holds for every initial condition x∗ +0 = (φ∗ +0, I∗ +0) ∈ Tn × G∗ for this +reason Ψ∗(Tn × {I∗ +0}) is an invariant torus of ˆH, with frequency vector +u′∗(I∗ +0). Observe that the energy on the torus is ˆH(Ψ∗(φ∗ +0, I∗ +0)) = h∗(I∗ +0). +The preserved invariant tori are completely determined by the trans- +formed actions I∗ +0 ∈ G∗. We are now going to characterize the preserved +tori by the original action coordinates. +First, let us see that u( ˆG) ⊂ F ∗. Recall that: +∆c,ˆq(k, α) = {J ∈ R such that |k ¯Bu(I) + k ¯ +A| < α}, +ˆG = {I ∈ G − 2γ +µ such that |k ¯Bu(I) + k ¯ +A| < +β +|k|τ +1 +}. +With this definition is obvious that if I ∈ ˆG then u(I) is +β +|k|τ +1 , K, c, ˆq- +non-resonant. Hence u(I) /∈ ∆c,ˆq(k, +β +|k|τ +1 ) for all k ̸= 0. Then u( ˆG) ⊂ F ∗. +We want to find a correspondence between the invariant tori of ˆh and +the invariant tori of the perturbed system ˆH = ˆh + R, or in the new +coordinates ˆh∗. +Recall +u′ = ¯Bu + ¯ +A, +u′∗ = ¯Bu∗ + ¯ +A = ( +1 +�m +i=1 +ci +Ii +1 +u∗ +1 + +�m +i=1 +ˆqi +Ii +1 +�m +i=1 +ci +Ii +1 +, u∗ +2, . . . , u∗ +n). +Observe u′∗(0, I2, . . . , In) = ˆqm +cm = +1 +K′ the inverse of the modular period, +hence u′∗ and u′ are not one-to-one at Z because they project the first +component of u∗ and u to +1 +K′ . +Let us define I∗ +0 = (u∗)−1(u(I0)), recall that u and u∗ are indeed +one-to-one even though u′ and u′∗ are not, so I∗ +0 is properly defined. +With this definition u∗(I∗ +0) = u(I0) and this implies u′∗(I∗ +0 ) = u′(I0). +Now, let us define T (φ0, I0) = Ψ∗(φ0, I∗ +0). +We obtain 6.13 because the set T (Tn × {I0}) is an invariant torus +of the hamiltonian flow of ˆH with frequency vector u′∗(I∗ +0 ) because Tn × +{I∗ +0} is an invariant torus for the hamiltonian flow of ˆh∗. And we have +seen that u′∗(I∗ +0) = u′(I0). In a nutshell, the original frequencies (of the +unperturbed system) u(I0) for I0 ∈ ˆG are in F ∗ and hence can be seen + +88 +6. A NEW KAM THEOREM +Dρ(G) = Wρ1(Tn × Vρ1(G)) +Wρ1/4(Tn) × G∗ +Wρ1/4(Tn) × G +Wρ1/4(Tn) × ˆG +ˆG +u( ˆG) ⊂ F +F ∗ +Ψ∗ +u∗ +i +i +π +u| ˆ +G +T +i +(u∗)−1 +Figure 2. Diagram of the different maps and sets used in the proof. +as frequencies of the unperturbed system in the new coordinates u∗(I∗ +0). +Hence we can conclude that for this I0 ∈ ˆG its new (perturbed) solution +is also linear in a torus (φ0 + u′∗t, I∗ +0) ∈ Ψ∗(Tn × {I∗ +0}) = T (Tn × {I0}). +And the new frequency vector u′∗ is such that u′∗ = u′. +Let us now prove 6.14. Let us write, for (φ0, I∗ +0) ∈ W ρ1 +4 (Tn) × G∗. +Ψ∗(φ0, I∗ +0) = (φ0 + Ψ∗ +φ(φ0, I∗ +0), I∗ +0 + Ψ∗ +I(φ0, I∗ +0 )). +And for (φ0, I0) ∈ W ρ1 +4 (Tn)× ˆ +G. +T (φ0, I0) = (φ0 + Tφ(φ0, I0), I0 + TI(φ0, I0)). +Then, for (φ0, I0) ∈ W ρ1 +4 (Tn)× ˆ +G: +Tφ(φ0, I0) = Ψ∗ +φ(φ0, I∗ +0), +and +TI(φ0, I0) = Ψ∗ +I(φ0, I∗ +0) + I0 − I∗ +0 . +Let us bound the norms of these terms: +|Ψ∗ +φ(φ0, I∗ +0)|∞ +≤ +1 +c1 |Ψ∗ − id|G∗,( ρ1 +4 ,0),c1 +≤ +16MKτ+1ρ1 +β +210Kτ ε +ν2ρ1β +≤ +214MK2τ+1ε +ν2β2 +, +where we used that c1 ≥ +β +16MKτ+1ρ1 . Then, +Ψ∗ +I(φ0, I∗ +o) +≤ +|Φ∗ − id|G∗,( ρ1 +4 ,0,c1) +≤ +210kτ ε +ν2ρ1β . +Now it only remains the term I∗ +0 − I0: +|I∗ +0 − I0| ≤ |(u∗)(−1) − (u)(−1)|F ∗ ≤ +� +s≥0 +ξs + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +89 +≤ +� +s≥0 +4MKτ+1ε +νβ2(τ+2)s ≤ 8MKτ+1ε +νβ2(τ+2) . +Let us put everything together and use ˆρ ≤ νρ1, K ≤ 2/ˆρ and β = +γ/L. +|Ψ∗ +φ(φ0, I∗ +0)|∞ +≤ +214M( 2 +ˆ +ρ )2τ+1ε +ν2( γ +L )2 +≤ +22τ+15ML2 +ν2 ˆρ2τ+1 +ε +γ2 +|Ψ∗(φ0, I∗ +0)| + |I∗ +0 − I0| +≤ +210( 2 +ˆ +ρ )τε +ν ˆρ( γ +L ) ++ +8M( 2 +ˆ +ρ )τ+1ε +ν( γ +L )2(τ+2) += +210+τ Lε +ν ˆρτ+1γ + +8M2τ+1Lε +ν ˆρτ+1γ2(τ+2) +≤ +220+τ Lε+M2τ+4Lε +ν ˆρτ+1γ +≤ 210+τL(1+M) +ν ˆρτ+1 +ε +γ +(7) Estimate of the measure +Finally, we carry out the estimate of part 3. Let us write +ˆG∗ = (u∗)−1(u( ˆG)). +The invariant tori fill the set +T (Tn × ˆG) = Ψ∗(Tn × ˆG∗) +i.e. all the tori inside T (Tn × ˆG) are invariant although there are more of +them. Because Ψ(q) are hamiltonian transformations, in particular, they +preserve the volume: +meas[Ψ(q)(Tn × ˆG∗)] = meas(Tn × ˆG∗) = (2π)nmeas( ˆG∗). +Now, let us consider the measure of the limit: +meas[Ψ∗(Tn × ˆG∗)]. +To do this we use the superior limit of sets: +∞ +� +n=q +∞ +� +j=q +(Ψ(j)(Tn × ˆG∗)). +Because Ψ(j)(Tn × ˆG∗) are compact and we have the bound +|Ψ∗ − Ψ(q)|G∗,( ρ1 +4 ,0),c1 ≤ +210Kτε +ν2ρ1β2(1+ν)q , +�∞ +j=q(Ψ(j)(Tn × ˆG∗)) is also compact. All the measures are well-defined +and we can say that +meas[Ψ∗(Tn × ˆG∗)] ≥ (2π)nmeas( ˆG∗). +Then, to bound the measure of the complement of the invariant set it +is enough to bound the measure of G \ ˆG∗. +But first, we are going to define some auxiliary sets. Let ˜β = 2γM +µ , +˜βq = (1− +1 +2νq )˜β. Note that ˜β ≥ β if and only if µ ≤ 2ML and we assumed +µ ≤ 2τ+6L2M. +Then, for q ≥ 0 we define + +90 +6. A NEW KAM THEOREM +˜Fq = (F − ˜βq) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, +˜βq +|k|τ +1 +), +˜Gq = (u(q))−1( ˜Fq) +and +˜F ∗ = +� +q≥0 +˜Fq = (F − ˜β) \ +� +k∈Zn\{0} +|k|1≤K +∆c,ˆq(k, +˜β +|k|τ +1 +), +˜G∗ = +� +q≥0 +˜Gq. +In order to prove the bounds, we need to prove previously the inclu- +sions ˜G∗ ⊂ ˆG∗ and ˜G0 ⊂ G. +(a) G ⊃ ˜G0 = (u(0))−1( ˜F0) = (u)−1(F − ˜β), but we know u(G) = F. +(b) ˜G∗ ⊂ ˆG∗. Take I ∈ ˜G∗, then I ∈ ˜Gq∀q ≥ 0. Hence ∃J ∈˜˜Fq∀q such +that u(q)(I). Then ∃J ∈ ˜F ∗ such that u∗(J) = I. +If we check that J ∈ u( ˜G) we obtain (u∗)−1(J) = I ∈ ˆG∗ and we will +be done. We want ˜F ∗ ⊂ u( ˆG). Because we are taking out all the +resonances in ˜F ∗ it is enough to see (F − ˜β) ⊂ u(G − 2γ +µ ). We only +need to use that | ∂u +∂I |G,ρ2 ≤ M. Then F − ˜β ⊂ u(G − 2γ +µ ). This holds +if and only if +˜β +M ≤ 2γ +µ which is true because ˆβ ≤ 2γM +µ . +Then, we proceed as follows +meas(G \ ˆG∗) +≤ +meas(G \ ˜G∗) +≤ +meas( ˜G0 \ ˜G∗) +≤ +�∞ +q=1 meas( ˜Gq−1 \ ˜Gq). +For q ≥ 1 we obtain the following estimate: +meas( ˜Gq−1 \ ˜Gq) ≤ +1 +| det( ∂u(q−1) +∂I +(I))| +meas( +u(q−1)( ˜ +Gq−1) +� �� � +˜Fq−1 +\( +u(q−1)( ˜ +Gq) +� +�� +� +˜Fq − εq−1)). +Where we have used lemma 6.32. Also det( ∂u(q−1) +∂I +(I)) ≥ µn +q−1 because +of the µq−1-non-degeneracy condition all the eigenvalues have to be greater +than µq−1. +meas( ˜Gq−1 \ ˜Gq) +≤ +1 +µn +q−1 meas( ˜Fq−1 \ ( ˜Fq − εq−1)) +≤ +2n +µn meas( ˜Fq−1 \ ( ˜Fq − εq−1)). +Now, we are going to apply lemma 6.30 with +˜Fq−1 = F(˜βq−1, ˜βq−1, Kq − 1) +and ˜Fq = F(˜βq, ˜βq, Kq). +Applying the lemma: + +6.3. +A KAM THEOREM ON bm-SYMPLECTIC MANIFOLDS +91 +meas( ˜Fq−1 \ ˜Fq) ≤ D(˜βq − ˜βq−1) ++2(diamF)n−1 + + + + +� +k∈Zn\{0} +|k|1≤Kq−1 +˜βq − ˜βq−1 +|k|τ +1|k|2,ω ++ +� +k∈Zn\{0} +Kq−1≤|k|1≤Kq +˜βq +|k|τ +1|k|2,ω + + + + +and +meas( ˜Fq \ ( ˜Fq − εq)) ≤ (D + 2n+1(diamF)n−1Kn)εq. +Putting everything together (and using that ˜β0 = 0), we get +(6.23) +meas(G \ ˆG∗) +≤ +2n +µn + +D ˜β + 2(diamF)n−1 +� +k∈Zn\{0} +˜β +|k|τ +1|k|2,ω ++D +∞ +� +q=1 +εq−1 + 2n+1(diamF)n−1 +∞ +� +q=1 +Kn +q εq−1 +� +. +We now only have to check that the series in the previous expression +converge. Let us check that they converge at Z first and then outside of +Z. Recall that at Z we take the vectors ¯k ̸= 0. +� +k∈Zn\{0} +¯k̸=0 +1 +|k|τ +1 |k|2,ω +≤ +� +k∈Zn\{0} +¯k̸=0 +1 +|k|τ +1 |¯k| +≤ +� +¯k∈Zn−1\{0} +¯k̸=0 +� +kn∈Z +√n +(|¯k|1+|kn|)τ|¯k|1 +≤ +√n2n−1 �∞ +j=1 +� +kn∈Z +jn−3 +j+|kn|)τ +where we used that the number of vectors ¯k ∈ Zn−1 with |¯k|1 = j ≥ 1 +can be bounded by 2n−1jn−2. This series can be bounded by comparing +it to an integral: +� +kn∈Z +1 +(j + |kn|)τ ≤ 1 +jτ + 2 +� ∞ +0 +dx +(j + x)τ += 1 +jτ + +2 +(τ + 1)jτ−1 ≤ τ + 1 +τ − 1 +1 +jτ+1 . +Where we used that τ > 1 because n ≥ 2. Then +� +k∈Zn\{0} +¯k̸=0 +1 +|k|τ +1|k|2,ω +≤ +√n2n−1(τ + 1) +τ − 1 +∞ +� +j=1 +1 +jτ−n+2 +which converges by the condition τ > n − 1. +Now let us check that it converges outside of Z. +� +k∈Zn\{0} +1 +|k|τ +1 |k|2,ω += +� +k∈Zn\{0} +¯k̸=0 +1 +|k|τ +1 |k|2,ω + � +k∈Zn\{0} +¯k=0 +1 +|k|τ +1 |k|2,ω += +� +k∈Zn\{0} +¯k̸=0 +1 +|k|τ +1 |¯k| + � +k1∈Z +1 +|k|τ +1 |k2 +1B(I1)2| + +92 +6. A NEW KAM THEOREM +We have seen before that the first term converges. The second term: +� +k1∈Z +1 +|k1|τ|kτ +1B(I1)2| = +1 +B(I1)2 +� +k1∈Z +1 +kτ+2 +1 +, +which converges ∀I1 ̸= 0, i.e. outside of Z. +Now we go back to the expression 6.23. +The other terms of that +expression can be bounded simultaneously inside and outside Z. Now we +only have to check that the third series converges, because if the third +converges so does the second. We only have to check that �∞ +q=1 Kn +q εq−1 +converges. We will use that ε ≤ +8ε +νρ12(2τ+2)(q−1) . +�∞ +q=1 Kn +q εq−1 += +Kn �∞ +q=1 2n(q−1)εq−1 += +Kn �∞ +q=1 +8ε2n(q−1) +νρ12(2τ+2)(q−1) += +Kn 8ε +νρ1 +�∞ +q=1 +1 +2(2τ+2−n)(q−1) . +Which converges if and only if 2τ + 2 − n ≥ 1. And we are done +because 2τ ≥ n − 1 since τ ≥ n − 1 by hypothesis. +Putting everything together: +meas(G \ ˆG∗) +≤ 2n +µn + +D2 2γM +µ ++ 2(diamF)n−1 2γM +µ +√n2n−1(τ + 1) +τ − 1 +∞ +� +j=1 +1 +jτ−n+2 +D 8ε +νρ1 +∞ +� +q=1 +1 +2(2τ + 2)(q − 1) ++2n+1(diamF)n−1Kn 8ε +νρ1 +∞ +� +q=1 +1 +2(2τ+2−n)(q−1) +� +Now using that +ε ≤ +ν2µ2β2 +2τ+30L4M 3K2τ+1 ≤ 2τ−18 · 8MKτ+1ρ2 +LMK2τ+2 +γ ≤ 2τ−15ρ2 +LKτ+1 γ +We can write meas(G \ ˆG∗) ≤ C′γ where C′ depends only on n, µ, D, +diamF, M, τ, ρ1, ρ2, L, K and if we efine C = (2π)nC′. Hence, +meas[(Tn × G) \ T (Tn ˆG)] ≤ Cγ. +□ + +CHAPTER 7 +Desingularization of bm-integrable systems +In this chapter, we follow [GMW17], for the definition of the desingularization +of the bm-symplectic form. +Definition 7.1. The fǫ-desingularization ωǫ form of ω = dx +xm ∧ +��m−1 +i=0 xiαm−i +� ++ +β is: +ωǫ = dfǫ ∧ +�m−1 +� +i=0 +xiαm−i +� ++ β. +Where in the even case, fǫ(x) is defined as ǫ−(2k−1)f(x/ǫ). And f ∈ C∞(R) is an +odd smooth function satisfying f ′(x) > 0 for all x ∈ [−1, 1] and satisfying outside +that +(7.1) +f(x) = +� +−1 +(2k−1)x2k−1 − 2 +for +x < −1, +−1 +(2k−1)x2k−1 + 2 +for +x > 1. +And in the odd case, fǫ(x) = ǫ−(2k)f(x/ǫ). And f ∈ C∞(R) is an even smooth +positive function which satisfies: f ′(x) < 0 if x < 0, f(x) = −x2 + 2 for x ∈ [−1, 1], +and +(7.2) +f(x) = +� +−1 +(2k+2)x2k+2 − 2 +if k > 0, x ∈ R \ [−2, 2] +log(|x|) +if k = 0, x ∈ R \ [−2, 2]. +Remark 7.2. With the previous definition, we obtain smooth symplectic (in +the even case) or smooth folded symplectic (in the odd case) forms that agree +outside an ǫ-neighbourhood with the original bm-forms. Moreover, there is a con- +vergence result in terms of m. See [GMPS17] for the details. +To simplify notation, we introduce F m−i +ǫ +(x) = ( d +dxfǫ(x))xi, and hence F i +ǫ(x) = +( d +dxfǫ(x))xm−i. With this notation the desingularization ωǫ is written: +ωǫ = +m−1 +� +i=0 +F m−i +ǫ +(x)dx ∧ αm−i + β. +Definition 7.3. The desingularization for (M, ω, µ) is the triple (M, ωε, µǫ) +where ωε is defined as above and µε is: +µ �→ µǫ = +� +f1ǫ = +m +� +i=1 +ˆciGi +ǫ(x), f2(˜I, ˜φ), . . . , fn(˜I, ˜φ) +� +, +where +µ = +� +f1 = c0 log(x) + +m−1 +� +i=1 +ci +1 +xi , f2(I, φ) . . . , fn(I, φ) +� +93 + +94 +7. DESINGULARIZATION OF bm-INTEGRABLE SYSTEMS +Gi +ǫ(x) = +� x +0 +F i +ǫ(τ)dτ, +and ˆc1 = c0 and ˆci−1 = −ici if i ̸= 0. Also + + + + + + + + + + + +˜I = (˜I1, I2, . . . , In), +˜I1 += +� I1 +0 +��m +i=1 KˆciF i +ε(τ) +�m +i=1 +Kˆcj +τj +� +dτ +˜φ1 = (˜φ1, φ2, . . . , φn), +˜φ1 += + + +�m +i=1 KˆciF i +ε(I1) +�m +i=1 +Kˆcj +Ij +1 + + φ1 +Remark 7.4. Observe that with the last definition, when ǫ tends to 0, µǫ tends +to µ. +Theorem C. The desingularization transforms a bm-integrable system into an +integrable system for m even on a symplectic manifold. For m odd the desingular- +ization transforms it into a folded integrable system. The integrable systems are +such that: +Xω +fj = Xωǫ +fjǫ. +Proof. Let us first check the singular part, i.e. let us check that that Xω +f1 = +Xωǫ +f1ǫ. Let us compute the two equations that define each one of the vector fields. +We have to impose −df1 = ιXω +f1 ω and −df1ǫ = ιXωǫ +f1ǫ ωǫ. But observe first that we +can rewrite ω = �m +i=1 +1 +xi dx ∧ αi + β and ωǫ = �m +i=1 F i +ǫdx ∧ αi + β. The conditions +translate as: +− +m +� +i=1 +ˆci +1 +xi dx = ιXω +f1 +� m +� +i=1 +1 +xi dx ∧ αi + β +� +, +− +m−1 +� +i=0 +ˆciF i +ǫ(x)dx = ιXωǫ +f1ǫ +�m−1 +� +i=0 +F i +ǫ(x)dx ∧ αi + β +� +. +Since the toric action leaves the form ω invariant, in particular, the singular +set is invariant, and then Xωǫ +f1ǫ and Xω +f1 are in the kernel of dx. Moreover, since β +is a symplectic form in each leaf of the foliation and Xωǫ +f1ǫ and Xω +f1 are transversal +to this foliation, they are also in the kernel of β. +− +m−1 +� +i=0 +ˆci +1 +xi dx = +m−1 +� +i=0 +1 +xi dx ∧ αi(Xω +f1), +− +m−1 +� +i=0 +ˆciF i +ǫ(x)dx = +m−1 +� +i=0 +F i +ǫ(x)dx ∧ αi(Xωǫ +f1ǫ). +Then, the conditions over Xω +f1 and Xωǫ +f1ǫ are respectively: +−ˆci = αi(Xω +f1), +−ˆci = αi(Xωǫ +f1ǫ). +Then, the two vector fields have to be the same. +Let us now see Xω +fj = Xωǫ +fjǫ for j > 1. Assume now we have the bm-symplectic +form in action-angle coordinates ω = �m +i=1 +Kˆci +Ii +1 dI1 ∧ dφ1 + �n +i=1 dIi ∧ dφi. +The differential of the functions are + +7. DESINGULARIZATION OF bm-INTEGRABLE SYSTEMS +95 +df ε +i += +∂f ε +i +∂I1 dI1 + ∂f ε +i +∂φ1 dφ1 + �n +j=2 +� +∂f ε +i +∂Ij dIj + ∂f ε +i +∂φj dφj +� += +∂fi +∂I1 +��m +i=1 KˆciF i +ε(τ) +�m +i=1 +Kˆcj +τj +� +dI1 + ∂fi +∂φ1 +��m +i=1 KˆciF i +ε(τ) +�m +i=1 +Kˆcj +τj +� +dφ1 ++ �n +j=2 +� +∂f ε +i +∂Ij dIj + ∂f ε +i +∂φj dφj +� +. +On the other hand, the desingularized form is: +ωε = +m +� +j=1 +KˆciF j +ε (I1)dI1 ∧ dφ1 + +m +� +j=2 +dIj ∧ dφj. +Hence, one can see that the expression for both Xω +fj and Xωǫ +fjǫ is +Xω +fj = Xωǫ +fjǫ = +∂fi +∂I1 +�m +i=1 +Kˆci +Ii +1 +∂ +∂φ1 +− +∂fi +∂φ1 +�m +i=1 +Kˆci +Ii +1 +∂ +∂I1 ++ +n +� +j=2 +�∂f ε +i +∂Ij +dIj + ∂f ε +i +∂φj +dφj +� +□ +Remark 7.5. The previous lemma tells us that the dynamics of the desingu- +larized system are identical to the dynamics of the original bm-integrable system in +the bm-symplectic manifold. +Hence the desingularized bm-form goes to a folded symplectic form in the case +m = 2k + 1 and to symplectic for m = 2k. And the bm-integrable system goes to +a folded integrable system (see [CM22]) in the case m = 2k + 1 and to a standard +integrable system for n = 2k. + + +CHAPTER 8 +Desingularization of the KAM theorem on +bm-symplectic manifolds +The idea of this section is to recover some version of the classical KAM theorem +by “desingularizing the bm-KAM theorem”, as well as a new version of a KAM +theorem that works for folded symplectic forms. Observe that no KAM theorem +is known for folded symplectic forms. The best that is known is a KAM theorem +for presymplectic structures that was done in [AdlL12]. Desingularizing the KAM +means applying the bm-KAM in the bm-manifold and then translating the result to +the desingularized setting. +To be able to obtain proper desingularized theorems we need to identify which +integrable systems can be obtained as a desingularization of a bm-integrable system. +To simplify computations we are going to use a particular case of bm-integrable +systems, where f1 = +1 +Im−1 +1 +. We call these systems simple. Observe that by taking +a particular case of bm-integrable systems we will not get all the systems that can +be obtained by desingularizing a bm-integrable system, but some of them. +(1) Even case m = 2k. +F = (f1 = +1 +I2k−1 +1 +, f2, . . . , fn), ω = +1 +Im +1 dI1 ∧ dφ1 + �n +j=1 dIj ∧ dφj. +Observe that close to Z in the even case we can assume f(I1) = cI1 +for some c > (2 − +1 +22k−1 ). Then fε(I1) = +1 +ε(2k−1) +cI1 +ε += c′I, hence ωε = +c′dI1 ∧ dφ1 + �n +j=1 dIj ∧ dφj. Also F m +ε (I1) = c′, Gm +ε (I1) = c′I1. Then, +� ˜I1 += +� I1 +0 +c′ +1/τ m dτ = +� I1 +0 c′τ mdτ = c′ Im+1 +1 +m+1 , +˜φ1 += +c′ +1/Im +1 φ1 = c′Im +1 φ1 +(8.1) +F ε = ((m − 1)cm−1c′I1, f2(˜I, ˜φ), . . . fn(˜I, ˜φ)). +Hence, the systems in this form can be viewed as a desingularization +of a bm-integrable system. +Theorem D (Desingularized KAM for symplectic manifolds). Con- +sider a neighborhood of a Liouville torus of an integrable system Fε as +in 8.1 of a symplectic manifold (M, ωε) semilocally endowed with coor- +dinates (I, φ), where φ are the angular coordinates of the torus, with +ωε = c′dI1∧dφi+�n +j=1 dIj∧dφj. Let H = (m−1)cm−1c′I1+h(˜I)+R(˜I, ˜φ) +be a nearly integrable system where +� +˜I1 += +c′ Im+1 +1 +m+1 , +˜φ1 += +c′Im +1 φ1, +97 + +98 +8. bm-KAM DESINGULARIZATION +and +� ˜I += +(˜I1, I2, . . . , In), +˜φ += +(˜φ1, φ2, . . . , φn). +Then the results for the bm-KAM theorem 6.3 applied to Hsing = +1 +I2k−1 +1 ++ +h(I) + R(I, φ) hold for this desingularized system. +Remark 8.1. This theorem is not as general as the standard KAM, +but we also know extra information about the dynamics. For instance, +the perturbation of trajectories in tori inside of Z will be trajectories lying +inside of Z. In this sense, the theorem is new because it leaves invariant +an hypersurface of the manifold. +(2) Odd case m = 2k + 1. +F = (f1 = +1 +I2k +1 , f2, . . . , fn) and ω = +1 +I2k+1 +1 +dI1 ∧ dφ1 + �n +j=1 dIj ∧ dφj. +Before continuing we need the following notions defined in [CM22]. +Definition 8.2. A function f : M → R in a folded symplectic mani- +fold (M, ω) is folded if df|Z(v) = 0 for all v ∈ V = kerω|Z. +Definition 8.3. An integrable system in a folded symplectic manifold +(M, ω) with critical surface Z is a set of functions F = (f1, . . . , fn) such +that they define Hamiltonian vector fields which are independent (df1 ∧ +. . . ∧ dfn ̸= 0 in the folded cotangent bundle) on a dense subset of Z and +M, and commute with respect to ω. +Note that we need to prove that the desingularized functions in this +case are folded. +Observe that close to Z in the odd case we can assume f(I1) = −I2 +1+2. +Then fε(I1) = ε−(2k)f( I1 +ε ) = +1 +ε2k (−( I1 +ε )2 + 2) = cI2 +1 + +2 +ε2k . Then +ωε = 2cI1dI1 ∧ dφ1 + +n +� +j=1 +dIj ∧ dφj. +Also F m +ε (I1) = 2cI1, Gm +ε (I1) = cI2 +1. Then, +� +˜I1 += +� I1 +0 +2cτ +1/τ m dτ = 2c I(m+2) +1 +(m+2) , +˜φ1 += +2cIm+1 +1 +φ1 +Then the desingularized moment map becomes +(8.2) +F ε = ((m − 1)cm−1cI2 +1, f2(˜I, ˜φ), . . . fn(˜I, ˜φ)). +It is a simple computation to check that these functions are actually +folded and hence they form a folded integrable system. Note that the +systems of the form 8.2 can be viewed as a desingularization of a bm- +integrable system. Then, as we proceeded in the even case: +Theorem E (Desingularized KAM for folded symplectic manifolds). +Consider a neighborhood of a Liouville torus of an integrable system Fε as +in 8.2 of a folded symplectic manifold (M, ωε) semilocally endowed with +coordinates (I, φ), where φ are the angular coordinates of the Torus, with + +8. bm-KAM DESINGULARIZATION +99 +ωε = 2cI1dI1 ∧ dφ1 + �m +j=2 dIj ∧ dφj. Let H = (m − 1)cm−1cI2 +1 + h(˜I) + +R(˜I, ˜φ) a nearly integrable system with +� +˜I1 += +2c Im+2 +1 +m+2 , +˜φ1 += +2cIm+1 +1 +φ1, +and +� ˜I += +(˜I1, I2, . . . , In), +˜φ += +(˜φ1, φ2, . . . , φn). +Then the results for the bm-KAM theorem 6.3 applied to Hsing = +1 +I2k +1 ++ +h(I) + R(I, φ) hold for this desingularized system. +Remark 8.4. The last two theorems can be improved if we consider +bm-integrable systems not necessarily simple. + + +CHAPTER 9 +Potential applications to Celestial mechanics +All the theory developed in this monograph would not be fertile if we could not +envisage applications of perturbation theory to actual physical systems. We provide +several examples from Celestial mechanics and conclude with potential applications +of our KAM theory to detect periodic trajectories. +In this chapter we present several examples appearing in Celestial Mechanics +where singular symplectic forms show up. Some of these examples are contained +in [MDD+19]. +Most of the singularities appear as a consequence of applying +regularization techniques. We invite the reader to consult the book [Kna18] for a +pedagogical approach to the study of regularization. +This list of examples is of special relevance for this booklet as the theoretical +results that we obtain such as action-angle coordinates or KAM can be, de facto, +applied to the list of problems considered below. +Structures that are symplectic almost everywhere can arise as the result of +changes of coordinates which do not preserve the canonical symplectic structure. +For instance: For the Kepler problem given a configuration space R2 and phase +space T ∗R2, the traditional (canonical) Levi-Civita transformation described as +follows: identify R2 ∼= C so that T ∗R2 ∼= T ∗C ∼= C2 and treat (q, p) as complex +variables (q1 + iq2 := u, p1 + ip2 := v). Take the following change of coordinates +(q, p) = (u2/2, v/¯u), where ¯u denotes the complex conjugation of u. The resulting +coordinate change can easily be seen to preserve the canonical symplectic form. +However, this canonical change of coordinates can make the Hamiltonian equations +more complicated making more difficult to study the dynamics of the system. This +is why it is often interesting to consider other changes of coordinates where the +symplectic form is not preserved. Some of them induce new singular forms where +our geometrical and dynamical techniques can be applied. +Other examples are discussed in [DKM17]. +9.1. The Kepler Problem +In suitable coordinates in T ∗ � +R2 \ {0} +� +, the Kepler problem has Hamiltonian +(9.1) +H(q, p) = ∥p∥2 +2 +− +1 +∥q∥. +With the canonical Levi-Civita transformation (q, p) = (u2/2, v/¯u), this expression +becomes +(9.2) +H(u, v) = ∥v∥2 +2∥¯u∥2 − +1 +∥u∥2 . +101 + +102 +9. +POTENTIAL APPLICATIONS TO CELESTIAL MECHANICS +Changes of coordinates preserving the canonical symplectic form leads to more +complicated equations. So we propose a new reciipe: leave the momentum un- +changed and examine the transformation (q, p) = (u2/2, p) instead. This can result +in a simpler Hamiltonian. The transformation is not a symplectomorphism and the +symplectic form on T ∗R2 pulls-back under the transformation to a two-form which +is symplectic almost everywhere, but degenerates on a hypersurface of T ∗R2 +Namely, the Liouville one-form p1dq1 + p2dq2 = ℜ(pd¯q) pulls back to +θ = ℜ +� +pd +� ¯u2 +2 +�� += +ℜ (p¯ud¯u) += +p1(u1du1 − u2du2) + p2(u2du1 + u1du2) +and the associated 2-form −dθ yields a form that is almost everywhere symplectic +ω = u1du1 ∧ dp1 − u2du1 ∧ dp2 + u2du2 ∧ dp1 + u1du2 ∧ dp2. +In order to test the nature of this form we wedge the form with itself and we +find +ω ∧ ω = (u2 +1 − u2 +2)du1 ∧ dp1 ∧ du2 ∧ dp2 +which is degenerate along the hypersurface given by u1 = ±u2. +We now consider the restriction of the form to the critical set. It does not have +maximal rank so it is not a folded symplectic structure. This form is degenerately +folded and the folding hypersurface is not regular and is described by the equations +u1 = ±u2. +9.2. The Problem of Two Fixed Centers +We now regularize the problem of two fixed centers. +The problem of two fixed centers is associated to the motion of a satellite moving +in a gravitational potential generated by two fixed massive bodies. We assume also +that the motion of the satellite is restricted to the plane in R3 containing the two +massive bodies. +The Hamiltonian function in suitable coordinates reads: +(9.3) +H = p2 +2m − µ +r1 +− 1 − µ +r2 +where µ is the mass ratio of the two bodies (i.e. µ = +m1 +m1+m2 ). +The integrability of this problem was first proved by Euler via elliptic coordi- +nates, where the coordinate lines are confocal ellipses and hyperbola. +Explicitly, consider a coordinate system in which the two centers are placed at +(±1, 0), in which the (Cartesian) coordinates are given by (q1, q2). Then the elliptic +coordinates of the system are given by +q1 = sinh λ cos ν +(9.4) +q2 = cosh λ sin ν +(9.5) +for (λ, ν) ∈ R × S1. Thus lines of λ = c and ν = c are given by confocal hyperbola +and ellipses in the plane, respectively. Similar to the Levi-Civita transformation +this results in a double-branched covering with branch points at the centers of +attraction. +Pulling back the canonical symplectic structure ω = dq ∧ dp we find +(9.6) +ω = cosh λ cos ν(dλ ∧ dp1 + dν ∧ dp2) − sinh λ sin ν(dν ∧ dp1 + dλ ∧ dp2) + +9.3. DOUBLE COLLISION AND MCGEHEE COORDINATES +103 +which is degenerate along the hypersurface (λ, ν) satisfying cosh λ cos ν = sinh λ sin λ. +9.3. Double Collision and McGehee coordinates +In this section, we describe another example of b-symplectic structure appearing +quite naturally in physical dynamical systems. From this example, it would seem +natural that a collection of different examples for bm-symplectic models or even +bm-folded models would follow. +But one finds a major problem while pursuing +these examples. Understanding why this example does not extend to construct +bm-symplectic models of bm-folded for any m gives a general pattern. +First let us introduce the McGehee coordinate change for the problem of double +collision. +The system of two particles moving under the influence of the generalized po- +tential U(x) = −|x|−α, α > 0, where |x| is the distance between the two particles, +is studied by McGehee in [McG81]. We fix the center of mass at the origin and +hence can simplify the problem to the one of a single particle moving in a central +force field. +The equation of motion can be written as, +(9.7) +¨x = −∇U(x) = −α|x|−α−2x +where the dot represents the derivative with respect to time. In the Hamiltonian +formalism, this equation becomes +(9.8) +˙x += +y, +˙y += +−α|x|−α−2x. +To study the behavior of this system, the following change of coordinates is sug- +gested in [McG81]: +(9.9) +x += +rγeiθ, +y += +r−βγ(v + iw)eiθ +where the parameters β and γ are related with α as follows: +(9.10) +β += +α/2, +γ += +1/(1 + β). +Identifying the plane R2 with the complex plane C, we can write the symplectic +form of this problem as ω = ℜ(dx ∧ dy). +Remark 9.1. To check that a form ω is actually a bm-symplectic form, it is not +enough to check that the multi-vector field dual to ω∧ω is a section of �2n(bmT M) +which is transverse to the zero section. One has to check additionally that the +Poisson structure dual to ω itself is a proper section of �2(bmT M). +Proposition 9.2. Under the coordinate change (9.9), the symplectic form ω +is sent to a b-symplectic structure for α = 2. +Proof. The proof of this proposition is a straightforward computation. Ob- +serve that the change is not a smooth change, so we are not working with standard +De Rham forms. But, at the end of the computation it will become clear that the +form is a b-symplectic form and hence the computations are legitimate. If one does + +104 +9. +POTENTIAL APPLICATIONS TO CELESTIAL MECHANICS +the change of variables, we obtain: +(9.11) +¯y += +rβγ(v − iw)e−iθ. +dx += +γrγ−1eiθdr + rγeiθidθ. +d¯y += +r−βγ−1(−βγ)(v − iw)e−iθdr + r−βγe−iθdv ++rβγ(v − iw)e−iθ(−i)dθ. +By wedging the previous two forms, we obtain: +(9.12) +dx ∧ d¯y += +dr ∧ dv(γrγ−1−βγ) ++ +dr ∧ dw(γrγ−1−βγ) ++ +dr ∧ dθ(γrγ−1−βγ(−iv − w)) ++ +dθ ∧ dr(irγ−1−βγ(−βγ)(v − iw)) ++ +dθ ∧ dv(irγ−βγ) ++ +dθ ∧ dw(irγ−βγ(−i)). +Now we can take the real part of this form and use that γ − 1 − βγ = −αγ. In the +new coordinates, the form reads. +(9.13) +ω = ℜ(dx ∧ d¯y) += +γr−βγ+γ−1dr ∧ dv − γ(1 − β)r−βγ+γ−1wdr ∧ dθ +− +r−βγ+γdw ∧ dθ. +Moreover, we can use that γ(1 + β) = 1 to simplify the previous expression further +to: +(9.14) +ω = (dr ∧ dv + dr ∧ dw)γr−αγ + dr ∧ dθ(wr−αγ) + dθ ∧ dw(r−αγ+1). +In order to classify this structure, we wedge it with itself and look at the structure +of the form in the singular set. Wedging this form, we obtain +(9.15) +ω ∧ ω += +−γr−2βγ+2γ−1dr ∧ dv ∧ dθ ∧ dw += +−γr +2−3α +2+α dr ∧ dv ∧ dθ ∧ dw. +where we use (9.10). Let us set f(α) = 2−3α +2+α . This function does not take values +lower than −3 or higher than 1. When α = 2 this gives us a b-symplectic structure: +ω ∧ ω = −γrdr ∧ dv ∧ dθ ∧ dw. +The section of �4(bT M) given by the dual structure of ω ∧ ω is clearly transverse +to the zero section. +On the other hand if α = 2, then β = 1 and hence: +ω = γr−1dr ∧ ω ∧ dv, +and its dual Poisson structure is clearly also a proper section of �2(bT M). +□ +Remark 9.3. One may ask if for other values of α it is possible to obtain +other bm-symplectic structures for different m. For example for α = 6, as ω ∧ ω = +−γr−2dr ∧dv ∧dθ ∧dw, so it seems likely to obtain a b2-symplectic form. But from +the expression of ω it becomes clear that it is not a proper section of �2(b2T ∗M) + +9.4. POTENTIAL APPLICATIONS +105 +m1 = 1 − µ +m2 = µ +q +r2 = q − q2 +r1 = q − q1 +Center of mass +r +q1 +q2 +Figure 1. Scheme of the three-body problem. +9.4. The restricted three-body problem +In this last section of the monograph, we catch up with the circular planar +restricted three-body problem. +The restricted elliptic 3-body problem is a simplified version of the 3-body +problem. It describes the trajectory of body with negligible mass moving in the +gravitational field of two massive bodies called primaries, orbiting in elliptic Kep- +lerian motion. The restricted planar version assumes that all motion occurs in a +plane. +The associated Hamiltonian of the particle can be written as: +(9.16) +H(q, p) = ∥p∥2 +2 ++ +1 − µ +∥q − q1∥ + +µ +∥q − q2∥ = T + U +wit µ the reduced mass of the system. +As it was observed in [KMS16b], it is possible to associate a singular struc- +ture to this problem. Consider the symplectic form on T∗R2 in polar coordinates, +After making a change to polar coordinates (q1, q2) = (r cos α, r sin α) and the +corresponding canonical change of momenta we find the Hamiltonian function +(9.17) +H(r, α, Pr, Pα) = P 2 +r +2 + P 2 +α +2r2 + U(r cos α, r sin α) +where Pr, Pα are the associated canonical momenta and with potential energy: +U(r cos α, r sin α) +The McGehee change of coordinates is used to examine the behavior of orbits +near infinity, see also [DKdlRS19]: +(9.18) +r = 2 +x2 . +The corresponding change for the canonical momenta is easily seen to be +(9.19) +Pr = −x3 +4 Px. + +106 +9. +POTENTIAL APPLICATIONS TO CELESTIAL MECHANICS +The Hamiltonian is transformed to +(9.20) +H(r, α, Pr, Pα) = x6P 2 +x +32 ++ x4P 2 +α +8 ++ U(x, α). +By transforming the position coordinate (9.18) without modifying the momentum +associated to r, we are left with a simpler Hamiltonian, however, the pull-back of +the symplectic form is no longer symplectic, but exhibits a singularity of order 3 +and it is called b3-symplectic: +(9.21) +ω = 4 +x3 dx ∧ dPr + dα ∧ dPα. +Adding the line at infinity provides a description of the dynamics within the +critical set Z = {x = 0}. From the change of coordinates implemented, we might +think that the dynamics within Z may have no physical meaning, but its interplay +with the dynamics close to Z gives information about the behaviour of escape orbits +sometimes identified as singular periodic orbits (see [MO21] and [MOPS22]). +Given an autonomous Hamiltonian system of a symplectic manifold of dimen- +sion 2n, the level sets of the Hamiltonian function are often endowed with a contact +structure ( a contact structure is given by a one form α satisfying a condition of +type α ∧ (dα)n−1 ̸= 0). +In [MO18, MO21] applications of the b-apparatus are discussed in this con- +text. In particular, the notion of bm-contact structures is introduced by translating +the condition above for bm-forms. The classical notions in the contact realm such +as Reeb vector fields can also be introduced in this set-up. +By considering the Mc Gehee change as we did in the contact context, in +[MO21] it is proved: +Theorem 9.4. After the McGehee change, the Liouville vector field Y = p ∂ +∂p is +a b3-vector field that is everywhere transverse to the level sets of the Hamiltonian Σc +for c > 0 and the level-sets (Σc, ιY ω) for c > 0 are b3-contact manifolds. Topologi- +cally, the critical set of this contact manifold is a cylinder (which can be interpreted +as a subset of the line at infinity) and the Reeb vector field admits infinitely many +non-trivial periodic orbits on the critical set. +One of the possible applications of our KAM theorem would be to find new +periodic orbits of the restricted three body problem close to infinity by perturbing +the periodic orbits described above. This old technique of perturbation theory is +probably due to Poincar´e (Poincar´e’s continuation method, see [MO17]). +This +opens the door to new investigations which will be considered elsewhere. + +Bibliography +[AA81] +V. I. Arnol’d and V. I. Arnol’d, Singularity theory : selected papers / [edited by] v.i. +arnold, Cambridge University Press Cambridge [Cambridgeshire] ; New York, 1981 +(English). +[AdlL12] +Hassan Najafi Alishah and Rafael de la Llave, Tracing KAM tori in presymplectic +dynamical systems, J. Dynam. Differential Equations 24 (2012), no. 4, 685–711. +MR 3000600 +[Arn89] +V. I. Arnol’d, Poisson structures on the plane and other powers of volume forms, +Journal of Soviet Mathematics 47 (1989), no. 3, 2509–2516. +[BMO22] +Joaquim Brugu´es, Eva Miranda, and C´edric Oms, The arnold conjecture for singular +symplectic manifolds, arXiv:2212.01344 (2022). +[CM22] +Robert Cardona and Eva Miranda, Integrable Systems on Singular Symplectic Mani- +folds: From Local to Global, Int. Math. Res. Not. IMRN (2022), no. 24, 19565–19616. +MR 4523256 +[DG96] +Amadeu Delshams and Pere Guti´errez, Effective stability and KAM theory, J. Dif- +ferential Equations 128 (1996), no. 2, 415–490. 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Melrose, The atiyah-patodi-singer index theorem, Research Notes in Mathematics, +CRC Press, 1993. +[Mel93b] +, The atiyah-patodi-singer index theorem, Research Notes in Mathematics, +CRC Press, 1993. +[MM22] +Anastasia Matveeva and Eva Miranda, Reduction theory for singular symplectic +manifolds and singular forms on moduli spaces, arXiv:2205.12919 (2022). +[MO17] +Kenneth R. Meyer and Daniel C. Offin, Introduction to Hamiltonian dynamical sys- +tems and the N-body problem, third ed., Applied Mathematical Sciences, vol. 90, +Springer, Cham, 2017. MR 3642697 +[MO18] +Eva Miranda and C´edric Oms, The geometry and topology of contact structures with +singularities, arXiv:1806.05638 (2018). +[MO21] +Eva Miranda and C´edric Oms, The singular weinstein conjecture, Advances in Math- +ematics 389 (2021), 107925. +[MOPS22] +Eva Miranda, C´edric Oms, and Daniel Peralta-Salas, On the singular Weinstein con- +jecture and the existence of escape orbits for b-Beltrami fields, Commun. 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Swan, Vector bundles and projective modules, Transactions of the Amer- +ican Mathematical Society 105 (1962), 264–277. + diff --git a/1NAyT4oBgHgl3EQfbff_/content/tmp_files/load_file.txt b/1NAyT4oBgHgl3EQfbff_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4b7f83a504ccc11469793af65729a79769a6d3a --- /dev/null +++ b/1NAyT4oBgHgl3EQfbff_/content/tmp_files/load_file.txt @@ -0,0 +1,2626 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf,len=2625 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='00266v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='SG] 31 Dec 2022 Action-angle coordinates and KAM theory for singular symplectic manifolds Eva Miranda Arnau Planas Laboratory of Geometry and Dynamical Systems, Department of Mathematics & IMTech, Universitat Polit`ecnica de Catalunya, Barcelona and CRM, Centre de Recerca Matem`atica, Bellaterra Current address: UPC-Edifici P, Avinguda del Doctor Maran´on, 44-50, 08028, Barcelona, Spain Email address: evamiranda@upc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='edu Department of Mathematics, Universitat Polit`ecnica de Catalunya, Barcelona Email address: arnau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='planas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='bahi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='com 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 53D05, 53D20, 70H08, 37J35, 37J40 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 37J39, 58D19 Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' amsbook, AMS-LATEX Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016 and ICREA Academia Prize 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Eva Miranda is also supported by the Spanish State Research Agency, through the Severo Ochoa and Mar´ıa de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Eva Miranda also acknowledges partial support from the grant “Computational, dynamical and geometrical complexity in fluid dynamics”, Ayudas Fundaci´on BBVA a Proyectos de Investigaci´on Cient´ıfica 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Both authors are supported by the project PID2019-103849GB-I00 of the Spanish State Agency AEI /10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Dedicated to the memory of Amelia Galcer´an Sorribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Contents Preface vii Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Introduction and preliminaries 1 Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Structure and results of this monograph 3 Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A primer on singular symplectic manifolds 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' b-Poisson manifolds 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On bm-Symplectic manifolds 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Desingularizing bm-Poisson manifolds 12 Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A crash course on KAM theory 15 Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Action-angle coordinates and cotangent models 19 Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' An action-angle theorem for bm-symplectic manifolds 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Basic definitions 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On bm-integrable systems 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Examples of bm-integrable systems 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Looking for a toric action 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Action-angle coordinates on bm-symplectic manifolds 32 Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Reformulating the action-angle coordinate via cotangent lifts 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Cotangent lifts and Arnold-Liouville-Mineur in Symplectic Geometry 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The case of bm-symplectic manifolds 38 Part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A KAM theorem for bm-symplectic manifolds 41 Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A new KAM theorem 43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the structure of the proof 43 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Technical results 50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A KAM theorem on bm-symplectic manifolds 69 Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Desingularization of bm-integrable systems 93 Chapter 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Desingularization of the KAM theorem on bm-symplectic manifolds 97 Chapter 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Potential applications to Celestial mechanics 101 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Kepler Problem 101 v vi CONTENTS 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Problem of Two Fixed Centers 102 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Double Collision and McGehee coordinates 103 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The restricted three-body problem 105 Bibliography 107 Preface I confess I envy the planets — they’ve got their own orbits and nothing stands on their way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Intermezzo, Mykhailo Kotsi- ubynsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This monograph explores classification and perturbation problems for inte- grable systems on a class of Poisson manifolds called bm-Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is the first class of Poisson manifolds for which perturbation theory is established outside the symplectic category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Even if the class of bm-Poisson manifolds is not ample enough to represent the wild set of Poisson manifolds, this investigation can be seen as a first step for the study of perturbation theory for general Poisson man- ifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In view of the work of the first author with Nest, the theorems established in this monograph constitute more than a mild generalization in Poisson Geometry and, this toy example, sets the path to consider KAM theory in the general realm of Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Reduction theorems and bm-symplectic manifolds have been recently explored in [MM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This monograph contributes to the theory opening the investigation of perturbation theory on these manifolds thus completing other facets in the study of their dynamics as the recent work on the Arnold conjecture [BMO22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Symplectic geometry has been the common language of physics as the position- momentum tandem can be modelled over a cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Cotangent bundles are naturally endowed with a symplectic form which is a non-degenerate closed 2-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The symplectic form of the cotangent bundle is given by the differential of the Liouville one-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' bm-Poisson manifolds are manifolds that are symplectic away from a hyper- surface along which they satisfy some transversality properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' They often model problems on symplectic manifolds with boundary such as the study of their de- formation quantization and celestial mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As on the complementary of the critical set the manifolds are symplectic, extending the investigation of Hamiltonian dynamics to this realm is key to understand Hamiltonian Dynamics on compact- ification of symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Several regularization transformations used in celestial mechanics (as McGehee or Moser regularization) provide examples of such compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' One of the interesting properties of bm-Poisson manifolds is that their investi- gation can be achieved considering the language of bm-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' That is to say, we can work with forms that are symplectic away from the critical set and admit a smooth extension as a form over a Lie algebroid generalizing De Rham forms as form over the standard Lie algebroid of the tangent bundle of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To consider bm-forms the standard tangent bundle is replaced by the bm-tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This vii viii PREFACE allows us to mimic symplectic geometry by replacing the cotangent bundle by the dual of the bm-tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' However, Poisson geometry leaves its footprint and new invariants which can be identified as the modular class of the Poisson structure arise already at the semilocal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Contrary to the initial expectations, several of the results for bm-symplectic manifolds do not resemble the b-case so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Considering these more general singu- larities yields a better understanding of the general Poisson case and the different levels of complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As an illustration of this phenomena: in the study of quan- tization of those systems an interesting pattern makes the quantization radically different in the even and odd case [GMW18b, GMW21] and the resulting model is finite-dimensional in the b-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Understanding how the different degrees m are related is a hard task: The desingularization technique introduced by Guillemin- Miranda-Weitsman in [GMW17] turned out to have important applications in the investigation of complexity properties of toric bm-symplectic manifolds [GMW18a] and to the study of the Arnold conjecture in this set-up [BMO22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this mono- graph we explore a new facet of these manifolds: that of perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the second part of the monograph we consider integrable systems on these manifolds turn out to have associated generalized Hamiltonian actions of tori in a neighbourhood of a Liouville torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We use this generalized Hamiltonian group action to prove existence of action-angle coordinates in a neighborhood of a Liouville torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The action-angle coordinate theorem that we prove gives a semilocal normal form in the neighbourhood of a Liouville torus for the bm-symplectic structure which depends on the modular weight of the connected component of the critical set in which the Liouville torus is lying and the modular weights of the associated toric action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This action-angle theorem allows us to identify a neighborhood of the Liouville torus with the bm-cotangent lift of the action of a torus acting by translations on itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This interpretation of the action-angle theorem as cotangent lift allows us to identify the modular weight as their only semilocal invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In doing so, we compare this action-angle coordinate theorem with the classical action- angle coordinate theorems for symplectic manifolds and an action-angle theorem for folded symplectic manifolds ([CM22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In part 3 of the monograph we study perturbation theory in this new set-up and examine some potential applications to physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In particular, we prove a KAM theorem for bm-Poisson manifolds which clearly refines and improves the one obtained for b-Poisson manifolds in [KMS16a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As an outcome of this result together with the extension of the desingularization techniques of Guillemin- Miranda-Weitsman to the realm of integrable systems, we obtain a KAM theorem for folded symplectic manifolds where KAM theory has never been considered be- fore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the way, we also obtain a brand-new KAM theorem for symplectic manifolds where the perturbation keeps track of a distinguished hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In celestial me- chanics, this distinguished hypersurface can be the line at infinity or the collision set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Barcelona, December 2022, Eva Miranda and Arnau Planas Part 1 Introduction and preliminaries CHAPTER 1 Introduction Both symplectic and Poisson geometry emerge from the study of classical me- chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Both are broad fields widely studied and with powerful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' But as Poisson structures are far more general than the symplectic ones, most outstanding results in symplectic geometry do not translate well to Poisson manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Here is where bm-Poisson structures come to play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' bm-Poisson structures (or bm-symplectic structures) lie somewhere between these two worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' They extend symplectic struc- tures but in a really controlled way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is why fundamental results in symplectic geometry still work in bm-symplectic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' However, an adaptation of these theories like deformation or Moser theory requires some work (see [GMPS15a] and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The study of bm-Poisson geometry sparked from the study of symplectic man- ifold with boundary [Mel93a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the last years the interest in this field increased after the classification result for b-Poisson structures obtained in [Rad02].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Later on, [GMP14] translated these structures to the language of forms and started applying symplectic tools to study them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A lot of papers in the following years studied different aspects of these structures: [GMP10], [GMP14], [GMPS15b], [GMW17], [MOT14] and [GLPR17] are some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Inspired by the study of manifolds with boundary, we work on a pair of mani- folds (M, Z) where Z is an hypersurface and call this pair b-manifold In this context, [Sco16] generalized the b-symplectic forms by allowing higher degrees of degeneracy of the Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The bm-symplectic structures inherit most of the properties of b-symplectic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This booklet focuses on different aspects of the investigation of bm-symplectic structures covering mainly integrable systems and KAM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' First, we present some preliminary notions necessary to address the problem of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We present an action-angle theorem for bm-Poisson structures and state and prove the KAM theory equivalent in manifolds with bm-symplectic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Structure and results of this monograph 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Part 1: Introduction and Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the preliminaries, we give the basic notions that lead to the questions we are addressing in this booklet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the first part, we introduce the concept of b-Poisson manifolds or b-symplectic manifolds, a class of Poisson manifold which is symplectic outside a critical hyper- surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It study comes motivated by the investigation of manifolds with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Next, we talk about a generalization of these structures, that allow a higher degree of degeneracy of the structure: the bm-symplectic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' These structures are the main focus of our investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A key concept that will play an important work in this book is the study of the desingularization of these singular structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 3 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' INTRODUCTION Finally, we give a short introduction to KAM theory, a theory that will be gener- alized in the setting of bm-manifolds in the last chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Motivation comes from several examples of singular symplectic structures ap- pearing naturally in classical problems of celestial mechanics which are discussed on the last chapter of the monograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We also describe the difficulties of finding these examples, and the subtleties of dealing with these singular structures in the exploration of conservative systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Part 2: Action-angle coordinates and cotangent models for bm- integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this Chapter we define the concept of bm-functions and bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We present several examples of bm-integrable systems that come from classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' After all this we present a version of the action-angle theorem for bm-symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M, x, ω, F) be a bm-integrable system, where F = (f1 = a0 log(x)+�m−1 j=1 aj 1 xj , f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let m ∈ Z be a regular point, and such that the integral manifold through m is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let Fm be the Liouville torus through m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, there exists a neighborhood U of Fm and coordinates (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , θn, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , σn) : U → Tn × Bn such that: (1) We can find an equivalent integrable system F = (f1 = a′ 0 log(x) + �m−1 j=1 a′ j 1 xj ) such that a′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , a′ m−1 ∈ R, (2) ω|U = \uf8eb \uf8ed m � j=1 c′ j c σj 1 dσ1 ∧ dθn \uf8f6 \uf8f8 + n � i=2 dσi ∧ dθi where c is the modular period and c′ j = −(j − 1)a′ j−1, also (3) the coordinates σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , σn depend only on fn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Part 3: KAM theory on bm-symplectic manifolds and applica- tions to Celestial Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this chapter we provide several KAM theorem for (singular) symplectic manifolds including bm-symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We begin by considering perturbation theory for bm-symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we give an outline of how to construct the bm-symplectomorphism that will be the main character of the proof of the KAM theorem for bm-symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' After this, we show some technical results that are needed for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' These technical results even if quite similar to the standard KAM equivalents, have some subtleties that need to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We end the chapter with the proof of the bm-KAM theorem and several applications to establish KAM theorems in other singular sit- uations (folded symplectic manifolds) and on symplectic manifolds with prescribed invariant hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The first KAM theorem is the following: Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let G ⊂ Rn, n ≥ 2 be a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let H(φ, I) = ˆh(I) + f(φ, I), where ˆh is a bm-function ˆh(I) = h(I) + q0 log(I1) + �m−1 i=1 qi Ii 1 defined on Dρ(G), with h(I) and f(φ, I) analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ˆu = ∂ˆh ∂I and u = ∂h ∂I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume | ∂u ∂I |G,ρ2 ≤ M, |u|ξ ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume that u is µ non-degenerate (| ∂u ∂I | ≥ µ|v| for some µ ∈ R+ and I ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Take a = 16M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume that u is one-to-one on G and its range F = u(G) is a D-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let τ > n − 1, γ > 0 and 0 < ν < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' STRUCTURE AND RESULTS OF THIS MONOGRAPH 5 (1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1) ε := ∥f∥G,ρ ≤ ν2µ2ˆρ2τ+2 24τ+32L6M 3 γ2, (2) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) γ ≤ min(8LMρ2 ν ˆρτ+1 , L K′ ) (3) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3) µ ≤ min(2τ+5L2M, 27ρ1L4Kτ+1, βντ+122τ+1ρτ 1), where ˆρ := min � νρ1 12(τ+2), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Define the set ˆG = ˆGγ := {I ∈ G− 2γ µ |u(I) is τ, γ, c, ˆq− Dioph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, there exists a real continuous map T : W ρ1 4 (Tn) × ˆG → Dρ(G) an- alytic with respect the angular variables such that (1) For all I ∈ ˆG the set T (Tn ×{I}) is an invariant torus of H, its frequency vector is equal to u(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (2) Writing T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)) with estimates |Tφ(φ, I)| ≤ 22τ+15ML2 ν2ˆρ2τ+1 ε γ2 |TI(φ, I))| ≤ 210+τL(1 + M) ν ˆρτ+1 ε γ (3) meas[(Tn ×G)\\T (Tn× ˆG)] ≤ Cγ where C is a really complicated constant depending on n, µ, D, diamF, M, τ, ρ1, ρ2, K and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also, we obtain a way to associate a standard symplectic integrable system or a folded integrable system to a bm-integrable system, depending on the parity of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is done in such a way that the dynamics of the desingularized system are the same than the dynamics of the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' So it defines a honest desingularization of the integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The desingularization transforms a bm-integrable system into an integrable system on a symplectic manifold for even m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For m odd, the desin- gularization associates to it a folded integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The integrable systems satisfy: Xω fj = Xωǫ fjǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This allows us to obtain two new KAM theorems using this desingularization combined with the former bm-KAM theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The first of these theorems is a KAM theorem for standard symplectic manifolds, where the perturbation has a particu- lar expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This result is more restrictive than the standard KAM theorem but allow us to guarantee that the perturbations leave a given hypersurface invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This means that the tori belonging to that hypersurface remain on the hypersur- face after the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This can be interesting for a number of reasons and situations such as problems in Celestial mechanics where it is convenient to keep track of a particular hypersurface such as the line at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The higher order singularities allow to consider perturbations that are tangent to the hypersurface up to a certain order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' INTRODUCTION Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider a neighborhood of a Liouville torus of an integrable system Fε as in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1 of a symplectic manifold (M, ωε) semilocally endowed with coordinates (I, φ), where φ are the angular coordinates of the torus, with ωε = c′dI1 ∧dφi +�n j=1 dIj ∧dφj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let H = (m−1)cm−1c′I1 +h(˜I)+R(˜I, ˜φ) be a nearly integrable system where � ˜I1 = c′ Im+1 1 m+1 , ˜φ1 = c′Im 1 φ1, and � ˜I = (˜I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , In), ˜φ = (˜φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , φn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then the results for the bm-KAM theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3 applied to Hsing = 1 I2k−1 1 + h(I) + R(I, φ) hold also for this desingularized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The second one is a KAM theorem for folded-symplectic manifolds, where KAM theory has not been considered to-date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider a neighborhood of a Liouville torus of an integrable system Fε as in 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2 of a folded symplectic manifold (M, ωε) semilocally endowed with coordinates (I, φ), where φ are the angular coordinates of the Torus, with ωε = 2cI1dI1 ∧ dφ1 + �m j=2 dIj ∧ dφj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let H = (m − 1)cm−1cI2 1 + h(˜I) + R(˜I, ˜φ) a nearly integrable system with� ˜I1 = 2c Im+2 1 m+2 , ˜φ1 = 2cIm+1 1 φ1, and � ˜I = (˜I1, I2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , In), ˜φ = (˜φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , φn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then the results for the bm-KAM theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3 applied to Hsing = 1 I2k 1 +h(I)+R(I, φ) also hold for this desingularized system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Last but not least, we illustrate the connection between bm-symplectic struc- tures and classical mechanics by providing several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Several potential applications to celestial mechanics are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' CHAPTER 2 A primer on singular symplectic manifolds In this first chapter of the booklet we introduce basic notions on singular sym- plectic structures, as well as some concepts on standard KAM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Those are the two main pillars of this monograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let M be a smooth manifold, a Poisson structure on M is a bilinear map {·, ·} : C∞(M) × C∞(M) → C∞(M) which is skew-symmetric and satisfies both the Jacobi identity and the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It is possible to express {f, g} in terms of a bivector field via the following equality {f, g} = Π(df ∧ dg) with Π a section of Λ2(T M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Π is the associated Poisson bivector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will use indistinctively the terminology of Poisson structure when referring to the bracket or the Poisson bivector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A b-Poisson bivector field on a manifold M 2n is a Poisson bivector such that the map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1) F : M → 2n � T M : p �→ (Π(p))n is transverse to the zero section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, a pair (M, Π) is called a b-Poisson man- ifold and the vanishing set Z of F is called the critical hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that Z is an embedded hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This class of Poisson structures was studied by Radko [Rad02] in dimension two and considered in numerous papers in the last years: [GMP10], [GMP14], [GMPS15b], [GMW17], [MOT14] and [GLPR17] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' b-Poisson manifolds Next, we recall classification theorem of b-Poisson surfaces as presented by Olga Radko and the cohomological re-statement and proof given by Guillemin, Miranda and Pires in [GMP14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In what follows, (M, Π) will be a closed smooth surface with a b-Poisson struc- ture on it, and Z its critical hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let h be the distance function to Z as in [MOT14]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Liouville volume of (M, Π) is the following limit: V (Π) := limǫ→0 � |h|>ǫ ωn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The previous limit exists and it is independent of the choice of the defining function h of Z (see [Rad02] for the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For any (M, Π) oriented Poisson manifold, let Ω be a volume form on it, and let uf denote the Hamiltonian vector field of a smooth function 1Notice the difference with [Rad02] where h is assumed to be a global defining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2For surfaces n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 7 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS f : M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The modular vector field XΩ is the derivation defined as follows: f �→ Luf Ω Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Given γ a connected component of the critical set Z(Π) of a closed b-Poisson manifold (M, Π), the modular period of Π around γ is defined as: Tγ(Π) := period of XΩ|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The modular vector field XΩ of the b-Poisson manifold (M, Z) does not depend at Z on the choice of Ω because for different choices for volume form the difference of modular vector fields is a Hamiltonian vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that this Hamiltonian vector field vanishes on the critical set as Π vanishes there too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let Mn(M) = Cn(M)/ ∼ where Cn(M) is the space of dis- joint oriented curves and ∼ identifies two sets of curves if there is an orientation- preserving diffeomorphism mapping the first one to the second one and preserving the orientations of the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The following theorem classifies b-symplectic structures on surfaces using these invariants: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6 (Radko [Rad02]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider two b-Poisson structures Π, Π′ on a closed orientable surface M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Denote its critical hypersurfaces by Z and Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' These two b-Poisson structures are globally equivalent (there exists a global orientation preserving diffeomorphism sending Π to Π′) if and only if the following coincide: the equivalence classes of [Z] and [Z′] ∈ Mn(M), their modular periods around the connected components of Z and Z′, their Liouville volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' An appropriate formalism to deal with these structures was introduced in [GMP10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A b-manifold3 is a pair (M, Z) of a manifold and an embed- ded hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this way, the concept of b-manifold previously introduced by Melrose is generalized to consider additional geometric structures on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A b-vector field on a b-manifold (M, Z) is a vector field tangent to the hypersurface Z at every point p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A b-map from (M, Z) to (M ′, Z′) is a smooth map φ : M → M ′ such that φ−1(Z′) = Z and φ is transverse to Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that if x is a local defining function for Z and (x, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , xn−1) are local coordinates in a neighborhood of p ∈ Z then the C∞(M)-module of b-vector fields has the following local basis (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) {x ∂ ∂x, ∂ ∂x1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ∂ ∂xn−1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 3The ‘b’ of b-manifolds stands for ‘boundary’, as initially considered by Melrose (Chapter 2 of [Mel93b]) for the study of pseudo-differential operators on manifolds with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' b-POISSON MANIFOLDS 9 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Artistic representation of a b-function on a b-manifold near the critical hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In contrast to [GMP10], in this monograph we are not requiring the existence of a global defining function for Z and orientability of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' However, we require the existence of a defining function in a neighborhood of each point of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By relaxing this condition, the normal bundle of Z need not be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Given (M, Z) a b-manifold, [GMP10] shows that there exists a vector bundle, denoted by bT M whose smooth sections are b-vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This bundle is called the b-tangent bundle of (M, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The b-cotangent bundle bT ∗M is defined using duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A b-form is a section of the b-cotangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Around a point p ∈ Z the C∞(M)-module of these sections has the following local basis: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3) { 1 xdx, dx1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , dxn−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the same way we define a b-form of degree k to be a section of the bundle �k(bT ∗M), the set of these forms is denoted bΩk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Denoting by f the distance function4 to the critical hypersurface Z, we may write the following decomposition as in [GMP10] for any ω ∈b Ωk(M) : (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4) ω = α ∧ df f + β, with α ∈ Ωk−1(M) and β ∈ Ωk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This decomposition allows to extend the differential of the de Rham complex d to bΩ(M) by setting dω = dα ∧ df f + dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Degree 0 functions are called b-functions and and near Z can be written as c log |x| + g, where c ∈ R, g ∈ C∞, and x is a local defining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The associated cohomology is called b-cohomology and it is denoted by bH∗(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A b-symplectic form on a b-manifold (M 2n, Z) is defined as a non-degenerate closed b-form of degree 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=', ωp is of maximal rank as an element of Λ2( bT ∗ p M) for all p ∈ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The notion of b-symplectic forms is dual to the notion of b-Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The advantage of using forms rather than bivector fields is that symplectic tools can be ‘easily’ exported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4Originally in [GMP10] f stands for a global function, but for non-orientable manifolds we may use the distance function instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS Radko’s classification theorem [Rad02] can be translated into this language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This translation was already formulated in [GMP10]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='11 (Radko’s theorem in b-cohomological language, [GMP14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let S be a closed orientable surface and let ω0 and ω1 be two b-symplectic forms on (S, Z) defining the same b-cohomology class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=',[ω0] = [ω1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then there exists a diffeomorphism φ : S → S such that φ∗ω1 = ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On bm-Symplectic manifolds 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By relaxing the transversality condition allowing higher order singularities ([Arn89] and [AA81]) we may consider other symplectic structures with singularities as done by Scott [Sco16] with bm-symplectic struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let m be a positive integer a bm-manifold is a b-manifold (M, Z) together with a bm-tangent bundle attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The bm-tangent bundle is (by Serre-Swan theorem [Swa62]) a vector bundle, bmT M whose sections are given by, Γ(bmT M) = {v ∈ Γ(T M) : v(x) vanishes to order m at Z}, where x is a defining function for the critical set Z in a neighborhood of each connected component of Z and can be defined as x : M \\ Z → (0, ∞), x ∈ C∞(M) such that: x(p) = d(p) a distance function from p to Z for p : d(p) ≤ 1/2 x(p) = 1 on M \\ {p ∈ M such that d(p) < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5 (This definition of x allows us to extend the construction in [Sco16] to the non- orientable case as in [MOT14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=') We may define the notion of a bm-map as a map in this category (see [Sco16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The sections of this bundle are referred to as bm-vector fields and their flows define bm-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In local coordinates, the sections of the bm-tangent bundle are generated by: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5) {xm ∂ ∂x, ∂ ∂x1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ∂ ∂xn−1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proceeding mutatis mutandis as in the b-case one defines the bm-cotangent bundle (bmT ∗M), the bm-de Rham complex and the bm-symplectic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A Laurent Series of a closed bm-form ω is a decomposition of ω in a tubular neighborhood U of Z of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6) ω = dx xm ∧ ( m−1 � i=0 π∗(αi)xi) + β with π : U → Z the projection of the tubular neighborhood onto Z, αi a closed smooth de Rham form on Z and β a de Rham form on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In [Sco16] it is proved that in a neighborhood of Z, every closed bm-form ω can be written in a Laurent form of type (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6) having fixed a (semi)local defining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' bm-Cohomology is related to de Rham cohomology via the following theorem: 5Then a bm-manifold will be a triple (M, Z, x), but for the sake of simplicity we refer to it as a pair (M, Z) and we tacitly assume that the function x is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ON bm-SYMPLECTIC MANIFOLDS 11 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='12 (bm-Mazzeo-Melrose, [Sco16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M, Z) be a bm-manifold, then: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='7) bmHp(M) ∼= Hp(M) ⊕ (Hp−1(Z))m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The isomorphism constructed in the proof of the theorem above is non-canonical (see [Sco16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Moser path method can be generalized to bm-symplectic structures (see [MS21] for the generalization from surfaces in [Sco16] to general manifolds): Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='13 (Moser path method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ωt be a path of bm-symplectic forms defining the same bm-cohomology class [ωt] on (M 2n, Z) with M 2n closed and orientable then there exist a bm-symplectomorphism ϕ : (M 2n, Z) −→ (M 2n, Z) such that ϕ∗(ω1) = ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' An outstanding consequence of Moser path method is a global classification of closed orientable bm-symplectic surfaces `a la Radko in terms of bm-cohomology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='14 (Classification of closed orientable bm-surfaces, [Sco16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ω0 and ω1 be two bm-symplectic forms on a closed orientable connected bm- surface (S, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, the following conditions are equivalent: their bm-cohomology classes coincide [ω0] = [ω1], the surfaces are globally bm-symplectomorphic, the Liouville volumes of ω0 and ω1 and the numbers � γ αi for all connected components γ ⊆ Z and all 1 ≤ i ≤ m coincide (where αi are the one-forms appearing in the Laurent decomposition of the two bm-forms of degree 2, ω0 and ω1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The numbers [αi] = � γ αi are called modular weights for the connected components γ ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A relative version of Moser’s path method is proved in [GMW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As a corol- lary we obtain the following local description of a bm-symplectic manifold: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='16 (bm-Darboux theorem, [GMW17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ω be a bm-symplectic form on (M, Z) and p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we can find a coordinate chart (U, x1, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , xn, yn) centered at p such that on U the hypersurface Z is locally defined by x1 = 0 and ω = dx1 xm 1 ∧ dy1 + n � i=2 dxi ∧ dyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For the sake of simplicity sometimes we will omit any explicit reference to the critical set Z and we will talk directly about bm-symplectic struc- tures on manifolds M implicitly assuming that Z is the vanishing locus of Πn where Π is the Poisson vector field dual to the bm-symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Next, we present two lemmas that allow us to talk about bm-symplectic struc- tures and bm-Poisson as two different presentations of the same geometrical struc- ture on a b-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The lemma below shows that they are dual to each other and, thus, in one-to-one correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A PRIMER ON SINGULAR SYMPLECTIC MANIFOLDS Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ω be a bm-symplectic and Π its dual vector field, then Π is a bm-Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The quickest way to do this is to take the inverse, which is a bivector field, and observe that it is a Poisson structure (because dω = 0 implies [Π, Π] = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To see that it is bm-Poisson it is enough to check it locally for any point along the critical set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Take a point p on the critical set Z and apply the bm-Darboux theorem to get ω = dx1/xm 1 ∧ dy1 + � i>1 dxi ∧ dyi This means that in the new coordinate system Π = xm 1 ∂ ∂x1 ∧ ∂ ∂y1 + � i>1 ∂ ∂xi ∧ ∂ ∂yi and thus Π is a bm-Poisson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Conversely, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let Π be bm-Poisson and ω its dual vector field, then ω is a bm-symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If Π transverse `a la Thom on Z with singularity of order m then because of Weinstein’s splitting theorem we can locally write Π = xm 1 ∂ ∂x1 ∧ ∂ ∂y1 + � i>1 ∂ ∂xi ∧ ∂ ∂yi now its inverse is ω = dx1/xm 1 ∧ dy1 + � i>1 dxi ∧ dyi which is a bm-symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Hence we have a correspondence from bm-symplectic structures to bm-Poisson structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Desingularizing bm-Poisson manifolds In [GMW17] Guillemin, Miranda and Weitsman presented a desingularization procedure for bm-symplectic manifolds proving that we may associate a family of folded symplectic or symplectic forms to a given bm-symplectic structure depending on the parity of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Namely, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='20 (Guillemin-Miranda-Weitsman, [GMW17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ω be a bm-symplectic structure on a closed orientable manifold M and let Z be its critical hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If m = 2k, there exists a family of symplectic forms ωǫ which coincide with the bm-symplectic form ω outside an ǫ-neighborhood of Z and for which the family of bivector fields (ωǫ)−1 converges in the C2k−1-topology to the Poisson structure ω−1 as ǫ → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If m = 2k + 1, there exists a family of folded symplectic forms ωǫ which coincide with the bm-symplectic form ω outside an ǫ-neighborhood of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='20, any closed orientable manifold that supports a b2k-symplectic structure necessarily supports a symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In [GMW17] explicit formulae are given for even and odd cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us refer here to the even-dimensional case as these formulae will be used later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' DESINGULARIZING BM-POISSON MANIFOLDS 13 Let us briefly recall how the desingularization is defined and the main result in [GMW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall that we can express the b2k-form as: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8) ω = dx x2k ∧ �2k−1 � i=0 xiαi � + β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This expression holds on a ǫ-tubular neighborhood of a given connected com- ponent of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This expression comes directly from equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6, to see a proof of this result we refer to [Sco16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (S, Z, x), be a b2k-manifold, where S is a closed orientable manifold and let ω be a b2k-symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider the decomposition given by the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8) on an ǫ-tubular neighborhood Uǫ of a connected component of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let f ∈ C∞(R) be an odd smooth function satisfying f ′(x) > 0 for all x ∈ [−1, 1] and satisfying outside that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='9) f(x) = � −1 (2k−1)x2k−1 − 2 for x < −1, −1 (2k−1)x2k−1 + 2 for x > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let fǫ(x) be defined as ǫ−(2k−1)f(x/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The fǫ-desingularization ωǫ is a form that is defined on Uǫ by the following expression: ωǫ = dfǫ ∧ �2k−1 � i=0 xiαi � + β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This desingularization procedure is also known as deblogging in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Though there are infinitely many choices for f, we will assume that we choose one, and assume it fixed through the rest of the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It would be interesting to discuss the existence of an isotopy of forms under a change of function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because ωǫ can be trivially extended to the whole S in such a way that it agrees with ω (see [GMW17]) outside a neighborhood of Z, we can talk about the fǫ-desingularization of ω as a form on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' CHAPTER 3 A crash course on KAM theory The last part of this monograph is entirely dedicated to prove a KAM theorem for bm-symplectic structures and to find applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' So the aim of this section is to give a quick overview of the traditional KAM theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The setting of the KAM theorem is a symplectic manifold with action-angle coordinates and an integrable system in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The theorem says that under small perturbations of the Hamiltonian ”most” of the Liouville tori survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider Tn×G ⊂ Tn×Rn with action-angle coordinates in it (φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , φn, I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , In) and the standard symplectic form ω in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And assume the Hamiltonian function of the system is given by h(I) a function only depending on the action coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then the Hamilton equations of the system are given by ιXhω = dh where Xh is the vector field generating the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because h does not de- pend on φ the angular variables the system is really easy to solve, and the equations are given by x(t) = (φ(t), I(t)) = (φ0 + ut, I0), where u = ∂h/∂I is called the frequency vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' These motions for a fixed initial condition are inside a Liouville torus, and are called quasi-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The KAM theorem studies what happens to such systems when a small per- turbation is applied to the Hamiltonian function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' we consider the evolution of the system given by the Hamiltonian h(I) + R(I, φ), where we think of the term R(I, φ) as the small perturbation in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this in mind, the Hamilton equations can be written as ˙φ = u(I) + ∂ ∂I R(I, φ), ˙I = − ∂ ∂φR(I, φ), Another important concept to have in mind is the concept of rational depen- dency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A frequency u is rationally dependent if ⟨u, k⟩ = 0 for some k ∈ Zn, if there exists no k satisfying the condition then the vector u is called rationally indepen- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' There is a stronger concept of being rationally independent and that is the concept of being Diophantine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A vector u is γ,τ-diophantine if ⟨u, k⟩ ≥ γ |k|τ 1 for all k ∈ Zn \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' γ > 0 and τ > n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The KAM theorem states that the Liouville tori with frequency vector satisfying the diophantine condition survive under the small perturbation R(I, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' There are conditions relating the size of the perturbation with γ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also, the set of tori satisfying the Diophantine condition has measure 1 − Cγ for some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Now we give a proper statement of the theorem as was given in [DG96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 15 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A CRASH COURSE ON KAM THEORY Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1 (Isoenergetic KAM theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let G ⊂ Rn, n > 2, a compact, and let H(φ, I) = h(I) + f(φ, I) real analytic on Dρ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ω = ∂h/∂I, and assume the bounds: ���� ∂2h ∂I2 ���� G,ρ2 ≤ M, |ω|G ≤ L and |ωn(I)| ≥ l∀I ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume also that ω is µ-isoenergetically non-degenerate on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For a = 16M/l2, assume that the map Ω = Ωω,h,a is one-to-one on G, and that its range F = Ω(G) is a D-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let τ > n − 1, γ > 0 and 0 < ν < 1 given, and assume: ε := ∥f∥G,ρ ≤ ν2l6µ2ˆρ2τ+2 24τ+32L6M 3 · γ2, γ ≤ min �8LMρ2 νlˆρτ+1 , l � , where we write ρ := min � νρ1 12(τ+2), 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Define the set ˆG = ˆGγ := � I ∈ G − 2γ µ : ω(I)isτ, γ − Diophantine � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, there exists a real continuous map T : W ρ1 4 (Tn) × ˆG → Dρ(G), analytic with respect to the angular variables, such that: (1) For every I ∈ ˆG, the set T (Tn × {I}) is an invariant torus of H, its frequency vector is colinear to ω(I) and its energy is h(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (2) Writing T (φ, I) = (φ + Tφ(φ, I), I + TI(φ, I)), one has the estimates |Tφ| ˆ G,( ρ1 4 ,0),∞ ≤ 22τ+15L2M ν2l2ˆρ2τ+1 ε γ2 , |TI| ˆ G,( ρ1 4 ,0) ≤ 2τ+16L3M νl3µˆρτ+1 ε γ (3) meas[(Tn ×G)\\T (Tn × ˆG)] ≤ Cγ, where C is a very complicated constant depending on n, τ, diamF, D, ˆρ, M, L, l, µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This version of the KAM theorem is the isoenergetic one, this version ensures that the energy of the Liouville Tori identified by the diffeomorphism after the perturbation remains the same as before the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Our version of the bm-KAM is not isoenergetic for the sake of simplifying the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also, we should outline that the KAM theorem has already been explored in singular symplectic manifolds before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In [KMS16a] the authors proved a KAM theorem for b-symplectic manifolds, for a particular kind of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3 (KAM Theorem for b-Poisson manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let Tn × Bn r be en- dowed with standard coordinates (ϕ, y) and the b-symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider a b-function H = k log |y1| + h(y) on this manifold, where h is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let y0 be a point in Bn r with first component equal to zero, so that the corresponding level set Tn × {y0} lies inside the critical hypersurface Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume that the frequency map ˜ω : Bn r → Rn−1, ˜ω(y) := ∂h ∂˜y (y) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A CRASH COURSE ON KAM THEORY 17 has a Diophantine value ˜ω := ˜ω(y0) at y0 ∈ Bn and that it is non-degenerate at y0 in the sense that the Jacobian ∂˜ω ∂˜y (y0) is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then the torus Tn × {y0} persists under sufficiently small perturbations of H which have the form mentioned above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' they are given by ǫP, where ǫ ∈ R and P ∈b C∞(Tn × Bn r ) has the form P(ϕ, y) = k′ log |y1| + f(ϕ, y) f(ϕ, y) = f1( ˜ϕ, y) + y1f2(ϕ, y) + f3(ϕ1, y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' More precisely, if |ǫ| is sufficiently small, then the perturbed system Hǫ = H + ǫP admits an invariant torus T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Moreover, there exists a diffeomorphism Tn → T close1 to the identity taking the flow γt of the perturbed system on T to the linear flow on Tn with frequency vector �k + ǫk′ c , ˜ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 1By saying that the diffeomorphism is “ǫ-close to the identity” we mean that, for given H, P and r, there is a constant C such that ∥ψ − Id∥ < Cǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Part 2 Action-angle coordinates and cotangent models In this part, we consider the semilocal classification for any bm-Poisson manifold in a neighbourhood of an invariant compact submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The compact subman- ifolds under consideration are the compact invariant leaves of the distribution D generated by the Hamiltonian vector fields Xfi of an integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' An in- tegrable system is given by a set of n functions on a 2n-dimensional symplectic manifold which we can order in a map F = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Historically, integrable systems were introduced to actually integrate Hamiltonian systems XH using the first-integrals fi and, classically, we identify H = f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It turns out that in the sym- plectic context the compact regular orbits of the distribution D coincide with the fibers F −1(F(p)) for any point p on these orbits/fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The fact that the orbit coincides with the connected fiber is part of the magic of symplectic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The same picture is reproduced for singular symplectic manifolds of bm-type or bm-Poisson manifolds as we will see in this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The study of action-angle coordinates has interest from this geometrical point of view of the classification of geometric structures in a neighbourhood of a compact submanifold of a bm-Poisson manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It also has interest from a dynamical point of view as these compact submanifolds now coincide with invariant subsets of the Hamiltonian system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' From a geometric point of view, the existence of action-angle coordinates deter- mines a unique geometrical model for the bm-Poisson (or bm-symplectic) structure in a neighbourhood of the invariant set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' From a dynamical point of view, the exis- tence of action-angle coordinates provides a normal form theorem that can be used to study stability and perturbation problems of the Hamiltonian systems (as we will see in the last chapter of this monograph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' An important ingredient that makes our action-angle coordinate theorem brand- new from the symplectic perspective is that the system under consideration is more general than Hamiltonian, it is bm-Hamiltonian as the first-integrals of the system can be bm-functions which are not necessarily smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Dynamically, this means that we are adding to the set of Hamiltonian invariant vector fields, the modular vector field of the integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In contrast to the standard action-angle coordinates for symplectic manifolds, our action-angle theorem comes with m additional invariants associated with the modular vector field which can be interpreted in cohomological terms as the pro- jection of the bm-cohomology class determined by the modular vector field on the first cohomology group of the critical hypersurface under the Mazzeo-Melrose cor- respondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The strategy of the proof of the action-angle coordinate systems is the search of a toric action (so this takes us back to the motivation of the use of symmetries in this monograph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In contrast to the symplectic case, it is not enough that this action is Hamiltonian as then a direction of the Liouville torus would be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We need the toric action to be bm-Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The structure of this proof looks like the one in [KMS16a] but encounters serious technical difficulties as in order to check that the natural action to be considered is bm-Hamiltonian we need to go deeper inspired by [Sco16] in the relation between the geometry of the modular vector field and the coefficients of the Taylor series ci of one of the first-integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This allows us to understand new connections between the geometry and analysis of bm-Poisson structures not explored before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 21 Once we prove the existence of this bm-Hamiltonian action the proof looks very close to the one in [KMS16a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the second chapter of this part we re-state the action-angle theorem as a cotangent lift theorem with the following mantra: Every integrable system on a bm-Poisson manifold looks like a bm-cotangent lift in a neighborhood of a Liouville torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' CHAPTER 4 An action-angle theorem for bm-symplectic manifolds 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Basic definitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On bm-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The definition of the analogue of b-functions in the bm-setting is somewhat delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The set of bmC∞(M) needs to be such that for all the functions f ∈bm C∞(M), its differential df is a b-form, where d is the bm-exterior differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall that a form in bmΩk(M) can be locally written as α ∧ dx xm + β where α ∈ Ωk−1(M) and β ∈ Ωk(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall also that d � α ∧ dx xm + β � = dα ∧ dx xm + dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We need df to be a well-defined bm-form of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let f = g 1 xk−1 , then df = dg 1 xk−1 − g k−1 xk dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This from can only be a bm-form if and only if g only depends on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If f = g log(x), then dg log(x) + g 1 xdx, which imposes dg = 0 and hence g to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With all this in mind, we make the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The set of bm-functions is defined recursively according to the formula bmC∞(M) = x−(m−1)C∞(x) + bm−1C∞(M) with C∞(x) the set of smooth functions in the defining function x and bC∞(M) = {g log |x| + h, g ∈ R, h ∈ C∞(M)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A bmC∞(M)-function can be written as f = a0 log x + a1 1 x + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' + am−1 1 xm−1 + h where ai, h ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' From this chapter on we are only considering bm-manifolds (M, x, Z) with x defined up to order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' we can think of x as a jet of a function that coincides up to order m to some defining function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is the orig- inal viewpoint of Scott in [Sco16] which we adopt from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The difference with respect to the other chapters is that we do not fix an specific function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We say that two bm-integrable systems F1, F2 are equivalent if there exists ϕ, a bm-symplectomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' a diffeomorphism preserving both ω and the critical set Z (“up to order m”1), such that ϕ ◦ F1 = F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 1I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' it preserves the jet x 23 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Hamiltonian vector field associated to a bm-function f is a smooth vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us compute it locally using the bm-Darboux theorem: Π = xm 1 ∂ ∂x1 ∧ ∂ ∂y1 + m � i=2 ∂ ∂xi ∧ ∂ ∂yi and f = a0 log x1 + m−1 � i=1 ai 1 xi 1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then if we compute df = c1 ���� a0 1 x1 + m−1 � i=1 ci � �� � (a′ i − (i − 1)ai−1) 1 xi 1 dx1 − cm � �� � (m − 1)am−1 1 xm 1 dx1 + dh = m � i=1 ci xi 1 dx1 + dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1) Xf = Π(df, ·) = m � i=1 cixm−i 1 ∂ ∂y1 + Π(dh, ·), we obtain a smooth vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On bm-integrable systems In this section we present the definition of a bm-integrable system as well as some observations about these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M 2n, Z, x) be a bm-manifold, and let Π be a bm-Poisson structure on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' F = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn)2 is a bm-integrable system3 if: (1) df1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , dfn are independent on a dense subset of M and in all the points of Z where independent means that the form df1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∧ dfn is non-zero as a section of Λn(bmT ∗(M)), (2) the functions f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn Poisson commute pairwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The points of M where df1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , dfn are independent are called regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The next remarks will lead us to a normal form for the first function f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that df1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , dfn are independent on a point if and only if Xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn are independent at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is because the map bmT M →bm T ∗M : u �→ ωp(u, ·) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The condition of df1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , dfn being independent must be under- stood as df1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∧ dfn being a non-zero section of �n( bmT ∗M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2fi are bm-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 3In this monograph we only consider integrable systems of maximal rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ON bm-INTEGRABLE SYSTEMS 25 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8 the vector fields Xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn have to be in- dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This implies that one of the f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn has to be a singular (non- smooth) bm-function with a singularity of maximal degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If we write fi = c0,i log(x1) + �m−1 j=1 cj,i xj 1 + ˜f1 Xfi = m � j=1 xm−j 1 ˆcj,i ∂ ∂y1 + X ˜ fi where ˆcj,i(x) = d(cj,i) dx − (j − 1)cj−1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If there is no bm-function with a singularity of maximum degree all the terms in the ∂/∂y1 direction become 0 at Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And hence Xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn cannot have maximal rank at Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let F = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) a bm-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If f1 has a singular- ity of maximal degree, there exists an equivalent integrable system F ′ = (f ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , f ′ n) where f ′ 1 has a singularity of maximal degree and no other f ′ i has singularity of any degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let fi = c0,i log(x1) + m−1 � j=1 cj,1 xj 1 � �� � ζi(x1) + ˜fi = ζi(x1) + ˜fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='104, Xfi = m � i=1 xm−j 1 ˆcj,i � �� � gi(x1) ∂ ∂y1 + X ˜ fi = gi(x1) ∂ ∂y1 + X ˜ fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that gi(x1) = gi(0) = ˆcm,i at Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us look at the distribution given by the Hamiltonian vector fields Xfi = gi(x1) ∂ ∂y1 +X ˜ fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This distribution is the same that the one given by: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) {Xf1, Xf2 − g2(x1) g1(x1)Xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn − gn(x1) g1(x1) Xf1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that for i > 1, Xfi − gi(x1) g1(x1)Xf1 = X ˜ fi + g2(x1) g1(x1)X ˜ f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also g1(x1) is different from 0 close to Z because at Z g1(x1) = ˆcm,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Since the distribution given by these vector fields is the same, an integrable system that has Hamiltonian vector fields 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2 would be equivalent to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' From the expression 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2 it is clear that the new vector fields commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And it is also true that this new vector fields are Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us take F ′ the set of functions that have as Hamiltonian vector fields 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ From now on we will assume the integrable system to have only one singular function and this function to be f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because we asked Xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn to be linearly independent at all the points of Z and using the previous remarks cm := cm,1 ̸= 0 at all the points of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4Here have used the bm-Darboux theorem to do the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS Furthermore, we can assume f1 to have a smooth part equal to zero as sub- tracting the smooth part of f1 to all the functions gives an equivalent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also, we can assume that cm is 1 because dividing all the functions of the bm-integrable system by cm also gives us an equivalent system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As a summary, we can assume f1 = a0 log(x)+a11/x+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='+am−21/xm−2+ 1/xm−1 and f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn to be smooth, a0 ∈ R and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , am−2 ∈ C∞(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also we are going to state lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2 in [GMPS17], because we are going to use it later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The result states that if we have a toric action on a bm- symplectic manifold (which we will prove in a neighbourhood of a Liouville torus), then we can assume the coefficients a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , am−2 to be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' More precisely Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' There exists a neighborhood of the critical set U = Z × (−ε, ε) where the moment map µ : M → t∗ is given by µ = a1 log |x| + m � i=2 ai x−(i−1) i − 1 + µ0 with ai ∈ t0 L and µ0 is the moment map for the TL-action on the symplectic leaves of the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Examples of bm-integrable systems The following example illustrates why it is necessary to use the definition of bm- function as considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' There are natural examples of changes of coordinates in standard integrable systems on symplectic manifolds that yield bm-symplectic manifolds but do not give well-defined bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider a time change in the two body problem, to obtain a b2-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the classical approach to solve the 2-body problem the following two conserved quantities are obtained: f1 = µy2 2 + l2 2µr2 − k r , f2 = l, with symplectic form ω = dr ∧ dy + dl ∧ dα, where r is the distance between the two masses and l is the angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We also know that l is constant along the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because l is a constant of the movement, we can do a symplectic reduction on its level sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The form on the symplectic reduction becomes dr ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To simplify the notation, we will use x instead of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then ω = dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With hamiltonian function given by f = µ 2 y2 + l 2µ 1 x2 − k 1 x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, the equations are: ˙x = ∂f ∂y , ˙y = − ∂f ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Doing a time change τ = x3t then dx dτ = 1 x3 dx dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this time coordinate, the equations become: ˙x = 1 x3 ∂f ∂y , ˙y = − 1 x3 ∂f ∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' These equations can be viewed as the motion equations given by a b3-symplectic form ω = 1 x3 dx ∧ dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us check that this is actually a bm-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' EXAMPLES OF bm-INTEGRABLE SYSTEMS 27 All the functions Poisson commute is immediate because we only have one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' df = µydy +( k x2 − l µ 1 x3 )dx is a b3-form because the term with dx does not depend on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' All the functions are independent, this is true because df does not vanish as a b3-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the paper [Mar19] the author builds an action of SL(2, R) over (P, ωP ) where P = {ξ ∈ C|i(¯ξ − ξ) > 0} is the complex semi-plane, with moment map JP (ξ) = R ξim ((|ξ|2 + 1), 2ξr, ±(|ξ|2 + 1)), where the ± sign depends on the choice of the hemisphere projected by the stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' From now on we will take the sign +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also the symplectic form ωP has the following expression: ωP = ± R ξ2 im dξr ∧ dξim In order to simplify the notation we identify P with the real half-plane P = {x, y ∈ R2|y > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this identification, the moment map becomes Jp(x, y) = R y (x2 + y2 + 1, 2x, x2 + y2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Obviously, this moment map does not give an integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The symplectic form writes as: ωP = R y2 dy ∧ dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This form can be viewed as a b2-form if we extend P including the line {y = 0} as its singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us consider only one of the components of JP as bm- function and let us see if it gives a bm-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' First we will try with f1 = R y (x2 + y2 + 1) and then f2 = R y (2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (1) f1 = R y (x2 + y2 + 1) We have to check three things to see if this gives a b2-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (a) All the functions Poisson commute is immediate because we only have one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (b) All the functions are bm-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This point does not hold because df1 = R y2 (2xydx + (y2 − x2 − 1)dy) and the first component makes no sense as a section of Λ1(b2T ∗M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (c) All the functions are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this case, we need to check that df1 does not vanish, but since it is not a bm-form it makes no sense to be a non-zero section of Λ1(b2T ∗M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (2) f2 = R y (2x) (a) Same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (b) All the functions are bm-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This point does not hold because df2 = 2R y dx − 2Rx y2 dy and the first component makes no sense as a section of Λ1(b2T ∗M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (c) Same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Toric actions give natural examples of integrable systems where the component functions are given by the moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the case of surfaces: S1-actions on surfaces give natural examples of bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Only torus and spheres admit circle actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS In the picture below two integrable systems on the 2-sphere depending on the degree m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the right the image of the moment map that defines the integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The action is by rotations along the central axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Namely consider the sphere S2 as a bm-symplectic manifold having as critical set the equator: (S2, Z = {h = 0}, ω = dh hm ∧ dθ), with h ∈ [−1, 1] and θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For m = 1: The computation ι ∂ ∂θ ω = − dh h = −d(log |h|), tells us that the function µ(h, θ) = log |h| is the moment map and defines a b-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For higher values of m: ι ∂ ∂θ ω = − dh hm = −d(− 1 (m−1)hm−1 ), and the moment map is µ(h, θ) = − 1 (m−1)hm−1 which defines a bm-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' µ, m = 1 µ, m = 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Integrable systems associated to the moment map of an S1-action by rotations on a bm-symplectic 2-sphere S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider now as b2-symplectic manifold the 2-torus (T2, Z = {θ1 ∈ {0, π}}, ω = dθ1 sin2 θ1 ∧ dθ2) with standard coordinates: θ1, θ2 ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that the critical hypersurface Z in this example is not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It is the union of two disjoint circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider the circle action of rotation on the θ2-coordinate with fundamental vector field ∂ ∂θ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' As the following computation holds, ι ∂ ∂θ2 ω = − dθ1 sin2 θ1 = d �cos θ1 sin θ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The fundamental vector field of the S1-action defines b2C∞-integrable system given by the function − cos θ1 sin θ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The former example can be made general to produce examples of bm-integrable systems on a bm-symplectic manifold for any integer m (T2, Z = {θ1 ∈ {0, π}}, ω = dθ1 sinm θ1 ∧ dθ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then ι ∂ ∂θ2 ω = − dθ1 sinm θ1 = d � | cos θ1| cos θ1 2F1 � 1 2, 1−m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 3−m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' sin2(θ1) � (1 − m) sinm−1 θ1 � , with 2F1 the hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' LOOKING FOR A TORIC ACTION 29 µ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Integrable system given by an S1-action on a b2-torus T2 and its associated moment map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Thus, the associated S1-action has as bmC∞-Hamiltonian the function −| cos θ1| cos θ1 2F1 � 1 2, 1−m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 3−m 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' sin2(θ1) � (1 − m) sinm−1 θ1 which defines a bm-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Now we give a couple of examples of bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This example uses the product of bm-integrable systems on a bm-symplectic manifold with an integrable system on a symplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Given (M 2n1 1 , Z, x, ω1) a bm-symplectic manifold with f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn1 a bm-integrable system and (M 2n2 2 , ω2) a symplectic manifold with g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , gn2 an integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then (M1×M2, Z×M2, x, ω1+ω2) is a bm-symplectic manifold and (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn1, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , gn2) is a bm-integrable system on the higher dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In particular by combining the former examples of bm-integrable systems on surfaces and arbitrary integrable systems on symplectic manifolds we obtain exam- ples of bm-integrable systems in any dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (From integrable systems on cosymplectic manifolds to bm-integrable systems:) Using the extension theorem (Theorem 50) of [GMP14] we can extend any integrable system (f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) to an integrable system in a neighbourhood of a cosymplectic manifold (Z, α, ω) by just adding a bm-function f1 to the integrable system so that the new integrable system is (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) and considering the associated bm-symplectic form: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3) ˜ω = p∗α ∧ dt tm + p∗ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (t is the defining function of Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Looking for a toric action In this section we pursue the proof of action-angle coordinates for bm-integrable systems by recovering a torus group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This action is associated to the Hamil- tonian vector fields associated to Xfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is the same strategy used for b-integrable systems in [KMS16a]- One of the main difficulties is to prove that the coefficients a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , an can be considered 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS as constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This makes it more difficult to prove the existence of a Tn- action in the general bm-case than in the b-case, but once we have it we can use the results in [GMW17] to assume that the coefficients a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , an are constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this section we provide some preliminary material that will be needed later: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M, Z, x, ω) be a bm-symplectic manifold such that Z is connected with modular period k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let π : Z → S1 ≃ R/kZ be the projection to the base of the corresponding mapping torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let γ : S1 = R/kZ → Z be any loop such that π ◦ γ is positively oriented and has constant velocity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then the following are equal: (1) The modular period of Z, (2) � γ ιLω, (3) The value am−1 for any bmC∞(M)-function f = a0 log(x) + m−1 � j=1 aj 1 xj + h such that the hamiltonian vector field Xf has 1-periodic orbits homotopic in Z to some γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us first prove that (1)=(2) and then that (2)=(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (1)=(2) Let us denote by Vmod the modular vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall from [GMW17] that ιL(Vmod) is the constant function 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let s : [0, k] → Z be the trajec- tory of the modular vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because the modular period is k, s(0) and s(k) are in the same leaf L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ˆs : [0, k + 1] → Z a smooth extension of s such that s|[k,k+1] is a path in L joining ˆs(k) = s(k) to ˆs(k+1) = s(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This way ˆs becomes a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, k = � k 0 1dt = � S ιLω = � ˆs ιLω = � γ ιLω (2)=(3) Let r : [0, 1] �→ Z be the trajectory of Xf the hamiltonian vector field of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall that Xf satisfies ιXf ω = m � j=1 cj dx xi + dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let xm ∂ ∂x be a generator of the linear normal bundle L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We know that Xf is 1-periodic and its trajectory is homotopic to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, k = � r ιLω = � 1 0 ιxm ∂ ∂x ω(Xf|r(t))dt = � 1 0 −( m � j=1 ci dx xi + dh) · (xm ∂ ∂x)|r(t)dt = −cm = −am−1 □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' LOOKING FOR A TORIC ACTION 31 We will also need a Darboux-Carath´eodory theorem for bm-symplectic mani- folds: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='22 (Darboux-Carath´eodory (bm-version)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M 2n, x, Z, ω) be a bm-symplectic manifold and m be a point on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn be a bm-integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then there exist bm-functions (q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , qn) around m such that ω = n � i=1 dfi ∧ dqi and the vector fields {Xfi, Xqj}i,j commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If f1 is not smooth (recall that f1 = a0 log(x) + �m−1 j=1 aj 1 xi with an ̸= 0 on Z and a0 ∈ R) the qi can be chosen to be smooth functions, and (x, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , qn) is a system of local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The first part of this proof is exactly as in [KMS16a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume now f1 = a0 log(x) + m−1 � j=1 aj 1 xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We modify the induction requiring also that µi (in addition to be in Ki) is also in T ∗M ⊆b T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We can also ask this extra condition while asking µi(Xfi) = 1, we only have to check that Xfi does not vanish in T M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is clear because Xfi does not vanish at bT M and 0 = {fn, fi} = � m � i=1 ˜ai dx xi � (Xfi) = � dx xm m � i=1 aixi � (Xfi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' All the terms in the last expression vanish except for the one of degree m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then dx/xm is in the kernel of Xfi, hence Xfi does not vanish on T M and the qi can be chosen to be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' {Xx, Xf2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn, Xq1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Xqn} commute because {Xfi, Xqi}i,j commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then dx ∧ df2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∧ dfn ∧ dq1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∧ dqn is a non-zero section of �n(bT M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And hence (x, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn−1, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , qn) are local coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Before proceeding with the proof of the action-angle coordinates, we need to prove that in a neighbourhood of a Liouville torus the fibration is semilocally trivial: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='23 (Topological Lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let m ∈ Z be a regular point of a bm- integrable system (M, x, Z, ω, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume that the integral manifold Fm through m is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then there exists a neighborhood U of Fm and a diffeomorphism φ : U ≃ Tn × Bn which takes the foliation F to the trivial foliation {Tn × {b}}b∈Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We follow the steps of [LGMV08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this case, the only extra step that must be checked is that the foliation given by the bm-hamiltonian vector fields of F = (f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) is the same as the one given by the level sets of ˜F := (x, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In our case f1 = a0 log(x) + �m−1 u=1 ai 1 xi , where a0 ∈ R, ai ∈ C∞(x), am−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence the foliations are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then as in [LGMV08], we take an 32 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Fibration by Liouville tori: The middle fiber of the point p ∈ Z in magenta, the neighbouring Liouville tori in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' arbitrary Riemannian metric on M and this defines a canonical projection ψ : U → Fm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us define φ := ψ × ˜F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We obtain the commutative diagram (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' U Tn × Bn Bn φ ˜ F p Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Commutative diagram of the construction of the iso- morphism of bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' which provides the necessary equivalence of bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Action-angle coordinates on bm-symplectic manifolds In a neighbourhood of one of our Liouville tori all we can assume about the form of our bm-symplectic structure is that is given by the Laurent series defined in [Sco16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' That is to say, we can assume that in a tubular neighborhood U of Z ω = m−1 � j=1 dx xi ∧ π∗(αi) + β, where π : U → Z is the projection of the tubular neighborhood onto Z, αi are closed smooth de Rham forms on Z and β a de Rham form on M of degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In [RBM, MM22] normal forms are given for group actions in a neighbour- hood of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Below we provide a normal for the integrable system in a neigh- bourhood of an orbit of the torus action associated to the integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This theorem is finer than the bm-symplectic slice theorem provided in [MM22] as it also gives information about the first integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' One of the non-trivial steps of the proof is to associate a toric action to the integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The connection to normal forms of group actions will become even more evident when we discuss the associated cotangent models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ACTION-ANGLE COORDINATES ON bm-SYMPLECTIC MANIFOLDS 33 Theorem A (Action-angle coordinates for bm-symplectic manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M, x, ω, F) be a bm-integrable system, where F = (f1 = a0 log(x) + �m−1 j=1 aj 1 xj , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) with aj for j > 1 functions in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let m ∈ Z be a regular point and let us assume that the integral manifold of the distribution generated by the Xfi through m is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let Fm be the Liouville torus through m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, there exists a neighborhood U of Fm and coordinates (θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , θn, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , σn) : U → Tn × Bn such that: (1) We can find an equivalent integrable system F = (f1 = a′ 0 log(x) + �m−1 j=1 a′ j 1 xj , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) such that the coefficients a′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , a′ m−1 of f1 are con- stants ∈ R, (2) ω|U = \uf8eb \uf8ed m � j=1 c′ j c σj 1 dσ1 ∧ dθ1 \uf8f6 \uf8f8 + n � i=2 dσi ∧ dθi where c is the modular period and c′ j = −(j − 1)a′ j−1, also (3) the coordinates σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , σn depend only on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The idea of this proof is to construct an equivalent bm-integrable system whose fundamental vector fields define a Tn-action on a neighborhood of Tn × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It is clear that all the vector fields Xf1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Xfn define a torus action on each Liouville tori Tn × {b} where b ∈ Bn, but this does not guarantee that their flow defines a toric action on all Tn × Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The proof is structured in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The first one is the uniformization of the periods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' we define an Rn-action on a neighborhood of Tn × {0} such that the lattice defined by its kernel at every point is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This allows to induce an actual action of a torus (as the periods are constant) of rank n: A Tn action by taking quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The second step consists in checking that this action is actually bm-Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And in the final step we apply theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='22 to obtain the expression of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (1) Uniformization of periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let Φs XF be defined as the joint flow by the Hamiltonian vector fields of the action: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4) Φ : Rn × (Tn × Bn) → (Tn × Bn) ((s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , sn), (x, b)) �→ Φs1 Xf1 ◦ · · · ◦ Φsn Xfn ((x, b)) this defines an Rn-action on Tn×Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For each b ∈ Bn at a single orbit Tn × {b} the kernel of this action is a discrete subgroup of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will denote the lattice given by this kernel Λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because the orbit is compact, the rank of Λb is maximal i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This lattice is known as the period lattice of Tn × {b} as we know by standard arguments in group theory that the lattice has to be of maximal rank so as to have a torus as a quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In general we can not assume that Λb does not depend on b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The process of uniformization of the periods modifies the action 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4 in such a way that Λb = Zn for all b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us consider the following Hamiltonian vector field �n i=1 kiXfi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The bm-function that generates this Hamiltonian vector field is: k1 \uf8eb \uf8eda0 log(x) + m−1 � j=1 aj 1 xj \uf8f6 \uf8f8 + n � i=2 kifi 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' AN ACTION-ANGLE THEOREM FOR bm-SYMPLECTIC MANIFOLDS where recall that am−1 is constant equal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that the coefficient multiplying 1/xm−1 is k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='21 k1 = c the modular period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this case c = [αm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, for b ∈ Bn−1 × {0} the lattice Λb is contained in Rn−1 × cZ ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Pick (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , λn) : Bn → Rn such that: (λ1(b), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , λn(b)) is a basis of Λb for all b ∈ Bn, λn i vanishes along Bn−1 × {0} at order m for i < n and λi is equal to c along Bn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the previous points, λj i denotes the j-th component of λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The first condition can be satisfied by using the implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' That is because Φ(λ, m) = m is regular with respect to the s coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The second condition is automatically true because Λb ⊆ Rn−1×cZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We define the uniformed flow as: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5) ˜Φ : Rn × (Tn × Bn) → (Tn × Bn) ((s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , sn), (x, b)) �→ Φ(�n i=1 siλi, (x, b)) (2) The Tn-action is bm-Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The objective of this step is to find bm-functions σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , σn such that Xσi are the fundamental vector fields of the Tn-action Yi = �n j=1 λj iXfj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By using the Cartan formula for a bm-symplectic form, we obtain: LYiLYiω = LYi(d(ιYiω) + ιYidω) = LYi(d(− �n j=1 λj idfi)) = −LYi(�n j=1 dλj i ∧ dfj) = 0 Note that λj i are constant on the level sets of F as Φ(λ, m) = m and the level sets of F are invariant by Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall that if Y is a complete periodic vector field and P is a bivector such that LY LY P = 0, then LY P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' So, the vector fields Yi are Poisson vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To show that each ιYiω has a bmC∞ primitive we will see that [ιYiω] = 0 in the bm-cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' One one hand, if i > 1, ιYiω vanishes at Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This holds because Yi has not any component ∂/∂Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall Proposition 6 from [GMP14]: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If ω ∈b Ω(M) with ω|Z = 0, then ω ∈ Ω(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In a similar way for bm-forms we have, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If ω ∈bm Ω(M) with ω|Z vanishing up to order m, then ω ∈ Ω(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Thus as ιYiω vanishes at Z, the bm-forms ιYiω are indeed smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Thus we can now apply the standard Poincar´e lemma and as these forms are closed they are locally exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This proves that all the vector fields Yi with i > 1 are indeed Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the other hand, the fact that ιY1ω = cdf1 is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, because we have a toric action that is Hamiltonian, we can use lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2 in [GMPS17], and we get an equivalent system such that ai are all constant and moreover ⟨a′ i, X⟩ = αi(Xω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that by dividing by a′ m−1, we can still assume a′ m−1 = 1 to be consistent with our notation, but we then have to multiply f1 · c in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ACTION-ANGLE COORDINATES ON bm-SYMPLECTIC MANIFOLDS 35 (3) Apply Darboux-Carath´eodory theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The construction above gives us some candidates σ1 = cf1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , σn for the action coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We now apply the Darboux-Carath´eodory theorem and express the form in terms of x: ω = \uf8eb \uf8ed m � j=1 c cj xj dx ∧ dq1 \uf8f6 \uf8f8 + n � i=2 dσi ∧ dqi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Since the vector fields Xσi = ∂ ∂qi are fundamental fields of the Tn- action the flow 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5 gives a linear action on the qi coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that the coordinate system is only defined in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It may not be valid at points outside U that may be in the orbit of points in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us see that the charts can be extended to these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Define U′ the union of all tori that intersect U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will see that the coordinates are valid at U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let {pi, θj} be the extension of {σi, qj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' It is clear that {pi, θj} = δij by its construction in the Darboux-Carath´eodory theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To see that {θi, θj} = 0 we take the flows by Xpk and extend the expression to the whole U′: Xpk({θi, θj}) = {{θi, θj}, pk} = {θi, δij} − {θj, δjk} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The fact that ω is preserved is obvious because Xpk are hamiltonian vector fields and thus they preserve the bm-symplectic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Moreover, t, θ1, p2, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , pn, θn are independent on U′ and hence are a coordinate system in a neighbourhood of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the proof we have seen that there exists an equivalent inte- grable system where the coefficients of the singular function are indeed constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' From now on, when considering a bm-integrable system we are going to make this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By means of the desingularization transformation we may ob- tain an action-angle coordinate theorem for folded manifolds as we do in Part 3 for the KAM theorem for folded symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This folded action-angle theorem is a particular case of the one obtained in [CM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' CHAPTER 5 Reformulating the action-angle coordinate via cotangent lifts The action-angle theorem for symplectic manifolds (also known as action-angle coordinate theorem) can be reformulated in terms of a cotangent lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall that given a Lie group action on any manifold its cotangent lifted action is automatically Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By considering the action of a torus on itself by translations this action can be lifted to its cotangent bundle and give a semilocal normal form theorem as the Arnold-Liouville-Mineur theorem for symplectic man- ifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If we now replace this cotangent lift to the cotangent bundle to a lift to the bm-cotangent bundle we obtain the semilocal normal form of the main theorem of this chapter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let start recalling the symplectic and b-symplectic case following [KM17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Cotangent lifts and Arnold-Liouville-Mineur in Symplectic Geometry Let G be a Lie group and let M be any smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Given a group action ρ : G × M −→ M, we define its cotangent lift as the action on T ∗M given by ˆρg := ρ∗ g−1 where g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We then have a commuting diagram T ∗M T ∗M M M ˆ ρg ˆπ π ρg Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Commutiative diagram of the construction of the iso- morphism of bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' where π is the canonical projection from T ∗M to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The cotangent bundle T ∗M is a symplectic manifold endowed with the exact symplectic form given by the differential of the Liouville one-form ω = −dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Lioville one-form can be defined intrinsically: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1) ⟨λp, v⟩ := ⟨p, (πp)∗(v)⟩ with v ∈ T (T ∗M), p ∈ T ∗M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A standard argument (see for instance [GS90]) shows that the cotangent lift ˆρ is Hamiltonian with moment map µ : T ∗M → g∗ given by ⟨µ(p), X⟩ := ⟨λp, X#|p⟩ = ⟨p, X#|π(p)⟩, 37 38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ACTION-ANGLE COORDINATES AND COTANGENT LIFTS where p ∈ T ∗M, X is an element of the Lie algebra g and we use the same symbol X# to denote the fundamental vector field of X generated by the action on T ∗M or M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This construction is known as the cotangent lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the special case where the manifold M is a torus Tn and the group is Tn acting by translations, we obtain the following explicit structure: Let θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , θn be the standard (S1-valued) coordinates on Tn and let (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , θn � �� � =:θ , t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , tn � �� � =:t be the corresponding chart on T ∗Tn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' we associate to the coordinates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) the cotangent vector � i tidθi ∈ T ∗ θ Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Liouville one-form is given in these coordinates by λ = n � i=1 tidθi and its negative differential is the standard symplectic form on T ∗Tn: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3) ωcan = n � i=1 dθi ∧ dti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Denoting by τβ the translation by β ∈ Tn on Tn, its lift to T ∗Tn is given by ˆτβ : (θ, t) �→ (θ + β, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The moment map µcan : T ∗Tn → t∗ of the lifted action with respect to the canonical symplectic form is (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4) µcan(θ, t) = � i tidθi, where the θi on the right hand side are understood as elements of t∗ in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Even simpler, if we identify t∗ with Rn by choosing the standard basis ∂ ∂θi of t then the moment map is just the projection onto the second component of T ∗Tn ∼= Tn × Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that the components of µ naturally define an integrable system on T ∗Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We can rephrase the Arnold-Liouville-Mineur theorem in terms of the symplec- tic cotangent model: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let F = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) be an integrable system on the symplectic manifold (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then semilocally around a regular Liouville torus the system is equivalent to the cotangent model (T ∗Tn)can restricted to a neighbourhood of the zero section (T ∗Tn)0 of T ∗Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The case of bm-symplectic manifolds Let us start by introducing the twisted bm-cotangent model for torus actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This model has additional invariants: the modular vector field of the connected component of the critical set and the modular weights of the associated toric action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider T ∗Tn be endowed with the standard coordinates (θ, t), θ ∈ Tn, t ∈ Rn and consider again the action on T ∗Tn induced by lifting translations of the torus Tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will now view this action as a bm-Hamiltonian action with respect to a suitable bm-symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In analogy to the classical Liouville one-form we define the following non-smooth one-form away from the hypersurface Z = {t1 = 0} : 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' THE CASE OF bm-SYMPLECTIC MANIFOLDS 39 � cc1 log |t1| + m � i=2 cci t−(i−1) 1 −(i − 1) � dθ1 + n � i=2 tidθi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' When differentiating this form we obtain a bm-symplectic form on T ∗Tn which we call (after a sign change) the twisted bm-symplectic form on T ∗Tn with invariants (cc1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ccm): (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5) ωtw,c := \uf8eb \uf8ed m � j=1 cj c tj 1 dt1 ∧ dθ1 \uf8f6 \uf8f8 + n � i=2 dti ∧ dθi, where c is the modular period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The moment map of the lifted action is then given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6) µtw,q0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=',qm−1) := (q0 log |t1| + m � i=2 qit−(i−1) 1 , t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , tn), where we are identifying t∗ with Rn and cj = −(j − 1)qj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We call this lift together with the bm-symplectic form 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5 the twisted bm- cotangent lift with modular period c and invariants (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that the components of the moment map define a bm-integrable system on (T ∗Tn, ωtw,(cc1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=',ccm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The model of twisted bm-cotangent lift allows us to express the action-angle coordinate theorem for bm-integrable systems in the following way: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let F = (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) be a bm-integrable system on the bm- symplectic manifold (M, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then semilocally around a regular Liouville torus T, which lies inside the critical hypersurface Z of M, the system is equivalent to the cotangent model (T ∗Tn)tw,(cc1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=',ccm) restricted to a neighbourhood of (T ∗Tn)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Here c is the modular period of the connected component of Z containing T and the constants (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , cm) are the invariants associated to the integrable system and its associated toric action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Part 3 A KAM theorem for bm-symplectic manifolds The KAM theorem explains how integrable systems behave under small per- turbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' More precisely, it studies how an integrable system in action-angle coordinates responds to a small perturbation on its Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The trajectories of an integrable system in action-angle coordinates can be seen as linear trajectories over a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The KAM theorem finds a way to transform these original trajectories to other linear trajectories over some transformed torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The KAM theorem states that most of these tori, and the linear solutions of the system on these tori, survive if the perturbation is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this part, we give a new KAM theorem for bm-symplectic manifolds with detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is contained in the first chapter of this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Moreover, we devote three more chapters to applications: (1) Desingularization of bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We present a way to use the desingularization of bm-symplectic manifolds presented in [GMW17] to construct standard smooth integrable systems from bm-integrable sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This desingularized integrable system is uniquely defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (2) Desingularization of the KAM theorem on bm-symplectic man- ifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this section we use the desingularization of bm-integrable sys- tems in conjunction with the KAM theorem for bm-symplectic manifolds to deduce the original KAM theorem as well as a completely new KAM theorem for folded symplectic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (3) Potential applications to Celestial mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We overview a list of motivating examples from Celestial mechanics where regularization trans- formations give rise to bm-symplectic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We discuss some potential applications of perturbation theory in this set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' CHAPTER 6 A new KAM theorem The objective of this chapter is to give a construction of KAM theory in the setting of bm-symplectic manifolds and with bm-integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The core of the chapter is the construction of the proper statement and the proof of the equivalent of the KAM theorem on bm-symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This chapter is divided different sections: (1) On the structure of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On this section we are going to present the main ideas that are going to appear in the proper statement and proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The idea of the theorem is to build a sequence of bm-symplectomorphisms such that its limit transforms the hamiltonian to only depend on the action coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (2) Technical results and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On this section we present some technical results and definitions that are key for the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (3) KAM theorem on bm-symplectic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On this section we present the statement and the proof of the main result of this chap- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The proof is structured in 6 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the first part we define the parameters that are going to be used to define the sequence of bm- symplectomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the second part we build precisely this sequence of bm-symplectomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the third part we see that the sequence of frequency maps of the transformed Hamiltonian functions at every step converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the fourth part we see that the sequence of bm-symplectomorphisms converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the fifth part we obtain results on the stability of the trajec- tories under the original perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the sixth part, we find bounds to explain how close the invariant tori are from the unperturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Finally, we obtain a bound for the measure of the set of invariant tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the structure of the proof The first thing we do is to reduce our study to the case the perturbation is not a bm-function but an analytic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is because any purely singular perturbation only affects the component in the direction of the modular vector field and can be easily controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The idea of the proof is really similar to the classical KAM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We want to build a diffeomorphism such that its transformed hamiltonian only depends on the action coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' But it is not possible to build this diffeomorphism in one step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' What we do, as it is done in the classical case, it is to build a sequence of diffeomorphisms such that the part of the hamiltonian depending on the angular variables decreases at every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The idea is to remove the first K terms of its Fourier expression at every step while making K rapidly increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is done by 43 44 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM assuming the diffeomorphism comes as the flow at time 1 generated by a Hamil- tonian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this way one can use the Lie Series in conjunction with the Fourier series to find the expression for the hamiltonian function that generates our diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The final diffeomorphism will be the composition of all the diffeomorphisms obtained at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' One of the main difficulties of the proof, as in the classical case, is to prove that these diffeomorphisms converge and to prove some bounds of its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We also note that for our bm-symplectic setting, the diffeomorphisms we con- sider leave the defining function of the critical set invariant up to order m, this will have an important role later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also observe in particular that the critical set can not be transformed by any perturbation given by a bm-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Next we give some technical definitions and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We define the norms we are going to use to do all the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We set the notation for the proof and the state- ment of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We define the notion of non-resonance for a neighborhood of the critical set of the bm-symplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We study the set of all possible non-resonant vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And we state the inductive lemma, which gives us estimates and constructions for every step of our sequence of diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' After all this discussion we are in conditions to properly state the bm-version of the KAM theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' One important difference to the classical KAM theorem is that we have to guarantee that at Z the set of non-resonant vectors does not become the whole set of frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This condition can be understood as the perturbation being smaller than some constant multiplied by the inverse of the modular period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The proof of the theorem is done in six different steps by following the structure on [DG96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Since we are going to use the inductive lemma at every step, first we define the parameters and sets to which we are going apply such lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we check that we can actually apply the lemma and obtain some extra estimates for the results of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' After this we see that the sequence of frequency vectors converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We do the same with the sequence of canonical transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we get some bounds for the size of the components of the final diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Next we characterize the tori that survive by the perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Finally we give some estimates for the measure of the set of these tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that our version of the bm-KAM theorem improves the one in [KMS16a] in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Firstly it is applicable to bm-symplectic structures not only for b-symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also we give several estimates that are not obtained in [KMS16a], this estimates have sense in a neighborhood of the critical set Z, while [KMS16a] only studied the behavior at Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Finally the type of perturbation we consider is far more general, since we do not have any condition of the form of the perturbation but only on its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Reducing the problem to an analytical perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the standard KAM, we assume to have an analytic Hamiltonian h(I) depending only on the action coordinates and we add to it a small analytical perturbation R(φ, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This perturbed system receives the name of nearly integrable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And then find a new coordinate system such that h(I) + R(φ, I) = ˜h(˜I) where most of the quasi- periodic orbits are preserved and can be mapped to the unperturbed quasi-periodic orbits by means of the coordinate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In our setting we may assume h(I) to not be analytical and be a bm-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also the perturbation R(φ, I) may as well be considered a bm-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ON THE STRUCTURE OF THE PROOF 45 following lines we justify without loss of generality that actually we can assume the perturbation to be analytical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us state this more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let (M, x, Z, ω, F) be a bm-manifold with a bm integrable system F on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider action angle coordinates on a neighborhood of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we can assume the expressions: ω = \uf8eb \uf8ed m � j=1 cj Ij 1 \uf8f6 \uf8f8 dI1 ∧ dφ1 + n � i=2 dIi ∧ dφi, and F = (q′ 0 log I1 + m−1 � i=1 q′ i 1 Ii 1 + h(I), f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn) where h, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , fn are analytical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let the Hamiltonian function of our system be the first component of the moment map ˆh′ = q′ 0 log I1 + �m−1 i=1 q′ i 1 Ii 1 + h = ζ′ + h, where ζ′ := q′ 0 log I1 + �m−1 i=1 q′ i 1 Ii 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that dζ′ = �m i=1 ˆq′ i 1 I′ 1 , where ˆq′ i = −(i − 1)q′ i−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that by the result of the previous chapter cj/ˆq′ j = K the modular period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In particular cm/ˆq′ m = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The hamiltonian system given by ˆh′ can be easily solved by φ = φ0 +u′t, I = I0 where u′ is going to be defined in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Consider now a pertur- bation of this system: ˆH′ = ˆh′(I) = ˆR(I, φ), where ˆR is a bm-function ˆR(I, φ) = Rζ(I1)+R(I, φ) where Rζ(I1) = (r0 log I1 +�m−1 i=1 ri 1 Ii 1 ) is the singular part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we can consider the perturbations Rζ(I1) and R(I, φ) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This way, we may consider Rζ(I) as part of ˆh′(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then we have a new hamiltonian ˆh(I) = (q′ 0 + r0) log I1 + m−1 � i=1 (q′ i + ri) 1 Ii 1 + h = q0 log I1 + m−1 � i=1 qi 1 Ii 1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Now, instead of the identity Kˆq′ j = cj we will have K(ˆqj − ˆrj) = cj, which implies K � 1 − ˆrj ˆq′ j+ˆrj � = cj ˆqj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In particular K � 1 − ˆrm ˆq′m + ˆrm � = cm ˆqm Let us define K′ = K � 1 − ˆrm ˆq′m+ˆrm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' So from now on we assume ˆh = q0 log I1 + �m−1 i=1 qi 1 Ii 1 + h, that the perturbation R(φ, I) is analytical, and we have the condi- tion cm ˆqm = K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that this system with only the singular perturbation is still easy to solve in the same way that the system previous to this perturbation was.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Looking for a bm-symplectomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume we have a Hamil- tonian function H = ˆh(I) + R(φ, I) in action-angle coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Where ˆh(I) is the singular component of the bm-integrable system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1) ˆh(I) = h(I) + q0 log(I1) + m−1 � i=1 qi 1 Ii 1 , 46 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM where h(I) is analytical1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume also that the bm-symplectic form ω2 in these coordinates is expressed as: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) ω = \uf8eb \uf8ed m � j=1 cj Ij 1 \uf8f6 \uf8f8 dI1 ∧ dφ1 + n � i=2 dIi ∧ dφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And finally, the expression for the frequency vector is: ˆu = ∂ˆh ∂I = ∂(h(I) + q0 log(I1) + �m−1 i=1 qi 1 Ii 1 ) ∂I = � u1 + m � i=1 ˆqi Ii 1 , u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , un � , where ˆq1 = q0 and ˆqi−1 = −iqi if i ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The objective is to follow the steps of the usual KAM construction (the steps followed are highly inspired in [DG96]) replacing the standard symplectic form for ω and taking as hamiltonian the bm-function ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The objective of the construction is to find a diffeomorphism (actually a bm-symplectomorphism) ψ such that H ◦ ψ = h(˜I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This is done induc- tively, by taking H ◦ ψ = H ◦ φ1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='◦ φq ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=', while trying to make R(φ, I) smaller at every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us focus in one single step Recall the classical formula: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' See [DG96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' f ◦ φt = ∞ � j=0 tj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='Lj W f, Lj W f = {Lj−1 W f, W} Where W is the Hamiltonian that generates the flow φt, and {·, ·} is the correspond- ing Poisson bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will denote rk(H, W, t) = �∞ j=k tj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Lj W H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 1If another component of the moment map is chosen to be the hamiltonian of the system, the result still holds: the computations can be replicated assuming ˆh(I) = h(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 2In classical KAM, ω is used to denote the frequency vector ∂h ∂I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We need ω to denote the bm-symplectic form so we are going to use u to denote the frequency vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ON THE STRUCTURE OF THE PROOF 47 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3) H ◦ φ = H ◦ φ|t=1 = ∞ � j=0 tj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Lj W H ���� ˆh+R ������ t=1 = ˆh + R{ˆh + R, W} + r2(H, W, 1) = ˆh + R + {ˆh, W} + {R, W} + r2(ˆh, W, 1) +r2(R, W, 1) = ˆh + R + {ˆh, W} � �� � We want to cancel this term as fast as we can +r2(ˆh, W, 1) + r2(R, W, 1) We want {ˆh, W} +R≤k = 0, equivalently {W, ˆh} = R≤k, where R≤k means the Fourier expression of R up to order K: R≤k = � k∈Rn |k|1≤K Rk(I)eik·φ Let us impose the condition {W, ˆh} = R≤K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us write the expression of the Poisson bracket associated to the bm-symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' {W, ˆh} = � 1 �m j=1 cj Ij 1 � � ∂W ∂φ1 ∂ˆh ∂I1 − ∂W ∂I1 ∂ˆh ∂φ1 � + n � i=2 � ∂W ∂φi ∂ˆh ∂Ii − ∂W ∂Ii ∂ˆh ∂φi � Because ˆh depends only on I, ∂ˆh ∂φi = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Moreover, the singular part of the bm-function only depends on I1 and hence its derivatives with respect to the other variables are also 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Using that ∂ˆh ∂I = u + �m i=1 ˆqi Ii 1 the previous expression can be simplified: {W, ˆh} = \uf8eb \uf8edu1 + �m i=1 ˆqi Ii 1 �m j=1 cj Ij 1 \uf8f6 \uf8f8 ∂W ∂φ1 + n � i=2 ∂W ∂φi ui To expand the expression further we develop W in its Fourier expression: W = � k∈Rn |k|1≤K Wk(I)eikφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Fourier expansion is added up to order K, because it is only necessary for the expressions to agree up to order K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this notations the condition becomes: {W, ˆh}≤K = \uf8eb \uf8edu1 + �m i=1 ˆqi Ii 1 �m j=1 cj Ij 1 \uf8f6 \uf8f8 ∂ ∂φ1 \uf8eb \uf8ec \uf8ec \uf8ed � k∈Rn |k|1≤K Wk(I)eikφ \uf8f6 \uf8f7 \uf8f7 \uf8f8 48 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM + n � j=2 uj ∂ ∂φj \uf8eb \uf8ec \uf8ec \uf8ed � k∈Rn |k|1≤K Wk(I)eikφ \uf8f6 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8edu1 + �m i=1 ˆqi Ii 1 �m j=1 cj Ij 1 \uf8f6 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed � k∈Rn |k|1≤K Wk(I)eikφik1 \uf8f6 \uf8f7 \uf8f7 \uf8f8 + n � j=2 uj \uf8eb \uf8ec \uf8ec \uf8ed � k∈Rn |k|1≤K Wk(I)eikφikj \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � k∈Rn |k|1≤K Wk(I)eikφ · \uf8eb \uf8edik1 \uf8eb \uf8edu1 + �m i=1 ˆqi Ii 1 �m j=1 cj Ij 1 \uf8f6 \uf8f8 + n � j=2 ikjuj \uf8f6 \uf8f8 = R≤K Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' it is possible to make the two sides of the equation equal by imposing the condition term by term: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4) Wk(I) = Rk(I) 1 i � k1 � u1+�m i=1 ˆ qi Ii 1 �m j=1 cj Ij 1 � + �n j=2 kjuj � = Rk(I) 1 i � k1 � u1+�m i=1 ˆ qi Ii 1 �m j=1 cj Ij 1 � + ¯k¯u �, where we adopted the notation �n j=2 kjuj = ¯k¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that the expression 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4 has no sense when k = ⃗0 and hence {W, h}0 = R03 can not be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let W0(I) = 0, then {h, W}≤K = R≤K − R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Plugging the results above into the equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3, one obtains: H ◦ φ = ˆh + R0 + R≥K + r2(ˆh, W, 1) + r1(R, W, 1) With this construction the diffeomorphism φ is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' But this is only the first of many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If q denotes the number of the iteration of this procedure, in general, we obtain: 3The zero term of the Fourier series can be seen as the angular average of the function 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ON THE STRUCTURE OF THE PROOF 49 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5) H(q) = H(q−1) ◦ φ(q) = ˆh(q−1) + R(q−1) 0 + R(q−1) ≥K +r2(h(q−1), W (q), 1) + r1(R(q−1), W (q), 1), and at every step: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6) �ˆh(q) = ˆh(q−1) + R(q−1) 0 R(q) = R(q−1) >K + r2(ˆh(q−1), W (q), 1) + r1(R(q−1), W (q), 1) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the change of the defining function under bm-symplectomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Note that since we are considering bm-manifolds it only makes sense to consider I1 up to order m, see [Sco16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' When talking about defining functions we are interested in [I1], its jet up to order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' By definition bm- maps preserve I1 up to order m and bm-vector fields X are such that LX(I1) = g·Im 1 for g ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let φt be the integral flow of X a bm-vector field, then φt is a bm-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We want I1 ◦ φt = I1 + Im 1 · g for some g ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will use 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' I1 ◦ φt = ∞ � j=0 tj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='Lj XI1 = I1 + LX(I1) + ∞ � j=2 tj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='Lj XI1 = I1 + Im 1 + ∞ � j=2 tj j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='Lj XI1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the other hand, let us prove by induction Lk XI1 = g(k)Im 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The first case is obvious, assume the case k holds and let us prove the case k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Lk+1 X I1 = {Lk XI1, X} = {g(k)Im 1 , X} = (LXg(q))Im 1 + g(k) · mIm−1 1 LXI1 = (LXg(k) + g(k) · m · Im−1 1 g)Im 1 = g(k+1)Im 1 where g(k+1) = LXg(k) + g(k) · m · Im−1 1 g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The Hamiltonian vector flow of some smooth hamiltonian function h is a bm-vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' At each point of Z the following identity holds LXhI1 = Im 1 ∂f ∂φ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The result can be extended at a neighborhood of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Observe that combining the two previous results we get that the hamiltonian flow of a function preserves I1 up to order m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Technical results As the non-singular part of our functions we will be considering analytic func- tions on T × G, G ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The easiest way to work with these functions is to consider them as holomorphic functions on some complex neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us define formally this neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Wρ1(Tn) := {φ : ℜφ ∈ Tn, |ℑφ|∞ ≤ ρ1}, Vρ2(G) := {I ∈ Cn : |I − I′| ≤ ρ2 for some I′ ∈ G}, Dρ(G) := Wρ1(Tn) × Vρ2(G), where | · |∞ denotes the maximum norm and | · |2 denotes de Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Now it is necessary to clarify the norms that are going to be used on these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let f be an action function (only depending on the I-coordinates), and F an action vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' |f|G,η := supI∈Vη(G) |f(I)|, |f|G := |f|G,0 |F|G,η,p := supI∈Vη(G) |F(I)|p, |F|G,η := |F|G,η,2 Now, assume f(I, φ) to be an action-angle function written using its Fourier ex- pansion as � k∈Zn fk(I)eik·φ, and F to be an action-angle vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' |f|G,ρ := sup(φ,I)∈Dρ(G) |f(I)|, ∥f∥G,ρ := � k∈Zn |fk|G,ρ2e|k|1ρ1 |F|G,ρ,p := � k∈Zn |Fk|G,ρ2,pe|k|1ρ1, ∥F∥G,ρ = ∥F∥G,ρ,2 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='7 (Cauchy Inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ���� ∂f ∂φ ���� G,(ρ1−δ1,ρ2),1 ≤ 1 eδ1 ∥f∥G,ρ ���� ∂f ∂I ���� G,(ρ1,ρ2−δ2),∞ ≤ 1 δ2 ∥f∥G,ρ Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If Df = ( ∂f ∂φ, ∂f ∂I ), ∥Df∥G,ρ,c := max � ∥∂f ∂φ∥G,ρ,1, c∥∂f ∂I ∥G,ρ,∞ � Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To simplify our notation, let us define: A(I1) = �m j=1 ˆqj Ij 1 �m j=1 cj Ij 1 and B(I1) = 1 �m j=1 cj Ij 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this notation, equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4 can be written as: Wk(I) = Rk(I) i(k1B(I1)u1 + ¯k¯u + k1A(I1)) Observe that A(I1) and B(I1) are analytic (holomorphic on the complex ex- tended domain) where the denominator does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We can assume that this does not happen by shrinking the domain G in the direction of I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe, in particular, that when I1 → 0, A(I1) → ˆqm/cm = 1/K′ the inverse of the modular period and B(I1) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this way, the norms of A(I1) and B(I1) are bounded and well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will denote these norms by KA and KB respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also, since 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' TECHNICAL RESULTS 51 A(I1) and B(I1) are analytic, their derivatives will also be bounded, and we will denote the norms of these derivatives by KA′ and KB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' To further simplify the notation in the following computations we introduce the definition: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ¯ A = � A 0 � and ¯B = � B 0 0 Idn−1,n−1 � Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this notation, equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='4 can be written as: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='7) Wk(I) = Rk(I) i(k ¯B(I1)u + k ¯ A(I1)) Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Having fixed ω, a bm-symplectic form (as in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2) and ˆh a bm-function (as in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1) as a hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Given an integer K and α > 0, F ⊂ Rn (or Cn) the space of frequencies is said to be α, K-non-resonant with respect to (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , cm) and (ˆq1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ˆqm) if |k ¯B(I1)u + k ¯ A(I1)| ≥ α, ∀k ∈ Z \\ {0}, |k|1 ≤ K, ∀u ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We are going to use the notation α, K, c, ˆq-non-resonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The non-resonance condition is established on u = ∂h/∂I, not on ˆu = ∂ˆh/∂I, because our non-resonance condition already takes into account the singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' In this way we can use the analytic character of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If �� ∂u ∂I �� G,ρ2 is bounded by M ′, then �� ∂ ∂I � ¯Bu + ¯ A ��� G,ρ2 is also bounded: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8) �� ∂ ∂I � ¯Bu + ¯ A ��� G,ρ2 ≤ ��� ∂ ¯ B ∂I u + ¯B ∂u ∂I + ∂ ¯ A ∂I ��� G,ρ2 ≤ KB′|u|G,ρ2 + KBM ′ + KA =: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' When we consider the standard KAM theorem, the frequency vector u is relevant because the solution to the Hamilton equations of the unper- turbed problem has the form: I = I0, φ = φ0 + ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us see what plays the role of u in our bm-KAM theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us find the coordinate expression of the solution to ιXˆhω = dˆh, where ω is a bm-symplectic form in action-angle coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Xˆh = ˙I1 ∂ ∂I1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' + ˙In ∂ ∂In , where ˙I1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ˙In are the functions we want to find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' dˆh = \uf8eb \uf8ed m � j=1 ˆqi 1 Ij 1 \uf8f6 \uf8f8 dI1 + dh, and hence, 52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM Xˆh = Π(dˆh, ·) = �m i=1 ˆqi Ii 1 �m i=1 cj Ij 1 ∂ ∂φi + Xh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence φ = φ0 + ( ¯Bu + ¯ A � �� � u′ )t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' So the frequency vector that we are going to be concerned about is going to be u′ instead of ˆu = ∂ ∂I ˆh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If u is one-to-one from G to its image then u′ = ¯Bu + ¯ A is also one-to-one from G′ to its image in a neighborhood of Z, while at Z it is the projection of u such that the first coordinate is sent to ˆqm cm = 1/K′ the inverse of the modular period, were G′ ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Because u′ = \uf8eb \uf8ed 1 �m j=1 cj Ij 1 u1 + �m j=1 ˆqj Ij 1 �m j=1 cj I1 , u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , un \uf8f6 \uf8f8 , and B is invertible outside I1 = 0, shrinking G if necessary in the first dimension the map is one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' But at the critical set {I1 = 0}, u′ is a projection of u where the first component is sent to the constant value ˆqm cm = 1 K′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If u(G) is α, K, c, ˆq-non-resonant, then u(Vρ2(G)) is α 2 , K, c, ˆq- non-resonant, assuming that ρ2 ≤ α 2MK and �� ∂u ∂I �� G,ρ2 ≤ M ′ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Fix k ∈ Z \\ {0}, we want to bound |k ¯B(I1)v + k ¯ A(I1)| where v ∈ u(Vρ2(G)) as a function on v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Given v ∈ u(Vρ2(G)) we ask whether there is any bound for the distance to some v′ ∈ u(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' v ∈ u(Vρ2(G)) ⇒ v = u(x), x ∈ Vρ2(G) ⇒ ∃y ∈ G such that |x − y| ≤ ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Take v′ = u(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' |v − v′| ≤ |x − y| ���� ∂u ∂I ���� G,ρ2 ≤ ρ2M ′ ≤ ρ2M/KB ≤ α 2MK M/KB = α 2KKB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Where we used equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='8 in the third inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' |k1B(I1)v1 + ¯k¯v + k1A(I1)| ≥ |k1B(I1)v′ 1 + ¯k ¯v′ + k1A(I1)| � �� � ≥α −|k1B(I1)(v1 − v′ 1) + ¯k(¯v − ¯v′)| ≥ α − KB |k · (v − v′)| � �� � ≤Kα/(2KKB) ≥ α − α/2 = α/2 □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let ˆh(I) be a bm-function as in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume h(I) and R(φ, I) be real analytic on Dρ(G), u(G) = ∂h ∂I (G) is α, K, c, ˆq-non-resonant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume also that | ∂ ∂I u|G,ρ2 ≤ M ′ and ρ2 ≤ α 2MK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let c > 0 given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then R0(φ, I), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' TECHNICAL RESULTS 53 W≤K(φ, I) given by the previous construction are both real analytic on Dρ(G) and the following bounds hold (1) ||DR0||G,ρ,c ≤ ||DR||G,ρ,c (2) ||D(R − R0)||G,ρ,c ≤ ||DR0||G,ρ,c (3) ||DW||G,ρ,c ≤ 2A α ||DR0||G,ρ,c Where A = 1 + 2Mc α Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Inequalities 1 and 2 are obvious because of the Fourier expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us prove inequality 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us expand R(φ, I) and W(φ, I) in their Fourier expression: R = � k∈Rn Rk(I)eik·φ, W = � k∈Rn Wk(I)eik·φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will bound this expression finding term-by-term bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∂R ∂φ = � k∈Rn Rk(I)eik·φik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, if we denote [ ∂R ∂φ ]k the k-th term of the Fourier expansion of ∂R ∂φ , we have: �∂R ∂φ � k = Rkik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us compute the derivative of Wk with respect to the I variables: ∂Wk ∂I = ∂ ∂I � Rk i(k ¯B(I1)u + k ¯ A(I1)) � = ∂Rk/∂I i(k ¯B(I1)u + k ¯ A(I1))) − Rki ∂ ∂I (k ¯B(I1)u + k ¯ A(I1))) [i(k ¯B(I1)u + k ¯ A(I1)))]2 = ∂Rk/∂I i(k ¯B(I1)u + k ¯ A(I1))) + Rkik ∂ ∂I ( ¯B(I1)u + ¯ A(I1))) [(k ¯B(I1)u + k ¯ A(I1)))]2 = ∂Rk/∂I i(k ¯B(I1)u + k ¯ A(I1))) + [ ∂Rk ∂φ ]k ∂ ∂I ( ¯B(I1)u + ¯ A(I1))) [(k ¯B(I1)u + k ¯ A(I1)))]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, we take norms (| · |G,ρ2,∞) on each side of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ���� ∂Wk ∂I ���� G,ρ2,∞ ≤ 2 α ���� ∂Rk ∂I ���� G,ρ2,∞ + 4M α2 ���� �∂Rk ∂φ � k ���� G,ρ2,∞ ≤ 2 α ���� ∂Rk ∂I ���� G,ρ2,∞ + 4M α2 ���� �∂Rk ∂φ � k ���� G,ρ2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Taking the supremum at the whole domain: ���� ∂Wk ∂I ���� G,ρ2,∞ ≤ 2 α ���� ∂Rk ∂I ���� G,ρ2,∞ + 4M α2 ���� �∂Rk ∂φ � k ���� G,ρ2,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Moreover, 54 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM ∂W(I) ∂φ = ∂ ∂φ � � k∈Rn Wk(I)eik·φ � = ∂ ∂φ � � k∈Rn ikWk(I)eik·φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, the k-th term of the Fourier series of ∂W ∂φ is �∂W ∂φ � k = Wkik = Rk i(k ¯B(I1)u + k ¯ A(I1)))ik = 1 i(k ¯B(I1)u + k ¯ A(I1))) �∂R ∂φ � k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Taking norms (∥ · ∥G,ρ,1) at each side: ���� ∂W ∂φ ���� G,ρ,1 ≤ 2 α ���� ∂W ∂φ ���� G,ρ,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, ∥DW∥G,ρ,c = max ����� ∂W ∂φ ���� G,ρ,1 , c ���� ∂W ∂I ���� G,ρ,∞ � ≤ max � 2 α ���� ∂R ∂φ ���� G,ρ,1 , c 2 α ���� ∂R ∂I ���� G,ρ2,∞ + c4M α2 ���� ∂R ∂φ ���� G,ρ2,1 � ≤ max � 2 α ���� ∂R ∂φ ���� G,ρ,1 , 2 α ∥DR∥G,ρ2,c + c4M α2 ∥DR∥G,ρ2,c � = max � 2 α ���� ∂R ∂φ ���� G,ρ,1 , 2 α � 1 + 2M α c � ∥DR∥G,ρ2,c � ≤ 2 α � 1 + 2M α c � ∥DR∥G,ρ2,c ≤ 2 αA ∥DR∥G,ρ2,c, where A is as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' □ Recall the Cauchy inequalities, see [P¨os93]: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='9) ���� ∂f ∂φ ���� G,(ρ1,ρ2),1 ≤ 1 eδ1 ∥f∥G,ρ ���� ∂f ∂I ���� G,(ρ1,ρ2−δ2),∞ ≤ 1 δ2 ∥f∥G,ρ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' TECHNICAL RESULTS 55 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let f, g be analytic functions on Dρ(G), where 0 < δ = (δ1, δ2) < ρ = (ρ1, ρ2) and c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Define ˆδc := min(cδ1, δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' The following inequalities hold: (1) ∥Df∥G,ρ−δ,c ≤ c ˆδc ∥f∥G,ρ (2) ∥{f, g}∥G,ρ ≤ 2 c∥Df∥G,ρ,c · ∥Dg∥G,ρ,c (3) ∥D(f>K)∥G,(ρ−δ1,ρ2),c ≤ e−Kδ1∥Df∥G,ρ,c Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us prove each point separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (1) Using the Cauchy inequalities one obtains the following: ���� ∂f ∂φ ���� G,ρ−δ,1 = ���� ∂f ∂φ ���� G,(ρ1−δ1,ρ2−δ2),1 ≤ ���� ∂f ∂φ ���� G,(ρ1−δ1,ρ2),1 ≤ 1 eδ1 ∥f∥G,ρ, ���� ∂f ∂I ���� G,ρ−δ,∞ = ���� ∂f ∂I ���� G,(ρ1−δ1,ρ2−δ2),∞ ≤ ���� ∂f ∂I ���� G,(ρ1,ρ2−δ2),∞ ≤ 1 δ1 ∥f∥G,ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Putting the two inequalities inside the definition of the norm: ∥Df∥G,ρ−δ,c = max ����� ∂f ∂φ ���� G,ρ−δ,1 , c ���� ∂f ∂I ���� G,ρ−δ,∞ � ≤ max � 1 eδ1 ∥f∥G,ρ, c δ2 ∥f∥G,ρ � ≤ max � 1 eδ1 c c, c δ2 � ∥f∥G,ρ ≤ max � c eˆδc , c ˆδc � ∥f∥G,ρ, where the last inequality holds because ˆδc = min(cδ1, δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (2) Let us find the expression of {f, g} for a bm-symplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' {f, g} = ω(Xf, Xg) where Xf and Xg are such that ιXf ω = df and ιXgω = dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let restrict the computations only to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' df = n � i=1 ∂f ∂φ1 dφ1, Xf = n � i=1 ai ∂ ∂φi + n � i=1 bi ∂ ∂φi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Where ai and bi are coefficients to be determined by imposing the following condition: ιXf ω = \uf8eb \uf8ed m � j=1 cj Ij 1 \uf8f6 \uf8f8 (a1dI1 − b1dφ1) + n � i=2 (aidIi − bidφi) = df.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, solving for the coefficients the following expressions are ob- tained: 56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM a1 = 1 ��m j=1 cj Ij 1 � ∂f ∂φ1 and ai = ∂f ∂φi for i ̸= 1, b1 = − 1 ��m j=1 cj Ij 1 � ∂f ∂φ1 and bi = − ∂f ∂φi for i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, the expression for the hamiltonian vector fields becomes: Xf = 1 ��m j=1 cj Ij 1 � � ∂f ∂φ1 ∂ ∂φ1 − ∂f ∂I1 ∂ ∂I1 � + n � i=1 � ∂f ∂φi ∂ ∂φi − ∂f ∂Ii ∂ ∂Ii � , Xg = 1 ��m j=1 cj Ij 1 � � ∂g ∂φ1 ∂ ∂φ1 − ∂g ∂I1 ∂ ∂I1 � + n � i=1 � ∂g ∂φi ∂ ∂φi − ∂g ∂Ii ∂ ∂Ii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then the Poisson bracket applied to the two functions: {f, g} = ω(Xf, Xg) = 1 ��m j=1 cj Ij 1 � � ∂f ∂I1 ∂g ∂φ1 − ∂f ∂φ1 ∂g ∂I1 � + n � i=2 � ∂f ∂Ii ∂g ∂φi − ∂f ∂φi ∂g ∂Ii � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' And hence the norm of the Poisson bracket becomes: ∥{f, g}∥G,ρ = ������� 1 ��m j=1 cj Ij 1 � � ∂f ∂I1 ∂g ∂φ1 − ∂f ∂φ1 ∂g ∂I1 � + n � i=2 � ∂f ∂Ii ∂g ∂φi − ∂f ∂φi ∂g ∂Ii ������ G,ρ ≤ ����� n � i=1 � ∂f ∂Ii ∂g ∂φi − ∂f ∂φi ∂g ∂Ii ������ G,ρ Where we assumed ����m j=1 cj Ij 1 ��� ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This assumption makes sense, because we are interested in the behaviour close the critical set Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Close enough to the critical set this expression holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, ∥{f, g}∥G,ρ ≤ n � i=1 ���� ∂f ∂Ii ���� G,ρ ���� ∂g ∂φi ���� G,ρ + n � i=1 ���� ∂f ∂φi ���� G,ρ ���� ∂g ∂Ii ���� G,ρ ≤ ���� ∂f ∂I ���� G,ρ,∞ ���� ∂g ∂I ���� G,ρ,1 + ���� ∂f ∂I ���� G,ρ,1 ���� ∂g ∂I ���� G,ρ,∞ ≤ 1 c |Df∥G,ρ,c∥Dg∥G,ρ,c + 1 c|Df∥G,ρ,c∥Dg∥G,ρ,c 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' TECHNICAL RESULTS 57 ≤ 2 c ∥Df∥G,ρ,c∥Dg∥G,ρ,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (3) Lastly, ∥D(f>K)∥G,(ρ1−δ1,ρ2),1 = max ����� ∂f>K ∂φ ���� G,(ρ1−δ1,ρ1),1 , c ���� ∂f>K ∂I ���� G,(ρ1−δ1,ρ1),∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' We will proceed by bounding each term separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On one hand: ���� ∂f ∂φ ���� G,(ρ1,ρ2),1 = ����� � k∈Zn ikfk(I)eikφ ����� G,(ρ1,ρ2),1 ≥ � k∈Zn k ∥fk(I)∥G,ρ2,1 e|k|1ρ1 ≥ � k∈Zn |k|1>K k ∥fk(I)∥G,ρ2,1 e|k|1(ρ1+δ1−δ1) ≥ eKδ1 � k∈Zn |k|1>K k ∥fk(I)∥G,ρ2,1 e|k|1(ρ1−δ1) = eKδ1 ���� ∂f>K ∂φ ���� G,(ρ1−δ1,ρ2),1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the other hand: ���� ∂f ∂I ���� G,(ρ1,ρ2),∞ = ����� � k∈Zn ∂fk(I) ∂I eikφ ����� G,(ρ1,ρ2),∞ ≥ � k∈Zn ���� ∂fk(I) ∂I ���� G,ρ2,∞ e|k|1ρ1 ≥ � k∈Zn |k|1>K ���� ∂fk(I) ∂I ���� G,ρ2,∞ e|k|1(ρ1+δ1−δ1) ≥ eKδ1 � k∈Zn |k|1>K ���� ∂fk(I) ∂I ���� G,ρ2,∞ e|k|1(ρ1−δ1) ≥ eKδ1 ���� ∂f>K ∂I ���� G,(ρ1−δ1,ρ2),∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence ∥D(f>k)∥G,(ρ1−δ1,ρ2),c ≤ e−Kδ1∥Df∥G,ρ,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM □ Now we define a norm that indicates how close a map Φ is to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let x = (φ, I) ∈ C2n, then |x|c := max(|φ|1, c|I|∞) Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' For a map Υ : Dρ(G) → C2n its norm and the norm of its derivative its defined as: |Υ|G,ρ,c := sup x∈Dρ(G) |Υ(x)|c, |DΥ|G,ρ,c := sup x∈Dρ(G) |DΥ(x)|c, where |DΥ(x)|c = sup y∈R2n |y|c=1 |DΥ(x) · y|c Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' If Υ is analytic on Dρ(G), then |DΥ|G,ρ−δ,C ≤ |Υ|G,ρ,c ˆδc Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Observe that if we consider ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='∥ any norm on Cn and a matrix A of size n × n, and ∥A∥ defines the induced norm of matrices i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∥A∥ = sup y∈C2n ∥y∥=1 ∥A · y∥ then one has that ∥(∥a1∥′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ∥an∥′)∥ ≤ ∥A∥ where aj denotes the j-th row of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Also note that ∥ · ∥′ can be a any norm consider the infinity norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' This can be easily proven in the following way: ∥A · y∥ = ������� \uf8eb \uf8ec \uf8ed a1 · y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' an · y \uf8f6 \uf8f7 \uf8f8 ������� ≤ ������� \uf8eb \uf8ec \uf8ed ∥a1∥′∥y∥′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' ∥an∥′∥y∥′ \uf8f6 \uf8f7 \uf8f8 ������� Where ∀y ∈ Cn such that ∥y∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let aj be the rows of DΥ(x), aj = �∂Υj ∂φ , ∂Υj ∂I � , and be ∥aj∥′ its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' With this property in mind we proceed as follows: |DΥ|G,ρ−δ,c = sup x∈Dρ−δ(G) |DΥ(x)|c ≤ sup x∈Dρ−δ(G) |(|a1|∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , |an|∞)|c ≤ ��� � supx∈Dρ−δ ∥DΥ1∥∞ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , supx∈Dρ−δ ∥DΥ2n∥∞ ���� c = ��� � ∥DΥ1∥G,ρ−δ,∞ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ∥DΥ2n∥G,ρ−δ,∞ ���� c ≤ ��� � 1 δ1 ∥Υ1∥G,ρ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , 1 δ1 ∥Υ2n∥G,ρ ���� c 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' TECHNICAL RESULTS 59 ≤ 1 ˆδc ���∥Υ1∥G,ρ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , ∥Υ2n∥G,ρ ��� c = 1 ˆδc supx∈Dρ(G) |Υ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' , Υ2n|c = 1 ˆδc supx∈Dρ(G) |Υ|c = 1 ˆδc |Υ|G,ρ,c □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let W be an analytic function on Dρ(G), ρ > 0 and let Φt be its Hamiltonian flow at time t (t > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let δ = (δ1, δ2) > 0 and c > 0 given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Assume that ∥DW∥G,ρ,c ≤ ˆδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, Φt maps Dρ−tδ(G) into Dρ(G) and one has: (1) |Φt − Id|G,ρ−tδ,c ≤ t∥DW∥G,ρ,c, (2) Φ(Dρ(G)) ⊃ Dρ−tδ(G) for ρ′ ≤ ρ − tδ, (3) Assuming that ∥DW∥G,ρ,c < ˆδc/2e, for any given function f analytic on Dρ(G), and for any integer m ≥ 0, the following bound holds: ∥rm(f, W, t)∥G,ρ−tδ ≤ ∞ � l=0 � 1 �l+m m � · �2e∥DW∥G,ρ,c ˆδc �l� tm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='∥Lm Wf∥G,ρ = γm �2e∥DW∥G,ρ,c ˆδc � tm∥Lm Wf∥G,ρ, where for 0 ≤ x ≤ 1 we define γm(x) := ∞ � l=0 l!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (l + m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='xl Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' During the proof we are going to denote Φs(φ0, I0) by (φ(s), I(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Let us find the coordinate expression of the hamiltonian flow for the expression 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content='2 of a bm-symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Recall that the equation for the hamiltonian flow is d dsφi(s) = {φi, W} and d dsIi(s) = {Ii, W}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' {φi, W} = 1 ��m j=1 cj Ij 1 � �∂φi ∂I1 ∂W ∂φ1 − ∂φi ∂φ1 ∂W ∂I1 � + n � j=2 �∂φi ∂Ij ∂W ∂φj − ∂φi ∂φj ∂W ∂Ij � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Hence, d dsφi(s) = − 1 ��m j=1 cj Ij 1 � ∂W ∂I1 if i = 1 and d dsφi(s) = −∂W ∂Ii if i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' On the other side, {Ii, W} = 1 ��m j=1 cj Ij 1 � � ∂Ii ∂I1 ∂W ∂φ1 − ∂Ii ∂φ1 ∂W ∂I1 � + n � j=2 � ∂Ii ∂Ij ∂W ∂φj − ∂Ii ∂φj ∂W ∂Ij � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' 60 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' A NEW KAM THEOREM Hence, d dsIi(s) = 1 ��m j=1 cj Ij 1 � ∂W ∂φ1 if i = 1 and d dsIi(s) = ∂W ∂φi if i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' (1) Assume now that 0 < s0 ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQfbff_/content/2301.00266v1.pdf'} +page_content=' Then, |φ(s0) − φ0|∞ ≤ s0 sup0 19.71a4/3. +Moreover, under the condition |t| > 86.58a4/3, we provide an explicit lower bound on the +expression ||qx|| for all large q ∈ Z. These results are based on the recently discovered +continued fractions of cubic irrationals [1] and improve the currently best-known bounds +of Wakabayashi. +Keywords: cubic irrationals, continued fractions, irrationality exponent, effective irrationality exponent +Math Subject Classification 2010: 11J68, 11J70, 11J82 +1 +Introduction +The irrationality exponent λ(x) of an irrational real number x is defined as the supremum of +real numbers λ such that the inequality +����x − p +q +���� < q−λ +(1) +has infinitely many rational solutions p/q. It follows from the classical Dirichlet theorem that +for all x ∈ R \ Q, λ(x) ⩾ 2. On the other hand, Khintchine’s theorem implies that almost +all x ∈ R with respect to the Lebesgue measure satisfy λ(x) = 2. In the first half of the +XX century, there was a big interest in estimating the irrationality exponent of real algebraic +numbers. It culminated in 1955 with the work of Roth [8], who established the best possible +result, i.e. that for any algebraic x ∈ R\Q, λ(x) = 2. Unfortunately, that result is ineffective, +i.e. for λ > 2 it does not allow us to find all rational p/q that satisfy (1). Therefore, for +example, it can not be used to solve the Thue equations +F(p, q) = c +in integer p, q, where F ∈ Z[x, y] is a homogeneous polynomial of degree d ⩾ 3 and c is +some integer parameter. Since then, many mathematicians were working on effective results +regarding the irratoinality exponents of algebraic numbers. +Given x ∈ R \ Q, by the effective irrationality exponent of x we define a positive real +number λeff(x) such that for all λ > λeff(x) there exists an effectively computable Q > 0 +such that all rational solutions of the inequality (1) in reduced form satisfy q ⩽ Q. +All known upper bounds on λeff(x) for algebraic real x are much weaker than in Roth’s +theorem. First of all, the classical theorem of Liouville states that λeff(x) ⩽ d where d is +the degree of x. Therefore any non-trivial bound on λeff(x) should be strictly smaller than +1 + +d. One of the notable improvements of Liouville’s bound is based on Feldman’s refinement of +the theory of linear forms in logarithms [6]. Its advantage is that it gives λeff(x) < d for all +algebraic numbers. However, the difference between λeff(x) and d is usually extremely small. +For state-of-the-art results regarding this approach, we refer to the book of Bugeaud [5]. For +other notable achievements on this problem, we refer to [3, 4] and the references therein. +In this paper, we focus on the case of cubic irrationals. +Multiplying by some integer +number and shifting by another rational number, we can always guarantee that the minimal +polynomial of the resulting cubic x is x3 + px + q for some p, q ∈ Z. Notice also that such a +transformation does not change the (effective) irrationality exponent of x. The first general +result about λeff(x) for these specific values x was achieved by Bombieri, van der Poorten +and Vaaler [4] in 1996. They showed that under the conditions |p| > e1000 and |p| ⩾ q2, one +has +λeff(x) ⩽ 2 log(|p|3)| +log(|p|3/q2) + +14 +(log(|p|3/q2))1/3 . +Later, Wakabayashi [9] improved that bound and showed that λeff(x) ⩽ λw(p, q). It becomes +non-trivial (i.e. smaller than 3) under the condition +|p| > 22/334|q|8/3 +� +1 + +1 +390|q|3 +�2/3 +(2) +and for large p and q it asymptotically behaves like λw(p, q) +∼ +2 + (4 log |q| + +2 log 108)/(3 log |p|). +In this paper we investigate what estimates on λeff(x) can be achieved with help of the +convergents of the recently discovered continued fractions [1] of cubic irrationals with the +minimal polynomial x3 − tx2 − a ∈ Z[x]. It was shown there that, as soon as |t|3 > 12a > 0, +the real root of this equation with the largest absolute value admits the continued fraction +x = K +� +t +3(12k + 1)(3k + 1)α +3(12k + 5)(3k + 2)α +3(12k + 7)(6k + 5)α +3(12k − 1)(6k + 1)α +(2i + 1)t2 +(2i + 1)t +2(2i + 1)t2 +(2i + 1)t +� +. +Here i is the index of the corresponding partial quotient and k = +� i +4 +� +. Notice that the change +of variables y = q +x transforms cubic x from [4, 9] to numbers in this paper with t = −p and +a = −q2. +The resulting upper bounds on λeff(x) (see Theorems 1 and 2) depend on prime factori- +sations of a and t but in any case they are better than those in [9]. Theorem 1 states that +under the condition |t|3 > 12|a| the largest real root of the above cubic equation satisfies +||qx|| ⩾ τ(t, a)q−λ(t,a)(log(8|t1|q))λ(t,a)−1/2, +for all q > Q0(t, a), +where all values for τ(t, a), λ(t, a) and q0(t, a) are explicitly provided. This inequality always +gives a non-trivial upper bound on λeff(x) under the condition |t| > 86.58a4/3. It translates +to the condition |p| > 86.58|q|8/3 which is better than (2), where even without the term in +brackets we have |p| > 128.57|q|8/3. Moreover, the parameter a does not have to be of the +form −q2, therefore many cubics from this paper do not plainly transfer to those in [4, 9]. +On top of that, Theorem 2 provides an even better upper bound on λeff(x) but does not +give an explicit lower bound on ||qx|| as above. That bound always becomes non-trivial as +soon as |t| > 19.71a4/3. These bounds between t and a are obtained in Section 3. +2 +Preliminaries and main results +Consider the real root of the equation x3 − tx2 − a = 0 that has the largest absolute value +among other roots of the same equation. Notice that a ̸= 0 because otherwise x is not cubic. +Also, by replacing x with −x if needed, we can guarantee that a > 0. +2 + +Next, by the standard CF transformations (see [1, Lemma 1]), we can cancel some +common divisors of t and 3a from the continued fraction of x. Let g1 := gcd(t2, 3a) and +g2 := gcd(t, 3a/g1). For convenience, we denote t2 = g1t2, t = g2t1 and 3a = g1g2a∗. We di- +vide the following partial quotients of x by g1: β1, a1, β2; β3, a3, β4; . . . , β2k−1, a2k−1, β2k,. . . . +After that we divide the following partial quotients by g2: +β0, a0, β1; β2, a2, β3; . . . ; +β2k, a2k, β2k+1,. . . The resulting continued fraction has the same limit x as the initial one. +To make β0 integer, we consider the number x/g2 instead of x. Its continued fraction is then +K +� +t1 +(12k + 1)(3k + 1)a∗ +(12k + 5)(3k + 2)a∗ +(12k + 7)(6k + 5)a∗ +(12k − 1)(6k + 1)a∗ +(2i + 1)t2 +(2i + 1)t1 +2(2i + 1)t2 +(2i + 1)t1 +· · · +� +(3) +We define the following notions: +c6 = c6(t1, t2, a∗) := + + + + + + + + + +√64t1t2 + 270a∗ +27/4ec1 +if t > 0 +� +64|t1t2| − 54a∗ +27/4ec1 +if t < 0, +(4) +c7 = c7(t1, t2, a∗) := + + + + + + + + + + + +27/4ec1(16t1t2 + 9a∗) +9a∗√64t1t2 + 270a∗ +if t > 0 +223/4ec1(|t1t2| − 3a∗) +9a∗� +64|t1t2| − 54a∗ +if t < 0, +(5) +where c1 = c1(a∗) is defined in (10). Next, +τ = τ(t1, t2, a∗) := g2 · (log c7)1/2 +8c8 +6(8|t1|) +log c6 +log c7 +; +q0 = q0(t1, t2, a∗) := +c4 +7 +8|t1|. +(6) +The main result of this paper is +Theorem 1 For all integer q ⩾ q0 one has +||qx|| > τq−λ log(8|t1|q)−λ−1/2 +where λ = log c6 +log c7. In particular, λeff(x) ⩽ λ. +As shown in Section 5, the constant c1 can be replaced by a bigger constant c2 defined +in (11). But in that case, writing such an explicit inequality as in Theorem 1 is much harder +(but theoretically possible). We do not provide it in the exact form here but only state the +following result. Let c∗ +6 and c∗ +7 be defined in the same way as c6 and c7 but with the constant +c2 instead of c1. +Theorem 2 The effective irrationality exponent of x satisfies +λeff(x) ⩽ λ∗ := log c∗ +6 +log c∗ +7 +. +3 +Analysis of the results +Theorem 1 provides a nontrivial lower bound for ||qx|| as soon as log c6 +log c7 is strictly less that 2 +or in other words, c6 < c2 +7. In view of (4) and (5), for t > 0 this is equivalent to +√64t1t2 + 270a∗ +27/4ec1 +< 27/2e2c2 +1(16t1t2 + 9a∗)2 +81a∗2(64t1t2 + 270a∗) +3 + +⇐⇒ +a∗2(64t1t2 + 270a∗)3/2 < 221/4e3c3 +1 +81 +(16t1t2 + 9a∗)2. +Define the parameter u such that 16t1t2 = ua∗4 − 9a∗. Also for convenience define τ := +221/4e3c3 +1 +81 +. Then the last inequality rewrites +a∗2(4ua∗4 + 234a∗)3/2 < τu2a∗8 +⇐⇒ +4ua∗3 + 234 < τ 2/3u4/3a∗3. +Notice that for u ⩾ +� +4 +τ 2/3 + +234 +64a∗3 τ 4/3�3 +one has +τ 2/3u4/3a∗3 ⩾ +� +4 + 234 +64a∗3 τ 2 +� +· ua∗3 ⩾ 4ua∗3 + 234τ 2 +64a∗3 · 43 +τ 2 a∗3 = 4ua∗3 + 234 +and the condition c6 < c2 +7 is satisfied. +Recall that t1t2 = t3/(g1g2) and a∗ = 3a/(g1g2). +Therefore, 16t1t2 > ua∗4 − 9a∗ is +equivalent to 16t3 > +81 +(g1g2)3 ua4 − 27a. +From the definition (10) we see that c1 and hence τ depends on the prime factorisation +of a∗. If 3 ∤ a∗ then we always get c1 > 0.0924 which in turn implies τ > 0.00744 and +� 4 +τ 2/3 + 234 +64a∗3 τ 4/3 +�3 +⩽ 104.973. +On the other hand, we have g1g2 ⩾ 3. +Finally, we get that for t > 0 the non-trivial bound on λeff(x) is always achieved if +t > 104.97 +3 +· +� 81 +16 +�1/3 a4/3 ≈ 60.08a4/3, however for many pairs a and t it is satisfied under +essentially weaker conditions. +In the case 3 | a∗ we get c1 > 0.13329, thus τ > 0.0223 and +� 4 +τ 2/3 + 234 +64a∗3 τ 4/3 +�3 +⩽ 50.423. +Then the non-trivial bound is achieved if t > 50.42 · +� 81 +16 +�1/3 a4/3 ≈ 86.57a4/3. +The case t < 0 is dealt analogously. The condition c6 < c2 +7 is equivalent to +� +64|t1t2| − 54a∗ +27/4ec1 +< 27/2e2c2 +1(16|t1t2| − 48a∗)2 +81a∗2(64|t1t2| − 54a∗) +⇐⇒ +a∗2(64|t1t2| − 54a∗)3/2 < τ(16|t1t2| − 48a∗)2. +Define u such that 16|t1t2| = ua∗4 + 48a∗. Then the last inequality rewrites +4ua∗3 + 138 < τ 2/3u4/3a∗3. +One can check that the last inequality is satisfied for u ⩾ +� +4 +τ 2/3 + 138τ 4/3 +64a∗3 +�3 +. As in the case of +positive t, the right hand side is always smaller than 104.933 in the case 3 ∤ a∗ and is smaller +than 50.423 in the case 3 | a∗. Therefore for 3 ∤ a∗ the condition c6 < c2 +7 is always satisfied +if |t|3 > 104.933·81 +33·16 +a4 + 9a which follows from |t| > 60.06a4/3. For 3 | a∗, similar computations +give us |t|3 > 50.423·81 +16 +a4 + 9a which follows from |t| > 86.58a4/3. +One can repeat the same analysis as above for Theorem 2. In that case, the constant +c1 in the computations should be replaced by c2. We have that if 3 ∤ a∗ it always satisfies +c2 ⩾ 0.1939, which in turn implies τ ⩾ 0.0688 and then +� 4 +τ 2/3 + 234 +64a∗3 τ 4/3 +�3 +⩽ 23.933. +4 + +If 3 | a∗, we have c2 ⩾ 0.2797, τ ⩾ 0.206 and +� 4 +τ 2/3 + 234 +64a∗3 τ 4/3 +�3 +⩽ 11.473. +Finally, for the case 3 ∤ a∗, the condition c6 < c2 +7 is always satisfied if |t|3 > 23.933·81 +33·16 +a4 +9a +which follows from |t| > 13.72a4/3. For the case 3 | a∗ similar calculations give |t| > 19.71a4/3. +4 +Nice, convenient and perfect continued fractions +Definition 1 Let x be a continued fraction given by +x = K +� β0 +β1 +β2 +a0 +a1 +a2 · · · +� +; +βi, ai ∈ Z, ∀i ∈ Z⩾0. +For given positive integers k, r with −1 ⩽ r ⩽ k we define +γk,r := +k+r +� +i=k−r +2|(i−k+r) +βi, +γk,−1 := 1. +We say that x is d-nice at index k where d, k ∈ N if d | ak and for all positive integer r ⩽ k +one has ak−rβk+rγk,r−2 ≡ −ak+rγk,r−1 (mod d). We call x (d, r)-perfect at index k, where +0 ⩽ r ⩽ k if it is d-nice at index k and βk−r ≡ βk+r+1 ≡ 0 (mod d). +We say that x is eventually d-nice at index k from position k0 if the same conditions +as above are satisfied for all 0 ⩽ r ⩽ k − k0. In this paper the value k0 will often be a fixed +absolute constant. If there is no confusion about its value we will omit it in the text. +Let pn/qn be the n’th convergent of x. Define the following matrices +Sn := +� +pn +qn +pn−1 +qn−1 +� +; +Cn := +� an +βn +1 +0 +� +. +From the theory of continued fractions we know that Sn = CnCn−1 · · · C0. To make this +product shorter, we use the usual notation but in the descending order: Sn = �0 +i=n Ci. +Lemma 1 Let the continued fraction x be eventually d-nice at index k from the position k0. +Then for all 0 ⩽ r ⩽ k − k0 one has +k−r +� +i=k+r +Ci ≡ +� +0 +γk,r +γk,r−1 +0 +� +(mod d). +Moreover, if x is (d, r)-perfect at index k then �k−r +i=k+r+1 Ci ≡ 0 (mod d). +Proof. +We prove by induction on r. For r = 0 the statement is straightforward. Suppose that +the statement is true for r and verify it for r + 1. +k−r−1 +� +i=k+r+1 +Ci +≡ +� ak+r+1 +βk+r+1 +1 +0 +� � +0 +γk,r +γk,r−1 +0 +� � ak−r−1 +βk−r−1 +1 +0 +� +≡ +� ak−r−1βk+r+1γk,r−1 + ak+r+1γk,r +βk−r−1βk+r+1γk,r−1 +γk,r +0 +� +(mod d). +5 + +By the conditions of d-nice CF at index k, the last matrix is congruent to +� +0 +γk,r+1 +γk,r +0 +� +. +If x is (d, r)-perfect at index k then γk,r ≡ 0 (mod d) and we have +k−r +� +i=k+r+1 +Ci ≡ +� ak+r+1 +0 +1 +0 +� � +0 +γk,r +γk,r−1 +0 +� +≡ 0 (mod d). +⊠ +Another two properties of d-nice continued fractions that easily follow from the definition +are +• Let d1, d2 be two coprime positive integer numbers. If a continued fraction is eventually +d1-nice and eventually d2-nice at the same index k for the same position k0 then it is +eventually d1d2-nice at index k. +• If a continued fraction is eventually d-nice at index k then it is also eventually e-nice +at the same index from the same position for all positive integer divisors e of d. +Definition 2 We say that the continued fraction x is (eventually) d-convenient at index k if +there exists a sequence (cr)0⩽r⩽⌊k/2⌋ of residues modulo m such that for all positive integers +r ⩽ k (resp. r ⩽ k − k0) one has +• βk+r+1 ≡ c⌊ r +2 ⌋βk−r (mod d); +• if r is odd then ak+r ≡ −c⌊ r +2⌋ak−r (mod d); +• if r is even then ak+r ≡ −ak−r (mod d). +Lemma 2 Let d > 2. Then any eventually d-convenient continued fraction at index k is +also eventually d-nice. For d = 2, any d-convenient continued fraction at index k such that +ak ≡ 0 (mod d) is also d-nice. +Proof. First of all, for d > 2 and r = 0 the condition ak+r ≡ −ak−r (mod d) implies that +ak ≡ 0 (mod d), which is the first condition of d-nice CF. +Secondly, one can check that the first condition of d-convenient CF implies that for odd +r, γk,r ≡ c⌊ r +2 ⌋γk,r−1βk−r ≡ γk,r−1βk+r+1 (mod d). Then we get +ak−r−1βk+r+1γk,r−1 ≡ ak−r−1γk,r ≡ −ak+r+1γk,r (mod d) +and the second condition of d-nice CF is verified. +Thirdly, for even r we get cr/2γk,r ≡ cr/2γk,r−1βk−r ≡ γk,r−1βk+r+1 (mod d) and therefore +ak−r−1βk+r+1γk,r−1 ≡ cr/2ak−r−1γk,r ≡ −ak+r+1γk,r (mod d). +Again, the second condition of d-nice CF is satisfied. +⊠ +5 +Divisibility of entries of Sn +Lemma 3 Let k ∈ N, k ⩾ 2 and d be any integer divisor of 2k+1. The continued fraction (3) +is eventually d-convenient at index k from the position 2. Additionally, the same statement +is true for k ≡ 3 (mod 4) and d = 2. +6 + +In the further discussion we will always deal with eventually d-convenient or d-nice contin- +ued fractions from the position 2. Therefore, to make the description shorter, we will omit the +words ‘eventually’ and ‘from the position 2’ and call the continued fraction (3) d-convenient +or d-nice. +Proof. We will check the conditions of d-convenient continued fraction separately for +each of the cases, depending on k modulo 4. +Case k = 4m + 1. Then m ≡ − 3 +8 (mod d) and we use (3) to compute +ak+4r ≡ 8rt2, +βk+4r = a∗(12m + 1 + 12r)(3m + 1 + 3r) ≡ a∗ +16(24r − 7)(24r − 1) (mod d); +ak+4r+1 ≡ (8r + 2)t1, +βk+4r+1 ≡ a∗ +16(24r + 1)(24r + 7) (mod d); +ak+4r+2 ≡ 2(8r + 4)t2, +βk+4r+2 ≡ a∗ +8 (24r + 5)(24r + 11) (mod d); +ak+4r−1 ≡ (8r − 2)t1, +βk+4r−1 ≡ a∗ +8 (24r − 11)(24r − 5) (mod d). +The conditions of d-convenient continued fraction at index k can now be easily checked where +cr is the constant 1 sequence. +We proceed the same way in all other cases. +Case k = 4m + 2. Then m ≡ − 5 +8 (mod d) and +ak+4r ≡ 8rt1, +βk+4r ≡ a∗ +16(24r − 5)(24r + 1) (mod d); +ak+4r+1 ≡ 2(8r + 2)t2, +βk+4r+1 ≡ a∗ +8 (24r − 1)(24r + 5) (mod d); +ak+4r+2 ≡ (8r + 4)t1, +βk+4r+2 ≡ a∗ +8 (24r + 7)(24r + 13) (mod d); +ak+4r−1 ≡ (8r − 2)t2, +βk+4r−1 ≡ a∗ +16(24r − 13)(24r − 7) (mod d). +One can easily check that for s ≡ 0, 1 (mod 4), βk+s+1 ≡ 2βk−s (mod d) and for s ≡ 2, 3 +(mod 4), βk+s+1 ≡ 2−1βk−s (mod d). Also, for s ≡ 1 (mod 4), ak+s ≡ 2ak−s (mod d) and +for s ≡ 3 (mod 4), ak+s ≡ 2−1ak−s (mod d). Hence, the conditions of d-convenient CF are +verified, where the sequence cs is periodic with the period 2, 2−1. +Case k = 4m + 3, d ̸= 2. Then m ≡ − 7 +8 (mod d) and +ak+4r ≡ 16rt2, +βk+4r ≡ a∗ +8 (24r − 7)(24r − 1) (mod d); +ak+4r+1 ≡ (8r + 2)t1, +βk+4r+1 ≡ a∗ +8 (24r + 1)(24r + 7) (mod d); +ak+4r+2 ≡ (8r + 4)t2, +βk+4r+2 ≡ a∗ +16(24r + 5)(24r + 11) (mod d); +ak+4r−1 ≡ (8r − 2)t1, +βk+4r−1 ≡ a∗ +16(24r − 11)(24r − 5) (mod d). +One can then check the conditions of d-convenient CF at index k for the constant 1 sequence +cr. +Case k = 4m + 3, d = 2. +in this case one can easily see that ak+4r ≡ 0 (mod 2), +ak+4r+1 ≡ ak+4r+3 ≡ t1 (mod 2), ak+4r+2 ≡ t2 (mod 2); βk+4r ≡ βk+4r+1 ≡ a∗ (mod 2) and +7 + +βk+4r+2 ≡ βk+4r−1 (mod 2). Therefore, the CF is 2-convenient at index k with the constant +1 sequence cr. +Case k = 4m. Then m ≡ − 1 +8 (mod d) and +ak+4r ≡ 8rt1, +βk+4r ≡ a∗ +8 (24r − 5)(24r + 1) (mod d); +ak+4r+1 ≡ (8r + 2)t2, +βk+4r+1 ≡ a∗ +16(24r − 1)(24r + 5) (mod d); +ak+4r+2 ≡ (8r + 4)t1, +βk+4r+2 ≡ a∗ +16(24r + 7)(24r + 13) (mod d); +ak+4r−1 ≡ 2(8r − 2)t2, +βk+4r−1 ≡ a∗ +8 (24r − 13)(24r − 7) (mod d). +One can then check the conditions of d-convenient CF with the periodic sequence cr with the +period 2−1, 2. +⊠ +Lemmata 2 and 3 show that x is d-nice at each index k ⩾ 2 for appropriately chosen d. +As the next step, we show that for almost every prime p, it is also (p, t)-perfect at infinitely +many carefully chosen indices k and t. This fact will allow us to show that all the entries of +CkCk−1 · · · C1 are multiples of some big power of p. +First of all, let’s consider the case p > 2 and p | a∗. Let s ∈ N be such that ps||a∗. +Consider q = pl for some 1 ⩽ l ⩽ s. +If we write q = 2m + 1 then we have q | αk for +k = m + rq = (2r+1)q−1 +2 +where r ∈ Z⩾0. One can easily see that for any such value of k, x is +(q, 0)-perfect at index k. In view of Lemma 1, we can then split the product Sk into +� +2k+q−1 +2q +� +groups such that all entries of the resulting product matrix in each group are multiples of q. +Finally, we combine this information for ql for all 1 ⩽ l ⩽ s and derive that all entries of the +product �2 +i=k Ci are divisible by +p +s� +i=1 +� +2k+pi−1 +2pi +� +. +Next, consider the case p = 2 and p | a∗. We have p | ak for all k ≡ 3 (mod 4) and one can +easily see that for all such k, x is (p, 0) perfect at index k. Then the analogous application +of Lemma 1 as in the previous case implies that all entries of �2 +i=k Ci are divisible by 2⌊k/4⌋. +For the case p = 2, p ∤ a∗ the result is slightly weaker. From (3) one can verify that +β8m+2 ≡ β8m+5 ≡ 0 (mod 2) for all m ∈ Z⩾0 and therefore x is (2, 1)-perfect at indices +8m + 3. Then Lemma 1 then implies that all entries of �2 +i=k Ci are divisible by 2⌊(k+3)/8⌋. +Finally, in the next lemma we consider the remaining case of p ∈ N that do not divide a∗. +Lemma 4 Let p ∈ N be such that gcd(p, 6) = 1. Then for all k ∈ Z all the entries of the +product of matrices �2 +i=k Ci are divisible by +p +� +3k+p−2 +3p +� +. +Proof. We prove by routinely considering all the cases, depending on p modulo 12. +Case p = 12m + 1. Then with help of (3) one can verify that for all r ∈ Z⩾0, +0 +≡ β4(m+rp)+1 ≡ β4(2m+rp) ≡ β4(4m+rp)+1 ≡ β4(5m+rp)+2 ≡ β4(7m+rp)+3 ≡ β4(8m+rp)+2 +≡ β4(10m+rp)+3 ≡ β4(11m+rp)+4 (mod p) +8 + +and 0 ≡ a6m+rp (mod p). In view of Lemma 3, we then derive that x is (p, 2m − 1)-perfect +at indices k = 6m + 2rp for all r ∈ Z⩾0 and is (p, 2m)-perfect at indices k = 6m + (2r + 1)p. +Lemma 1 then implies that all the entries of the following products of matrices are divisible +by p: +4m+2rp+1 +� +i=8m+2rp +Ci, +16m+4rp+1 +� +i=20m+2rp+2 +Ci. +Finally, one can easily check that for k = (n + 1)p − p−1 +3 , the product �2 +i=k Ci contains n + 1 +blocks of the above form. Therefore all its entries are divisible by pn+1. +The other cases are done analogously. +Case p = 12m + 5. One verifies that for all r ∈ Z⩾0, +0 +≡ β4(m+rp)+2 ≡ β4(2m+rp)+3 ≡ β4(4m+rp)+6 ≡ β4(5m+rp)+9 ≡ β4(7m+rp)+12 ≡ β4(8m+rp)+13 +≡ β4(10m+rp)+16 ≡ β4(11m+rp)+19 (mod p) +and 0 ≡ a6m+rp+2 (mod p). +Then Lemma 3 implies that x is (p, 2m)-perfect at indices +k = 6m + 2rp for all r ∈ Z and is (p, 2m − 1)-perfect at indices k = 6m + (2r + 1)p. Lemma 1 +then implies that all the entries of the following products are divisible by p: +4m+2rp+2 +� +i=8m+2rp+3 +Ci, +16m+4rp+6 +� +i=20m+2rp+9 +Ci. +For k ⩾ (n + 1)p − p−2 +3 +one can easily check that the product �2 +i=k Ci contains n + 1 blocks +of the above form. Therefore all its entries are divisible by pn+1. +Case p = 12m + 7. Then for all r ∈ Z⩾0, +0 +≡ β4(m+rp)+3 ≡ β4(2m+rp)+4 ≡ β4(4m+rp)+9 ≡ β4(5m+rp)+12 ≡ β4(7m+rp)+17 ≡ β4(8m+rp)+18 +≡ β4(10m+rp)+23 ≡ β4(11m+rp)+26 (mod p) +and 0 ≡ a6m+rp+3 (mod p). Lemmata 3 and 1 imply that all the entries of the following +products are divisible by p: +4m+2rp+3 +� +i=8m+2rp+4 +Ci, +16m+4rp+9 +� +i=20m+2rp+12 +Ci. +For k ⩾ (n + 1)p − p−1 +3 +one can easily check that the product �2 +i=k Ci contains n + 1 blocks +of the above form. Therefore all its entries are divisible by pn+1. +Case p = 12m + 11. Then for all r ∈ Z⩾0, +0 +≡ β4(m+rp)+4 ≡ β4(2m+rp)+7 ≡ β4(4m+rp)+14 ≡ β4(5m+rp)+19 ≡ β4(7m+rp)+26 ≡ β4(8m+rp)+29 +≡ β4(10m+rp)+36 ≡ β4(11m+rp)+41 (mod p) +and 0 ≡ a6m+rp+5 (mod p). Lemmata 3 and 1 imply that all the entries of the following +products are divisible by p: +4m+2rp+4 +� +i=8m+2rp+7 +Ci, +16m+4rp+14 +� +i=20m+2rp+19 +Ci. +For k ⩾ (n + 1)p − p−2 +3 +one can easily check that the product �2 +i=k Ci contains n + 1 blocks +of the above form. Therefore all its entries are divisible by pn+1. +9 + +In all four cases we have that for k ⩾ (n+1)p− p−2 +3 +all the entries of �2 +i=k Ci are divisible +by pn+1. Writing it in terms of k we get that this power of p is +� +k + p−2 +3 +p +� += +�3k + p − 2 +3p +� +. +⊠ +We combine all the divisibility properties of �1 +i=n Ci together and get the following +Proposition 1 Let the prime factorisation of a∗ be a∗ = 2σ0pσ1 +1 pσ2 +2 · · · pσd +d +where σ0 can +be equal to zero while the other powers σi are strictly positive. Define P1 := {p1, . . . , pd}, +P2 := P \ (P1 ∪ {2, 3}). If 2 | a∗ then +gcd(pn, qn) ⩾ 2 +� +n +4 +� � +pi∈P1 +p +σi +� +j=1 +� +2n+pj −1 +2pj +� +i +· +� +p∈P2 +p +� +3n+p−2 +3p +� +. +(7) +If 2 ∤ a∗ then +gcd(pn, qn) ⩾ 2 +� +n+3 +8 +� � +pi∈P1 +p +σi +� +j=1 +� +2n+pj −1 +2pj +� +i +· +� +p∈P2 +p +� +3n+p−2 +3p +� +. +(8) +We now provide shorter lower bounds for (7) and (8) and then provide slightly better +ones that, after some efforts, can still be made effective for large enough n. Observe that +� 2n+pj−1 +2pj +� +⩾ +� n +pj +� +and +� 3n+p−2 +3p +� +⩾ +� n +p +� +. For convenience, if 3 ∤ a∗ we still add 3 to the set P1 +by setting pd+1 := 3, σd+1 := 0. Then +2 +�∞ +i=1 +n +2i · +� +pj∈P1 +p +�σj +i=1 +� +n +pi +j +� +j +· +� +p∈P2 +p +� +n +p +� +· +� +pj∈P1 +p +�∞ +i=σj+1 +n +pi +j +j +· +� +p∈P2 +p +�∞ +i=2 +n +pi ⩾ n! ⩾ +√ +2πn +�n +e +�n +. +The last inequality infers that +gcd(pn, qn) ⩾ +√ +2πn(c1n)n +(9) +where c1 = c1(a∗) is defined as +c1 = + + + + + + + + + +2−3/4 exp +� +−1 − � +pj∈P1 +ln pj +p +σj +j (pj−1) − � +p∈P2 +ln p +p(p−1) +� +if 2 | a∗; +2−7/8 exp +� +−1 − � +pj∈P1 +ln pj +p +σj +j (pj−1) − � +p∈P2 +ln p +p(p−1) +� +if 2 ∤ a∗. +(10) +c1 reaches its minimal value in the case P1 = {3} with σ1 = 0. Then c1 ≈ 0.0924. However, if +a∗ = 3 then c1 ≈ 0.1333. In general, more squares of small prime numbers divide a∗, bigger +is the value of c1. +We can provide a better asymptotic lower estimate on gcd(pn, qn) for large enough n. The +exact condition on n can be effectively computed, however the computations will not be nice. +Consider a prime p ∈ P2. The term p +� +3n+p−2 +3p +� +has an extra power of p compared to p⌊n/p⌋ if +for some integer k, +n +p < k ⩽ 3n + p − 2 +3p +⇐⇒ +n +k < p ⩽ 3n − 2 +3k − 1. +10 + +We also have � +p∈P1 p ≍ 1 where the implied constants only depend on a∗ but not on n. +Define the set +K := +n� +k=1 +�n +k , 3n − 2 +3k − 1 +� +Then gcd(pn, qn) ⩾ T · +√ +2πn(c1n)n where +T ≍ +� +p∈P∩K +p = exp + + � +p∈K∩P +ln p + + = exp +� n +� +k=1 +� +θ +�3n − 2 +3k − 1 +� +− θ +�n +k +��� +, +where θ(x) is the first Chebyshev function. It is well known (see [7] for example) that for +large enough x, |θ(x) − x| < +x +2 ln x. Therefore for y > x one has θ(y) − θ(x) ⩾ y − x − +y +ln y. +This implies +√ +ln n +� +k=1 +� +θ +�3n − 2 +3k − 1 +� +− θ +�n +k +�� +⩾ +√ +ln n +� +k=1 +n − 2k +k(3k − 1) − O +� +n +√ +ln n +� += n +√ +ln n +� +k=1 +1 +k(3k − 1) − O +� +n +√ +ln n +� +. +For any ε > 0 and for large enough n, the last expression can be made bigger that (τ − ε)n +where τ := �∞ +k=1 +1 +k(3k−1) ≈ 0.74102. Therefore T ≫ e(τ−ε)n = γn(1−ε) where γ = eτ. Finally, +we get +gcd(pn, qn) ≫ ((c2 − δ)n)n, +where c2 = c1 · γ, +(11) +δ can be made arbitrarily small and the implied constant in the inequality only depends on +a∗ and δ but not on n. For the case a∗ = 1, when the constant c1 is minimal possible, we get +c2 ≈ 0.1939. Respectively, for a∗ = 6, c2 ≈ 0.2797. +6 +Lower and upper bounds on the denominators qn. +In this section we will get upper and lower bounds of the denominators qn, compared to qn−1. +Since the recurrent formulae between qn, qn−1 and qn−2 depend on n modulo 4, it makes sense +to compare q4k and q4k+4. +We adapt some notation from [1]. Denote +T4k := +� +p4k +q4k +p4k−4 +q4k−4 +� +Then [1, (69) and (70)] one has +T4k+4 = +� ak11 +ak12 +1 +0 +� +S4k +(12) +where ak11 and ak12 are the corresponding indices of C4k+4C4k+3C4k+2C4k+1. In view of (3), +one computes +ak11 = 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)(t1t2)2 ++6(8k + 5)(8k + 7)(36k2 + 55k + 16)a∗t1t2 ++(12k + 5)(12k + 11)(3k + 2)(6k + 7)a∗2 +; +(13) +ak12 = 2(12k + 1)(3k + 1)(8k + 7)a∗t1((8k + 5)(8k + 9)t1t2 + 2(36k2 + 63k + 25)a∗). +To make the notation shorter, we write ak12 = 2(12k + 1)(3k + 1)(8k + 7)a∗t1p(k) where p(k) +is a polynomial of k with parameters t1t2 and a∗. +Then an easy adaptation of the proof of [1, Lemma 16] gives +11 + +Lemma 5 Let a∗ ∈ N and t1, t2 ∈ Z satisfy 12a∗ ⩽ |t1t2|. Then q4k+4 and q4k satisfy the +relation +|q4k+4| > (8k + 3)(8k + 5)(8k + 7)(8k + 9)(t1t2 + 2a∗)2|q4k|. +(14) +Now we will provide an opposite inequality between the denominators q4k+4 and q4k. +Three consecutive denominators of this form are related by the equation [1, (72)]: +q4k+4 = ak11q4k + (dq4k−4 − bk21q4k)ak12 +bk22 +, +(15) +where bk21/d and bk22/d are the corresponding entries of C−1 +4k−2C−1 +4k−1C−1 +4k , i.e. +d = −(12k − 7)(12k − 5)(12k − 1)(3k − 1)(6k − 1)(6k + 1)a∗3, +(16) +bk21 = −(12k − 5)(6k − 1)a∗ − 2(8k − 3)(8k − 1)t1t2, +bk22 = 2(8k − 1)t1((8k − 3)(8k + 1)t1t2 + 2(36k2 − 9k − 2)a∗) =: 2(8k − 1)t1p(k − 1). +Lemma 6 Let a∗, t1, t2 be the same as in Lemma 5. Then q4k+4 and q4k satisfy the following +relations: +|q4k+4| ⩽ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) +� +t1t2 + 135 +32 a∗ +�2 +|q4k|, +if t1t2 > 0, +(17) +|q4k+4| ⩽ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) +� +t1t2 + 27 +32a∗ +�2 +|q4k|, +if t1t2 < 0. +(18) +Proof. First, we estimate the terms in (15). Since |t1t2| ⩾ 12a∗, we get for all k ⩾ 1 +that +1 +12(12k − 5)(6k − 1)(12a∗) < (8k − 3)(8k − 1)(12a∗) ⩽ (8k − 3)(8k − 1)|t1t2|. Therefore +|bk21| ⩽ 3(8k − 3)(8k − 1)|t1t2|. +(19) +Next, by Lemma 5 we have +|dq4k−4| ⩽ (12k − 7)(12k − 5)(12k − 1)(3k − 1)(6k − 1)(6k + 1)a∗3 +(8k − 5)(8k − 3)(8k − 1)(8k + 1)(t1t2 + 2a∗)2 +q4k. +Since |t1t2 + 2a∗| ⩾ 10a∗ and 12a∗ ⩽ |t1t2|, one can verify that +|dq4k−4| < (8k − 3)(8k − 1)|t1t2q4k|. +(20) +Next, we have (8k + 5)(8k + 9)|t1t2| > 12(36k2 + 63k + 25)a∗, therefore we always have +|p(k)| +(8k + 5)(8k + 9)|t1t2| ∈ +� � +1, 7 +6 +� +if t1t2 ⩾ 0; +� 5 +6, 1 +� +if t1t2 < 0. +The last inequality in turn implies that for k ⩾ 1 the ratio ak12/bk22 is always positive and +satisfies +ak12 +bk12 +⩽ 6(12k + 1)(3k + 1)(8k + 7)a∗ +8k − 1 +. +(21) +Assume that t1t2 ⩾ 0. In that case, the last inequality together with (19) and (20) imply +that +����(dq4k−4 − bk21q4k)ak12 +bk22 +���� ⩽ 24(8k − 3)(8k + 7)(12k + 1)(3k + 1)a∗t1t2q4k. +12 + +One can check that for all k ⩾ 1, +6(8k + 5)(8k + 7)(36k2 + 55k + 16) + 24(8k − 3)(8k + 7)(12k + 1)(3k + 1) +2(8k + 3)(8k + 5)(8k + 7)(8k + 9) +< 135 +16 +(22) +and +81 +256 < (12k + 5)(12k + 11)(3k + 2)(6k + 7) +2(8k + 3)(8k + 5)(8k + 7)(8k + 9) +⩽ 23 +66 < 1. +(23) +These bounds together with the formula (13) and equation (15) imply the inequality (17) +for k ⩾ 1. +Finally, this bound can be easily verified for k = 0 from the equation q4 = +a011q0 + a012q−1 and q−1 = 0. +Consider the case t1 < 0. One can check that for all k ⩾ 1, +321 +187 ⩾ 6(8k + 5)(8k + 7)(36k2 + 55k + 16) +2(8k + 3)(8k + 5)(8k + 7)(8k + 9) ⩾ 27 +16. +(24) +This together with the condition |t1t2| > 12a∗2 imply that ak11 > 0 and q4k and q4k+4 share +the same sign for all k ∈ N. Next, since (12k − 5)(6k − 1)a∗ < (8k − 3)(8k − 1)|t1t2|, we +have that bk21 > 0 and then in view of (20) and ak12 +bk22 > 0, the term (dq4k−4 − bk21q4k)ak12 +bk22 has +the opposite sign compared to ak11q4k. That all implies that |q4k+4| ⩽ |ak11q4k|. Finally, the +inequalities (23) together with (24) establish the bound (18). +⊠ +Lemma 6 immediately implies that for t1t2 > 0, +|q4k| ⩽ 2k +� +t1t2 + 135 +32 a∗ +�2k +(8k + 1)!! ⩽ 16k +� +8 · 21/4e−1 +� +t1t2 + 135 +32 a∗ +�4k +k4k. +The case of t1t2 < 0 can be dealt with in a similar way. Finally, we get the estimate +|q4k| ⩽ 16kc4k +3 k4k, +(25) +where +c3 = c3(t1, t2, a∗) = + + + + + +8 · 21/4e−1 +� +t1t2 + 135 +32 a∗ +if t1t2 > 0 +8 · 21/4e−1 +� +|t1t2| − 27 +32a∗ +if t1t2 < 0. +Lemma 7 Under the same conditions on a∗, t1, t2 as in the previous lemma, one has +|q4k+4| ⩾ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) +� +t1t2 + 9 +16a∗ +�2 +|q4k|, +if t1t2 > 0, +(26) +|q4k+4| ⩾ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) (t1t2 + 3a∗)2 |q4k|, +if t1t2 < 0. +(27) +Proof. If t1t2 > 0 we have q4k+4 > ak11q4k. Then the lower bound in (23) together with +the lower bound in (24) imply the bound (26). +Now assume that t1t2 < 0. Then, as we have shown in the proof of Lemma 6, bk21 > 0 and +dq4k−4 and bk21q4k have the opposite signs. This together with ak12 +bk22 > 0, the inequality (21) +and 0 < bk21 ⩽ 2(8k − 3)(8k − 1)|t1t2| in turn imply that +|q4k+4| ⩾ |ak11q4k| − bk21ak12 +bk22 +|q4k| ⩾ (ak11 + 12(8k − 3)(12k + 1)(3k + 1)(8k + 7)a∗t1t2)|q4k| +13 + +We need to show that the expression +ak11+12(8k−3)(12k+1)(3k+1)(8k+7)a∗t1t2−2(8k+3)(8k+5)(8k+7)(8k+9) (t1t2 + 3a∗)2 +is always positive. Notice that after substituting (13) into it and expanding the brackets, the +term with (t1t2)2 disappears. The term for a∗t1t2 then equals to +−6(8k + 7)(160k3 + 1532k2 + 1063k + 196)a∗t1t2 +and the term for a∗2 is +−(71136k4 + 212976k3 + 227970k2 + 102633k + 16240) +(we made these computations with Wolfram Mathematika). Finally, one can check that in +the case |t1t2| > 12a∗, the absolute value of the first term is always bigger than that of the +second term and therefore the whole expression is positive. +Remark. +By performing neater computations, one can make the coefficient 3 in +(t1t2 + 3a∗)2 slightly smaller. However we decide not to further complicate already tedious +calculations. +⊠ +Analogously to (25), one can find shorter lower bounds for |q4k|. With help of the known +inequality (8k + 1)!! ⩾ 8k(8k/e)2k, Lemma 7 infers +|q4k| ⩾ 8kc4k +4 k4k, +(28) +where +c4 = c4(t1, t2, a∗) = + + + +8 · 21/4e−1 +� +t1t2 + 9 +16a∗ +if t1t2 > 0 +8 · 21/4e−1� +|t1t2| − 3a∗ +if t1t2 < 0. +7 +Distance between x and the convergents +From [1, Lemma 17] we know that, under the condition 12a∗ ⩽ |t1t2|, one has +����x − p4k +q4k +���� < 2 +���� +p4k +q4k +− p4k+4 +q4k+4 +���� . +In order to estimate the right hand side, we use the matrix equation [1, (72)]: +Tk+1 = +� ak11 − ak12 +bk21 +bk22 +dak12 +bk22 +1 +0 +� +Tk. +Notice that the values of d in fact depends on k (see the formula (16)). to emphasize this +dependence, in this section we write d(k) for it. Then the above equation gives the following +formula: +���� +p4k +q4k +− p4k+4 +q4k+4 +���� = +��� +�k +i=1 +d(i)ai12 +bi22 +��� · |p0q4 − q0p4| +q4kq4k+4 +. +We first compute its product term: +����� +k +� +i=1 +d(i)ai12 +bi22 +����� = +k +� +i=1 +2(12i − 7)(12i − 5)(12i − 1)(3i − 1)(6i − 1)(6i + 1)(12i + 1)(3i + 1)(8i + 7)a∗4|t1p(i)| +2(8i − 1)|t1p(i − 1)| +14 + += (8k + 7)|p(k)| +7|p(0)| +· (3k + 1)(6k + 1)(12k + 1)a∗4k(12k)! +26k34k(4k)! +⩽ +√ +3(3k + 1)(6k + 1)(12k + 1)(8k + 7)|p(k)| +7|p(0)| +· +�122k2a∗ +2 +√ +2e2 +�4k +. +Next, from (12) for k = 0 we get that |p0q4 − p4q0| = 14a∗t1|p(0)|. Finally, we unite all these +bounds together with the lower bounds (26), (27) and (28) for |q4k| to get +����x − p4k +q4k +���� ⩽ +2 +√ +3(3k + 1)(6k + 1)(12k + 1)(8k + 7)|t1a∗p(k)| +2(8k + 3)(8k + 5)(8k + 7)(8k + 9)(|t1t2| − 3a∗)2 · 64k2 · +� +122a∗ +2 +√ +2e2c2 +4 +�4k +To simplify the right hand side, notice that (3k+1)(6k+1)(12k+1) +(8k+3)(8k+5)(8k+9) < 27 +64. Next, since |t1t2| ⩾ +12a∗, one has |p(k)| = |(8k + 5)(8k + 9)t1t2 + 2(36k2 + 63k + 25)a∗| ⩽ 2(8k + 5)(8k + 9)|t1t2| +which for all k ⩾ 1 is smaller than 442k2|t1t2|. Finally, (|t1t2| − 3a∗)2 ⩾ +9 +16(t1t2)2. Collecting +all of these inequalities together gives, +����x − p4k +q4k +���� ⩽ +√ +3 · 27 · 442|t2 +1t2a∗| +64 · (9/16) · 64(t1t2)2 · +� +72a∗ +√ +2e2c2 +4 +�4k +⩽ |t1|c4k +5 , +(29) +where +c5 = + + + +9a∗ +(16t1t2+9a∗) +if t1t2 > 0 +9a∗ +16(|t1t2|−3a∗) +if t1t2 < 0. +8 +Estimating the irrationality exponent +In this section we establish Theorems refth1 and 2. Consider p∗ +k := p4k/ gcd(p4k, q4k) and +q∗ +k := q4k/ gcd(p4k, q4k). Definitely, they are both integers and (25) together with (9) imply +|q∗ +k| ⩽ 4 +� +2k +π +� c3 +4c1 +�4k +=: 4 +� +2k +π · c4k +6 . +(30) +For arbitrary δ > 0 and large enough k, one can use the inequality (11) to get +|q∗ +k| ≪ 16k +� +c3 +4(c2 − δ) +�4k +≪ +� c3 +4c2 ++ δ1 +�4k +=: (c∗ +6 + δ1)4k +(31) +where δ1 > 0 can be made arbitrarily close to zero for large enough k. Denote the upper +bound for b∗ +k by Q(k, t, a). +Next, we combine the last two inequalities with (29) and get +||q∗ +kx|| ⩽ |t1|c4k +5 · 4 +� +2k +π +� c3 +4c1 +�4k +⩽ 4|t1| +√ +k +�c3c5 +4c1 +�4k +=: 4|t1| +√ +kc−4k +7 +(32) +or +||q∗ +kx|| ≪ +� 4c2 +c3c5 +− δ2 +�−4k +=: (c∗ +7 − δ2)−4k +(33) +where δ2 can be made arbitrarily small and k is large enough, depending on δ2. Denote the +upper bound of ||q∗ +kx|| by R(k, t, a). +15 + +Consider an arbitrary q ⩾ +1 +2R(1,t,a) = q0. We now impose the condition c7 > e1/4. In +this case, by examining the derivative of +√ +kc−4k +7 +, one can check that it strictly decreases for +k ⩾ 1. Therefore, there exists a unique k ⩾ 2 such that R(k, t, a) < +1 +2q ⩽ R(k − 1, t, a). Let +p ∈ Z be such that ||qx|| = |qx − p|. Since two vectors (p∗ +k, q∗ +k) and (p∗ +k+1, q∗ +k+1) are linearly +independent, at least one of them must be linearly independent with (p, q). Suppose that is +(p∗ +k, q∗ +k). Then we estimate the absolute value of the following determinant: +1 ⩽ +���� +q +q∗ +k +p +p∗ +k +���� ⩽ +���� +q +q∗ +k +p − qx +p∗ +k − q∗ +kx +���� ⩽ qR(k, t, a) + ||qx||Q(k, t, a). +Since qR(k, t, a) < 1 +2, we must have ||qx|| ⩾ (2Q(k, t, a))−1. Analogously, if (p, q) is linearly +independent with (p∗ +k+1, q∗ +k+1), we have ||qx|| ⩾ (2Q(k + 1, t, a))−1. The latter lower bound +is weaker. Now, we need to rewrite the right hand side of the inequality in terms of q rather +than k. +Since +1 +2q ⩽ R(k − 1, t, a), we have that +c4(k−1) +7 +8|t1| +√ +k − 1 ⩽ q +=⇒ +k − 1 ⩽ log(8|t1|q) + log log(8|t1|q) +4 log c7 +. +The last implication can be justified by standard techniques on working with logarithms, +see [1, (41)]. +Finally, substitute the last lower bound for k in ||qx|| ⩾ (2Q(k + 1, t, a))−1 and get +||qx|| ⩾ +√π +8 +� +2(k + 1)c4(k+1) +6 +⩾ +2√π(log c7)1/2 +8 +√ +6c8 +6(log(8|t1|q) + log log(8|t1|q))1/2 · (8|t1|q) +log c6 +log c7 (log(8|t1|q)) +log c6 +log c7 +. +⩾ +(log c7)1/2 +8c8 +6(8|t1|) +log c6 +log c7 +· q− log c6 +log c7 (log(8|t1|q))− log c6 +log c7 − 1 +2 = τ +g2 +q−λ(log(8|t1|q))−λ− 1 +2 . +To finish the proof of Theorem 1, we recall, that for convenience, we in fact worked with the +number x/g2 rather than x, i.e. the inequality above is for ||qx/g2||. Hence one needs to +multiply both sides by g2. +Regarding theorem 2, we use inequalities (31) and (33). in this case the computations are +much easier and we get for any δ3 > 0 and large enough integer q that +||qx|| ⩾ q +− +log c∗ +6 +log c∗ +7 −δ3 +or in other words λeff(x) ⩽ log c∗ +6 +log c∗ +7 . That completes the proof of Theorem 2. +References +[1] D. +Badziahin. +Continued +fractions +of +cubic +Laurent +series. +Preprint. +https://arxiv.org/pdf/2211.08663. +[2] A. Baker. Rational approximations to certain algebraic numbers. Proc. LMS 14 (1964), +No 3, 385–398. +[3] M. A. Bennett. Effective Measures of Irrationality for Certain Algebraic Numbers. J. +AustMS 62 (1997), 329–344. +16 + +[4] E. Bombieri, A.J. van der Poorten, J. D. Vaaler. Effective measures of irrationality for +cubic extensions of number fields. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996) +No 2, 211–248. +[5] Y. Bugeaud. Linear forms in logarithms and applications. European Mathematical So- +ciety (2018). +[6] N. I. Feldman. Improved estimate for a linear form of the logarithms of algebraic num- +bers. Mat. Sb. 77 (1968), 256–270 (in Russian). English translation in Math. USSR. Sb. +6 (1968), 393–406. +[7] J. B. Rosser, L. Schoenfeld. Approximate formulas for some functions of prime numbers. +Illinois J. Math., 6 (1962), 64–94. +[8] K. F. Roth. Rational approximations to algebraic numbers. Mathematika, 2 (1955), 337– +360. +[9] I. Wakabayashi, Cubic Thue inequalities with negative discriminant. J. Number Theory, +97 (2002), No 2, 225–251. +17 + diff --git a/49E0T4oBgHgl3EQfegBh/content/tmp_files/load_file.txt b/49E0T4oBgHgl3EQfegBh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7414b76622b5d7c8b39a81aab3171914988548d --- /dev/null +++ b/49E0T4oBgHgl3EQfegBh/content/tmp_files/load_file.txt @@ -0,0 +1,477 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf,len=476 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='02391v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='NT] 6 Jan 2023 On effective irrationality exponents of cubic irrationals Dzmitry Badziahin January 9, 2023 Abstract We provide an upper bound on the efficient irrationality exponents of cubic algebraics x with the minimal polynomial x3 − tx2 − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In particular, we show that it becomes non-trivial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' better than the classical bound of Liouville in the case |t| > 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='71a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Moreover, under the condition |t| > 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='58a4/3, we provide an explicit lower bound on the expression ||qx|| for all large q ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' These results are based on the recently discovered continued fractions of cubic irrationals [1] and improve the currently best-known bounds of Wakabayashi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Keywords: cubic irrationals, continued fractions, irrationality exponent, effective irrationality exponent Math Subject Classification 2010: 11J68, 11J70, 11J82 1 Introduction The irrationality exponent λ(x) of an irrational real number x is defined as the supremum of real numbers λ such that the inequality ����x − p q ���� < q−λ (1) has infinitely many rational solutions p/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' It follows from the classical Dirichlet theorem that for all x ∈ R \\ Q, λ(x) ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' On the other hand, Khintchine’s theorem implies that almost all x ∈ R with respect to the Lebesgue measure satisfy λ(x) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In the first half of the XX century, there was a big interest in estimating the irrationality exponent of real algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' It culminated in 1955 with the work of Roth [8], who established the best possible result, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' that for any algebraic x ∈ R\\Q, λ(x) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Unfortunately, that result is ineffective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' for λ > 2 it does not allow us to find all rational p/q that satisfy (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore, for example, it can not be used to solve the Thue equations F(p, q) = c in integer p, q, where F ∈ Z[x, y] is a homogeneous polynomial of degree d ⩾ 3 and c is some integer parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since then, many mathematicians were working on effective results regarding the irratoinality exponents of algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Given x ∈ R \\ Q, by the effective irrationality exponent of x we define a positive real number λeff(x) such that for all λ > λeff(x) there exists an effectively computable Q > 0 such that all rational solutions of the inequality (1) in reduced form satisfy q ⩽ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' All known upper bounds on λeff(x) for algebraic real x are much weaker than in Roth’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' First of all, the classical theorem of Liouville states that λeff(x) ⩽ d where d is the degree of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore any non-trivial bound on λeff(x) should be strictly smaller than 1 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One of the notable improvements of Liouville’s bound is based on Feldman’s refinement of the theory of linear forms in logarithms [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Its advantage is that it gives λeff(x) < d for all algebraic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' However, the difference between λeff(x) and d is usually extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For state-of-the-art results regarding this approach, we refer to the book of Bugeaud [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For other notable achievements on this problem, we refer to [3, 4] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In this paper, we focus on the case of cubic irrationals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Multiplying by some integer number and shifting by another rational number, we can always guarantee that the minimal polynomial of the resulting cubic x is x3 + px + q for some p, q ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Notice also that such a transformation does not change the (effective) irrationality exponent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The first general result about λeff(x) for these specific values x was achieved by Bombieri, van der Poorten and Vaaler [4] in 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' They showed that under the conditions |p| > e1000 and |p| ⩾ q2, one has λeff(x) ⩽ 2 log(|p|3)| log(|p|3/q2) + 14 (log(|p|3/q2))1/3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Later, Wakabayashi [9] improved that bound and showed that λeff(x) ⩽ λw(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' It becomes non-trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' smaller than 3) under the condition |p| > 22/334|q|8/3 � 1 + 1 390|q|3 �2/3 (2) and for large p and q it asymptotically behaves like λw(p, q) ∼ 2 + (4 log |q| + 2 log 108)/(3 log |p|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In this paper we investigate what estimates on λeff(x) can be achieved with help of the convergents of the recently discovered continued fractions [1] of cubic irrationals with the minimal polynomial x3 − tx2 − a ∈ Z[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' It was shown there that, as soon as |t|3 > 12a > 0, the real root of this equation with the largest absolute value admits the continued fraction x = K � t 3(12k + 1)(3k + 1)α 3(12k + 5)(3k + 2)α 3(12k + 7)(6k + 5)α 3(12k − 1)(6k + 1)α (2i + 1)t2 (2i + 1)t 2(2i + 1)t2 (2i + 1)t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Here i is the index of the corresponding partial quotient and k = � i 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Notice that the change of variables y = q x transforms cubic x from [4, 9] to numbers in this paper with t = −p and a = −q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The resulting upper bounds on λeff(x) (see Theorems 1 and 2) depend on prime factori- sations of a and t but in any case they are better than those in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Theorem 1 states that under the condition |t|3 > 12|a| the largest real root of the above cubic equation satisfies ||qx|| ⩾ τ(t, a)q−λ(t,a)(log(8|t1|q))λ(t,a)−1/2, for all q > Q0(t, a), where all values for τ(t, a), λ(t, a) and q0(t, a) are explicitly provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' This inequality always gives a non-trivial upper bound on λeff(x) under the condition |t| > 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='58a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' It translates to the condition |p| > 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='58|q|8/3 which is better than (2), where even without the term in brackets we have |p| > 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='57|q|8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Moreover, the parameter a does not have to be of the form −q2, therefore many cubics from this paper do not plainly transfer to those in [4, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' On top of that, Theorem 2 provides an even better upper bound on λeff(x) but does not give an explicit lower bound on ||qx|| as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' That bound always becomes non-trivial as soon as |t| > 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='71a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' These bounds between t and a are obtained in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 2 Preliminaries and main results Consider the real root of the equation x3 − tx2 − a = 0 that has the largest absolute value among other roots of the same equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Notice that a ̸= 0 because otherwise x is not cubic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Also, by replacing x with −x if needed, we can guarantee that a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 2 Next, by the standard CF transformations (see [1, Lemma 1]), we can cancel some common divisors of t and 3a from the continued fraction of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Let g1 := gcd(t2, 3a) and g2 := gcd(t, 3a/g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For convenience, we denote t2 = g1t2, t = g2t1 and 3a = g1g2a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We di- vide the following partial quotients of x by g1: β1, a1, β2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' β3, a3, β4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' , β2k−1, a2k−1, β2k,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' After that we divide the following partial quotients by g2: β0, a0, β1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' β2, a2, β3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' β2k, a2k, β2k+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The resulting continued fraction has the same limit x as the initial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' To make β0 integer, we consider the number x/g2 instead of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Its continued fraction is then K � t1 (12k + 1)(3k + 1)a∗ (12k + 5)(3k + 2)a∗ (12k + 7)(6k + 5)a∗ (12k − 1)(6k + 1)a∗ (2i + 1)t2 (2i + 1)t1 2(2i + 1)t2 (2i + 1)t1 · · � (3) We define the following notions: c6 = c6(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' a∗) := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 √64t1t2 + 270a∗ 27/4ec1 if t > 0 � 64|t1t2| − 54a∗ 27/4ec1 if t < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (4) c7 = c7(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' a∗) := \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 27/4ec1(16t1t2 + 9a∗) 9a∗√64t1t2 + 270a∗ if t > 0 223/4ec1(|t1t2| − 3a∗) 9a∗� 64|t1t2| − 54a∗ if t < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (5) where c1 = c1(a∗) is defined in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Next, τ = τ(t1, t2, a∗) := g2 · (log c7)1/2 8c8 6(8|t1|) log c6 log c7 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' q0 = q0(t1, t2, a∗) := c4 7 8|t1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (6) The main result of this paper is Theorem 1 For all integer q ⩾ q0 one has ||qx|| > τq−λ log(8|t1|q)−λ−1/2 where λ = log c6 log c7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In particular, λeff(x) ⩽ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' As shown in Section 5, the constant c1 can be replaced by a bigger constant c2 defined in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' But in that case, writing such an explicit inequality as in Theorem 1 is much harder (but theoretically possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We do not provide it in the exact form here but only state the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Let c∗ 6 and c∗ 7 be defined in the same way as c6 and c7 but with the constant c2 instead of c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Theorem 2 The effective irrationality exponent of x satisfies λeff(x) ⩽ λ∗ := log c∗ 6 log c∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 3 Analysis of the results Theorem 1 provides a nontrivial lower bound for ||qx|| as soon as log c6 log c7 is strictly less that 2 or in other words, c6 < c2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In view of (4) and (5), for t > 0 this is equivalent to √64t1t2 + 270a∗ 27/4ec1 < 27/2e2c2 1(16t1t2 + 9a∗)2 81a∗2(64t1t2 + 270a∗) 3 ⇐⇒ a∗2(64t1t2 + 270a∗)3/2 < 221/4e3c3 1 81 (16t1t2 + 9a∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Define the parameter u such that 16t1t2 = ua∗4 − 9a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Also for convenience define τ := 221/4e3c3 1 81 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then the last inequality rewrites a∗2(4ua∗4 + 234a∗)3/2 < τu2a∗8 ⇐⇒ 4ua∗3 + 234 < τ 2/3u4/3a∗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Notice that for u ⩾ � 4 τ 2/3 + 234 64a∗3 τ 4/3�3 one has τ 2/3u4/3a∗3 ⩾ � 4 + 234 64a∗3 τ 2 � ua∗3 ⩾ 4ua∗3 + 234τ 2 64a∗3 · 43 τ 2 a∗3 = 4ua∗3 + 234 and the condition c6 < c2 7 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Recall that t1t2 = t3/(g1g2) and a∗ = 3a/(g1g2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore, 16t1t2 > ua∗4 − 9a∗ is equivalent to 16t3 > 81 (g1g2)3 ua4 − 27a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' From the definition (10) we see that c1 and hence τ depends on the prime factorisation of a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If 3 ∤ a∗ then we always get c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='0924 which in turn implies τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='00744 and � 4 τ 2/3 + 234 64a∗3 τ 4/3 �3 ⩽ 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' On the other hand, we have g1g2 ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, we get that for t > 0 the non-trivial bound on λeff(x) is always achieved if t > 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='97 3 � 81 16 �1/3 a4/3 ≈ 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='08a4/3, however for many pairs a and t it is satisfied under essentially weaker conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In the case 3 | a∗ we get c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='13329, thus τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='0223 and � 4 τ 2/3 + 234 64a∗3 τ 4/3 �3 ⩽ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then the non-trivial bound is achieved if t > 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='42 · � 81 16 �1/3 a4/3 ≈ 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='57a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The case t < 0 is dealt analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The condition c6 < c2 7 is equivalent to � 64|t1t2| − 54a∗ 27/4ec1 < 27/2e2c2 1(16|t1t2| − 48a∗)2 81a∗2(64|t1t2| − 54a∗) ⇐⇒ a∗2(64|t1t2| − 54a∗)3/2 < τ(16|t1t2| − 48a∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Define u such that 16|t1t2| = ua∗4 + 48a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then the last inequality rewrites 4ua∗3 + 138 < τ 2/3u4/3a∗3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can check that the last inequality is satisfied for u ⩾ � 4 τ 2/3 + 138τ 4/3 64a∗3 �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' As in the case of positive t, the right hand side is always smaller than 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='933 in the case 3 ∤ a∗ and is smaller than 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='423 in the case 3 | a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore for 3 ∤ a∗ the condition c6 < c2 7 is always satisfied if |t|3 > 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='933·81 33·16 a4 + 9a which follows from |t| > 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='06a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For 3 | a∗, similar computations give us |t|3 > 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='423·81 16 a4 + 9a which follows from |t| > 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='58a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can repeat the same analysis as above for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In that case, the constant c1 in the computations should be replaced by c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We have that if 3 ∤ a∗ it always satisfies c2 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='1939, which in turn implies τ ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='0688 and then � 4 τ 2/3 + 234 64a∗3 τ 4/3 �3 ⩽ 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 4 If 3 | a∗, we have c2 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='2797, τ ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='206 and � 4 τ 2/3 + 234 64a∗3 τ 4/3 �3 ⩽ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, for the case 3 ∤ a∗, the condition c6 < c2 7 is always satisfied if |t|3 > 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='933·81 33·16 a4 +9a which follows from |t| > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='72a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For the case 3 | a∗ similar calculations give |t| > 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='71a4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 4 Nice, convenient and perfect continued fractions Definition 1 Let x be a continued fraction given by x = K � β0 β1 β2 a0 a1 a2 · · · � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' βi, ai ∈ Z, ∀i ∈ Z⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For given positive integers k, r with −1 ⩽ r ⩽ k we define γk,r := k+r � i=k−r 2|(i−k+r) βi, γk,−1 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We say that x is d-nice at index k where d, k ∈ N if d | ak and for all positive integer r ⩽ k one has ak−rβk+rγk,r−2 ≡ −ak+rγk,r−1 (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We call x (d, r)-perfect at index k, where 0 ⩽ r ⩽ k if it is d-nice at index k and βk−r ≡ βk+r+1 ≡ 0 (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We say that x is eventually d-nice at index k from position k0 if the same conditions as above are satisfied for all 0 ⩽ r ⩽ k − k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In this paper the value k0 will often be a fixed absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If there is no confusion about its value we will omit it in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Let pn/qn be the n’th convergent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Define the following matrices Sn := � pn qn pn−1 qn−1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Cn := � an βn 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' From the theory of continued fractions we know that Sn = CnCn−1 · · · C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' To make this product shorter, we use the usual notation but in the descending order: Sn = �0 i=n Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 1 Let the continued fraction x be eventually d-nice at index k from the position k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then for all 0 ⩽ r ⩽ k − k0 one has k−r � i=k+r Ci ≡ � 0 γk,r γk,r−1 0 � (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Moreover, if x is (d, r)-perfect at index k then �k−r i=k+r+1 Ci ≡ 0 (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We prove by induction on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For r = 0 the statement is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Suppose that the statement is true for r and verify it for r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' k−r−1 � i=k+r+1 Ci ≡ � ak+r+1 βk+r+1 1 0 � � 0 γk,r γk,r−1 0 � � ak−r−1 βk−r−1 1 0 � ≡ � ak−r−1βk+r+1γk,r−1 + ak+r+1γk,r βk−r−1βk+r+1γk,r−1 γk,r 0 � (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 5 By the conditions of d-nice CF at index k, the last matrix is congruent to � 0 γk,r+1 γk,r 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If x is (d, r)-perfect at index k then γk,r ≡ 0 (mod d) and we have k−r � i=k+r+1 Ci ≡ � ak+r+1 0 1 0 � � 0 γk,r γk,r−1 0 � ≡ 0 (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⊠ Another two properties of d-nice continued fractions that easily follow from the definition are Let d1, d2 be two coprime positive integer numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If a continued fraction is eventually d1-nice and eventually d2-nice at the same index k for the same position k0 then it is eventually d1d2-nice at index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If a continued fraction is eventually d-nice at index k then it is also eventually e-nice at the same index from the same position for all positive integer divisors e of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Definition 2 We say that the continued fraction x is (eventually) d-convenient at index k if there exists a sequence (cr)0⩽r⩽⌊k/2⌋ of residues modulo m such that for all positive integers r ⩽ k (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' r ⩽ k − k0) one has βk+r+1 ≡ c⌊ r 2 ⌋βk−r (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' if r is odd then ak+r ≡ −c⌊ r 2⌋ak−r (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' if r is even then ak+r ≡ −ak−r (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 2 Let d > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then any eventually d-convenient continued fraction at index k is also eventually d-nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For d = 2, any d-convenient continued fraction at index k such that ak ≡ 0 (mod d) is also d-nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' First of all, for d > 2 and r = 0 the condition ak+r ≡ −ak−r (mod d) implies that ak ≡ 0 (mod d), which is the first condition of d-nice CF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Secondly, one can check that the first condition of d-convenient CF implies that for odd r, γk,r ≡ c⌊ r 2 ⌋γk,r−1βk−r ≡ γk,r−1βk+r+1 (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then we get ak−r−1βk+r+1γk,r−1 ≡ ak−r−1γk,r ≡ −ak+r+1γk,r (mod d) and the second condition of d-nice CF is verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Thirdly, for even r we get cr/2γk,r ≡ cr/2γk,r−1βk−r ≡ γk,r−1βk+r+1 (mod d) and therefore ak−r−1βk+r+1γk,r−1 ≡ cr/2ak−r−1γk,r ≡ −ak+r+1γk,r (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Again, the second condition of d-nice CF is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⊠ 5 Divisibility of entries of Sn Lemma 3 Let k ∈ N, k ⩾ 2 and d be any integer divisor of 2k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The continued fraction (3) is eventually d-convenient at index k from the position 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Additionally, the same statement is true for k ≡ 3 (mod 4) and d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 6 In the further discussion we will always deal with eventually d-convenient or d-nice contin- ued fractions from the position 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore, to make the description shorter, we will omit the words ‘eventually’ and ‘from the position 2’ and call the continued fraction (3) d-convenient or d-nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We will check the conditions of d-convenient continued fraction separately for each of the cases, depending on k modulo 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case k = 4m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then m ≡ − 3 8 (mod d) and we use (3) to compute ak+4r ≡ 8rt2, βk+4r = a∗(12m + 1 + 12r)(3m + 1 + 3r) ≡ a∗ 16(24r − 7)(24r − 1) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+1 ≡ (8r + 2)t1, βk+4r+1 ≡ a∗ 16(24r + 1)(24r + 7) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+2 ≡ 2(8r + 4)t2, βk+4r+2 ≡ a∗ 8 (24r + 5)(24r + 11) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r−1 ≡ (8r − 2)t1, βk+4r−1 ≡ a∗ 8 (24r − 11)(24r − 5) (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The conditions of d-convenient continued fraction at index k can now be easily checked where cr is the constant 1 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We proceed the same way in all other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case k = 4m + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then m ≡ − 5 8 (mod d) and ak+4r ≡ 8rt1, βk+4r ≡ a∗ 16(24r − 5)(24r + 1) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+1 ≡ 2(8r + 2)t2, βk+4r+1 ≡ a∗ 8 (24r − 1)(24r + 5) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+2 ≡ (8r + 4)t1, βk+4r+2 ≡ a∗ 8 (24r + 7)(24r + 13) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r−1 ≡ (8r − 2)t2, βk+4r−1 ≡ a∗ 16(24r − 13)(24r − 7) (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can easily check that for s ≡ 0, 1 (mod 4), βk+s+1 ≡ 2βk−s (mod d) and for s ≡ 2, 3 (mod 4), βk+s+1 ≡ 2−1βk−s (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Also, for s ≡ 1 (mod 4), ak+s ≡ 2ak−s (mod d) and for s ≡ 3 (mod 4), ak+s ≡ 2−1ak−s (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Hence, the conditions of d-convenient CF are verified, where the sequence cs is periodic with the period 2, 2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case k = 4m + 3, d ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then m ≡ − 7 8 (mod d) and ak+4r ≡ 16rt2, βk+4r ≡ a∗ 8 (24r − 7)(24r − 1) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+1 ≡ (8r + 2)t1, βk+4r+1 ≡ a∗ 8 (24r + 1)(24r + 7) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+2 ≡ (8r + 4)t2, βk+4r+2 ≡ a∗ 16(24r + 5)(24r + 11) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r−1 ≡ (8r − 2)t1, βk+4r−1 ≡ a∗ 16(24r − 11)(24r − 5) (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can then check the conditions of d-convenient CF at index k for the constant 1 sequence cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case k = 4m + 3, d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' in this case one can easily see that ak+4r ≡ 0 (mod 2), ak+4r+1 ≡ ak+4r+3 ≡ t1 (mod 2), ak+4r+2 ≡ t2 (mod 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' βk+4r ≡ βk+4r+1 ≡ a∗ (mod 2) and 7 βk+4r+2 ≡ βk+4r−1 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore, the CF is 2-convenient at index k with the constant 1 sequence cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case k = 4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then m ≡ − 1 8 (mod d) and ak+4r ≡ 8rt1, βk+4r ≡ a∗ 8 (24r − 5)(24r + 1) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+1 ≡ (8r + 2)t2, βk+4r+1 ≡ a∗ 16(24r − 1)(24r + 5) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r+2 ≡ (8r + 4)t1, βk+4r+2 ≡ a∗ 16(24r + 7)(24r + 13) (mod d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ak+4r−1 ≡ 2(8r − 2)t2, βk+4r−1 ≡ a∗ 8 (24r − 13)(24r − 7) (mod d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can then check the conditions of d-convenient CF with the periodic sequence cr with the period 2−1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⊠ Lemmata 2 and 3 show that x is d-nice at each index k ⩾ 2 for appropriately chosen d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' As the next step, we show that for almost every prime p, it is also (p, t)-perfect at infinitely many carefully chosen indices k and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' This fact will allow us to show that all the entries of CkCk−1 · · · C1 are multiples of some big power of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' First of all, let’s consider the case p > 2 and p | a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Let s ∈ N be such that ps||a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Consider q = pl for some 1 ⩽ l ⩽ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If we write q = 2m + 1 then we have q | αk for k = m + rq = (2r+1)q−1 2 where r ∈ Z⩾0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can easily see that for any such value of k, x is (q, 0)-perfect at index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In view of Lemma 1, we can then split the product Sk into � 2k+q−1 2q � groups such that all entries of the resulting product matrix in each group are multiples of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, we combine this information for ql for all 1 ⩽ l ⩽ s and derive that all entries of the product �2 i=k Ci are divisible by p s� i=1 � 2k+pi−1 2pi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Next, consider the case p = 2 and p | a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We have p | ak for all k ≡ 3 (mod 4) and one can easily see that for all such k, x is (p, 0) perfect at index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then the analogous application of Lemma 1 as in the previous case implies that all entries of �2 i=k Ci are divisible by 2⌊k/4⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For the case p = 2, p ∤ a∗ the result is slightly weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' From (3) one can verify that β8m+2 ≡ β8m+5 ≡ 0 (mod 2) for all m ∈ Z⩾0 and therefore x is (2, 1)-perfect at indices 8m + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then Lemma 1 then implies that all entries of �2 i=k Ci are divisible by 2⌊(k+3)/8⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, in the next lemma we consider the remaining case of p ∈ N that do not divide a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 4 Let p ∈ N be such that gcd(p, 6) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then for all k ∈ Z all the entries of the product of matrices �2 i=k Ci are divisible by p � 3k+p−2 3p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We prove by routinely considering all the cases, depending on p modulo 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case p = 12m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then with help of (3) one can verify that for all r ∈ Z⩾0, 0 ≡ β4(m+rp)+1 ≡ β4(2m+rp) ≡ β4(4m+rp)+1 ≡ β4(5m+rp)+2 ≡ β4(7m+rp)+3 ≡ β4(8m+rp)+2 ≡ β4(10m+rp)+3 ≡ β4(11m+rp)+4 (mod p) 8 and 0 ≡ a6m+rp (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In view of Lemma 3, we then derive that x is (p, 2m − 1)-perfect at indices k = 6m + 2rp for all r ∈ Z⩾0 and is (p, 2m)-perfect at indices k = 6m + (2r + 1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 1 then implies that all the entries of the following products of matrices are divisible by p: 4m+2rp+1 � i=8m+2rp Ci, 16m+4rp+1 � i=20m+2rp+2 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, one can easily check that for k = (n + 1)p − p−1 3 , the product �2 i=k Ci contains n + 1 blocks of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore all its entries are divisible by pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The other cases are done analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case p = 12m + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One verifies that for all r ∈ Z⩾0, 0 ≡ β4(m+rp)+2 ≡ β4(2m+rp)+3 ≡ β4(4m+rp)+6 ≡ β4(5m+rp)+9 ≡ β4(7m+rp)+12 ≡ β4(8m+rp)+13 ≡ β4(10m+rp)+16 ≡ β4(11m+rp)+19 (mod p) and 0 ≡ a6m+rp+2 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then Lemma 3 implies that x is (p, 2m)-perfect at indices k = 6m + 2rp for all r ∈ Z and is (p, 2m − 1)-perfect at indices k = 6m + (2r + 1)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 1 then implies that all the entries of the following products are divisible by p: 4m+2rp+2 � i=8m+2rp+3 Ci, 16m+4rp+6 � i=20m+2rp+9 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For k ⩾ (n + 1)p − p−2 3 one can easily check that the product �2 i=k Ci contains n + 1 blocks of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore all its entries are divisible by pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case p = 12m + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then for all r ∈ Z⩾0, 0 ≡ β4(m+rp)+3 ≡ β4(2m+rp)+4 ≡ β4(4m+rp)+9 ≡ β4(5m+rp)+12 ≡ β4(7m+rp)+17 ≡ β4(8m+rp)+18 ≡ β4(10m+rp)+23 ≡ β4(11m+rp)+26 (mod p) and 0 ≡ a6m+rp+3 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemmata 3 and 1 imply that all the entries of the following products are divisible by p: 4m+2rp+3 � i=8m+2rp+4 Ci, 16m+4rp+9 � i=20m+2rp+12 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For k ⩾ (n + 1)p − p−1 3 one can easily check that the product �2 i=k Ci contains n + 1 blocks of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore all its entries are divisible by pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Case p = 12m + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then for all r ∈ Z⩾0, 0 ≡ β4(m+rp)+4 ≡ β4(2m+rp)+7 ≡ β4(4m+rp)+14 ≡ β4(5m+rp)+19 ≡ β4(7m+rp)+26 ≡ β4(8m+rp)+29 ≡ β4(10m+rp)+36 ≡ β4(11m+rp)+41 (mod p) and 0 ≡ a6m+rp+5 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemmata 3 and 1 imply that all the entries of the following products are divisible by p: 4m+2rp+4 � i=8m+2rp+7 Ci, 16m+4rp+14 � i=20m+2rp+19 Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For k ⩾ (n + 1)p − p−2 3 one can easily check that the product �2 i=k Ci contains n + 1 blocks of the above form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore all its entries are divisible by pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 9 In all four cases we have that for k ⩾ (n+1)p− p−2 3 all the entries of �2 i=k Ci are divisible by pn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Writing it in terms of k we get that this power of p is � k + p−2 3 p � = �3k + p − 2 3p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⊠ We combine all the divisibility properties of �1 i=n Ci together and get the following Proposition 1 Let the prime factorisation of a∗ be a∗ = 2σ0pσ1 1 pσ2 2 · · · pσd d where σ0 can be equal to zero while the other powers σi are strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Define P1 := {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' , pd}, P2 := P \\ (P1 ∪ {2, 3}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If 2 | a∗ then gcd(pn, qn) ⩾ 2 � n 4 � � pi∈P1 p σi � j=1 � 2n+pj −1 2pj � i � p∈P2 p � 3n+p−2 3p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (7) If 2 ∤ a∗ then gcd(pn, qn) ⩾ 2 � n+3 8 � � pi∈P1 p σi � j=1 � 2n+pj −1 2pj � i � p∈P2 p � 3n+p−2 3p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (8) We now provide shorter lower bounds for (7) and (8) and then provide slightly better ones that, after some efforts, can still be made effective for large enough n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Observe that � 2n+pj−1 2pj � ⩾ � n pj � and � 3n+p−2 3p � ⩾ � n p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For convenience, if 3 ∤ a∗ we still add 3 to the set P1 by setting pd+1 := 3, σd+1 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then 2 �∞ i=1 n 2i · � pj∈P1 p �σj i=1 � n pi j � j � p∈P2 p � n p � � pj∈P1 p �∞ i=σj+1 n pi j j � p∈P2 p �∞ i=2 n pi ⩾ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⩾ √ 2πn �n e �n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The last inequality infers that gcd(pn, qn) ⩾ √ 2πn(c1n)n (9) where c1 = c1(a∗) is defined as c1 = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2−3/4 exp � −1 − � pj∈P1 ln pj p σj j (pj−1) − � p∈P2 ln p p(p−1) � if 2 | a∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 2−7/8 exp � −1 − � pj∈P1 ln pj p σj j (pj−1) − � p∈P2 ln p p(p−1) � if 2 ∤ a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (10) c1 reaches its minimal value in the case P1 = {3} with σ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then c1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='0924.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' However, if a∗ = 3 then c1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='1333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In general, more squares of small prime numbers divide a∗, bigger is the value of c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We can provide a better asymptotic lower estimate on gcd(pn, qn) for large enough n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The exact condition on n can be effectively computed, however the computations will not be nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Consider a prime p ∈ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The term p � 3n+p−2 3p � has an extra power of p compared to p⌊n/p⌋ if for some integer k, n p < k ⩽ 3n + p − 2 3p ⇐⇒ n k < p ⩽ 3n − 2 3k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 10 We also have � p∈P1 p ≍ 1 where the implied constants only depend on a∗ but not on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Define the set K := n� k=1 �n k , 3n − 2 3k − 1 � Then gcd(pn, qn) ⩾ T · √ 2πn(c1n)n where T ≍ � p∈P∩K p = exp \uf8eb \uf8ed � p∈K∩P ln p \uf8f6 \uf8f8 = exp � n � k=1 � θ �3n − 2 3k − 1 � − θ �n k ��� , where θ(x) is the first Chebyshev function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' It is well known (see [7] for example) that for large enough x, |θ(x) − x| < x 2 ln x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore for y > x one has θ(y) − θ(x) ⩾ y − x − y ln y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' This implies √ ln n � k=1 � θ �3n − 2 3k − 1 � − θ �n k �� ⩾ √ ln n � k=1 n − 2k k(3k − 1) − O � n √ ln n � = n √ ln n � k=1 1 k(3k − 1) − O � n √ ln n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For any ε > 0 and for large enough n, the last expression can be made bigger that (τ − ε)n where τ := �∞ k=1 1 k(3k−1) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='74102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore T ≫ e(τ−ε)n = γn(1−ε) where γ = eτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, we get gcd(pn, qn) ≫ ((c2 − δ)n)n, where c2 = c1 · γ, (11) δ can be made arbitrarily small and the implied constant in the inequality only depends on a∗ and δ but not on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' For the case a∗ = 1, when the constant c1 is minimal possible, we get c2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='1939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Respectively, for a∗ = 6, c2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='2797.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 6 Lower and upper bounds on the denominators qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In this section we will get upper and lower bounds of the denominators qn, compared to qn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since the recurrent formulae between qn, qn−1 and qn−2 depend on n modulo 4, it makes sense to compare q4k and q4k+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We adapt some notation from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Denote T4k := � p4k q4k p4k−4 q4k−4 � Then [1, (69) and (70)] one has T4k+4 = � ak11 ak12 1 0 � S4k (12) where ak11 and ak12 are the corresponding indices of C4k+4C4k+3C4k+2C4k+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In view of (3), one computes ak11 = 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)(t1t2)2 +6(8k + 5)(8k + 7)(36k2 + 55k + 16)a∗t1t2 +(12k + 5)(12k + 11)(3k + 2)(6k + 7)a∗2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (13) ak12 = 2(12k + 1)(3k + 1)(8k + 7)a∗t1((8k + 5)(8k + 9)t1t2 + 2(36k2 + 63k + 25)a∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' To make the notation shorter, we write ak12 = 2(12k + 1)(3k + 1)(8k + 7)a∗t1p(k) where p(k) is a polynomial of k with parameters t1t2 and a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then an easy adaptation of the proof of [1, Lemma 16] gives 11 Lemma 5 Let a∗ ∈ N and t1, t2 ∈ Z satisfy 12a∗ ⩽ |t1t2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then q4k+4 and q4k satisfy the relation |q4k+4| > (8k + 3)(8k + 5)(8k + 7)(8k + 9)(t1t2 + 2a∗)2|q4k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (14) Now we will provide an opposite inequality between the denominators q4k+4 and q4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Three consecutive denominators of this form are related by the equation [1, (72)]: q4k+4 = ak11q4k + (dq4k−4 − bk21q4k)ak12 bk22 , (15) where bk21/d and bk22/d are the corresponding entries of C−1 4k−2C−1 4k−1C−1 4k , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' d = −(12k − 7)(12k − 5)(12k − 1)(3k − 1)(6k − 1)(6k + 1)a∗3, (16) bk21 = −(12k − 5)(6k − 1)a∗ − 2(8k − 3)(8k − 1)t1t2, bk22 = 2(8k − 1)t1((8k − 3)(8k + 1)t1t2 + 2(36k2 − 9k − 2)a∗) =: 2(8k − 1)t1p(k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 6 Let a∗, t1, t2 be the same as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then q4k+4 and q4k satisfy the following relations: |q4k+4| ⩽ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) � t1t2 + 135 32 a∗ �2 |q4k|, if t1t2 > 0, (17) |q4k+4| ⩽ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) � t1t2 + 27 32a∗ �2 |q4k|, if t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' First, we estimate the terms in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since |t1t2| ⩾ 12a∗, we get for all k ⩾ 1 that 1 12(12k − 5)(6k − 1)(12a∗) < (8k − 3)(8k − 1)(12a∗) ⩽ (8k − 3)(8k − 1)|t1t2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore |bk21| ⩽ 3(8k − 3)(8k − 1)|t1t2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (19) Next, by Lemma 5 we have |dq4k−4| ⩽ (12k − 7)(12k − 5)(12k − 1)(3k − 1)(6k − 1)(6k + 1)a∗3 (8k − 5)(8k − 3)(8k − 1)(8k + 1)(t1t2 + 2a∗)2 q4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since |t1t2 + 2a∗| ⩾ 10a∗ and 12a∗ ⩽ |t1t2|, one can verify that |dq4k−4| < (8k − 3)(8k − 1)|t1t2q4k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (20) Next, we have (8k + 5)(8k + 9)|t1t2| > 12(36k2 + 63k + 25)a∗, therefore we always have |p(k)| (8k + 5)(8k + 9)|t1t2| ∈ � � 1, 7 6 � if t1t2 ⩾ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' � 5 6, 1 � if t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The last inequality in turn implies that for k ⩾ 1 the ratio ak12/bk22 is always positive and satisfies ak12 bk12 ⩽ 6(12k + 1)(3k + 1)(8k + 7)a∗ 8k − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (21) Assume that t1t2 ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In that case, the last inequality together with (19) and (20) imply that ����(dq4k−4 − bk21q4k)ak12 bk22 ���� ⩽ 24(8k − 3)(8k + 7)(12k + 1)(3k + 1)a∗t1t2q4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 12 One can check that for all k ⩾ 1, 6(8k + 5)(8k + 7)(36k2 + 55k + 16) + 24(8k − 3)(8k + 7)(12k + 1)(3k + 1) 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) < 135 16 (22) and 81 256 < (12k + 5)(12k + 11)(3k + 2)(6k + 7) 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) ⩽ 23 66 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (23) These bounds together with the formula (13) and equation (15) imply the inequality (17) for k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, this bound can be easily verified for k = 0 from the equation q4 = a011q0 + a012q−1 and q−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Consider the case t1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' One can check that for all k ⩾ 1, 321 187 ⩾ 6(8k + 5)(8k + 7)(36k2 + 55k + 16) 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) ⩾ 27 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (24) This together with the condition |t1t2| > 12a∗2 imply that ak11 > 0 and q4k and q4k+4 share the same sign for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Next, since (12k − 5)(6k − 1)a∗ < (8k − 3)(8k − 1)|t1t2|, we have that bk21 > 0 and then in view of (20) and ak12 bk22 > 0, the term (dq4k−4 − bk21q4k)ak12 bk22 has the opposite sign compared to ak11q4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' That all implies that |q4k+4| ⩽ |ak11q4k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, the inequalities (23) together with (24) establish the bound (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⊠ Lemma 6 immediately implies that for t1t2 > 0, |q4k| ⩽ 2k � t1t2 + 135 32 a∗ �2k (8k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⩽ 16k � 8 · 21/4e−1 � t1t2 + 135 32 a∗ �4k k4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The case of t1t2 < 0 can be dealt with in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, we get the estimate |q4k| ⩽ 16kc4k 3 k4k, (25) where c3 = c3(t1, t2, a∗) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 8 · 21/4e−1 � t1t2 + 135 32 a∗ if t1t2 > 0 8 · 21/4e−1 � |t1t2| − 27 32a∗ if t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Lemma 7 Under the same conditions on a∗, t1, t2 as in the previous lemma, one has |q4k+4| ⩾ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) � t1t2 + 9 16a∗ �2 |q4k|, if t1t2 > 0, (26) |q4k+4| ⩾ 2(8k + 3)(8k + 5)(8k + 7)(8k + 9) (t1t2 + 3a∗)2 |q4k|, if t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' If t1t2 > 0 we have q4k+4 > ak11q4k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then the lower bound in (23) together with the lower bound in (24) imply the bound (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Now assume that t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then, as we have shown in the proof of Lemma 6, bk21 > 0 and dq4k−4 and bk21q4k have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' This together with ak12 bk22 > 0, the inequality (21) and 0 < bk21 ⩽ 2(8k − 3)(8k − 1)|t1t2| in turn imply that |q4k+4| ⩾ |ak11q4k| − bk21ak12 bk22 |q4k| ⩾ (ak11 + 12(8k − 3)(12k + 1)(3k + 1)(8k + 7)a∗t1t2)|q4k| 13 We need to show that the expression ak11+12(8k−3)(12k+1)(3k+1)(8k+7)a∗t1t2−2(8k+3)(8k+5)(8k+7)(8k+9) (t1t2 + 3a∗)2 is always positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Notice that after substituting (13) into it and expanding the brackets, the term with (t1t2)2 disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The term for a∗t1t2 then equals to −6(8k + 7)(160k3 + 1532k2 + 1063k + 196)a∗t1t2 and the term for a∗2 is −(71136k4 + 212976k3 + 227970k2 + 102633k + 16240) (we made these computations with Wolfram Mathematika).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, one can check that in the case |t1t2| > 12a∗, the absolute value of the first term is always bigger than that of the second term and therefore the whole expression is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' By performing neater computations, one can make the coefficient 3 in (t1t2 + 3a∗)2 slightly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' However we decide not to further complicate already tedious calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⊠ Analogously to (25), one can find shorter lower bounds for |q4k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' With help of the known inequality (8k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⩾ 8k(8k/e)2k, Lemma 7 infers |q4k| ⩾ 8kc4k 4 k4k, (28) where c4 = c4(t1, t2, a∗) = \uf8f1 \uf8f2 \uf8f3 8 · 21/4e−1 � t1t2 + 9 16a∗ if t1t2 > 0 8 · 21/4e−1� |t1t2| − 3a∗ if t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 7 Distance between x and the convergents From [1, Lemma 17] we know that, under the condition 12a∗ ⩽ |t1t2|, one has ����x − p4k q4k ���� < 2 ���� p4k q4k − p4k+4 q4k+4 ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In order to estimate the right hand side, we use the matrix equation [1, (72)]: Tk+1 = � ak11 − ak12 bk21 bk22 dak12 bk22 1 0 � Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Notice that the values of d in fact depends on k (see the formula (16)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' to emphasize this dependence, in this section we write d(k) for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then the above equation gives the following formula: ���� p4k q4k − p4k+4 q4k+4 ���� = ��� �k i=1 d(i)ai12 bi22 ��� · |p0q4 − q0p4| q4kq4k+4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We first compute its product term: ����� k � i=1 d(i)ai12 bi22 ����� = k � i=1 2(12i − 7)(12i − 5)(12i − 1)(3i − 1)(6i − 1)(6i + 1)(12i + 1)(3i + 1)(8i + 7)a∗4|t1p(i)| 2(8i − 1)|t1p(i − 1)| 14 = (8k + 7)|p(k)| 7|p(0)| (3k + 1)(6k + 1)(12k + 1)a∗4k(12k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 26k34k(4k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⩽ √ 3(3k + 1)(6k + 1)(12k + 1)(8k + 7)|p(k)| 7|p(0)| �122k2a∗ 2 √ 2e2 �4k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Next, from (12) for k = 0 we get that |p0q4 − p4q0| = 14a∗t1|p(0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, we unite all these bounds together with the lower bounds (26), (27) and (28) for |q4k| to get ����x − p4k q4k ���� ⩽ 2 √ 3(3k + 1)(6k + 1)(12k + 1)(8k + 7)|t1a∗p(k)| 2(8k + 3)(8k + 5)(8k + 7)(8k + 9)(|t1t2| − 3a∗)2 · 64k2 · � 122a∗ 2 √ 2e2c2 4 �4k To simplify the right hand side, notice that (3k+1)(6k+1)(12k+1) (8k+3)(8k+5)(8k+9) < 27 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Next, since |t1t2| ⩾ 12a∗, one has |p(k)| = |(8k + 5)(8k + 9)t1t2 + 2(36k2 + 63k + 25)a∗| ⩽ 2(8k + 5)(8k + 9)|t1t2| which for all k ⩾ 1 is smaller than 442k2|t1t2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, (|t1t2| − 3a∗)2 ⩾ 9 16(t1t2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Collecting all of these inequalities together gives, ����x − p4k q4k ���� ⩽ √ 3 · 27 · 442|t2 1t2a∗| 64 · (9/16) · 64(t1t2)2 · � 72a∗ √ 2e2c2 4 �4k ⩽ |t1|c4k 5 , (29) where c5 = \uf8f1 \uf8f2 \uf8f3 9a∗ (16t1t2+9a∗) if t1t2 > 0 9a∗ 16(|t1t2|−3a∗) if t1t2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 8 Estimating the irrationality exponent In this section we establish Theorems refth1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Consider p∗ k := p4k/ gcd(p4k, q4k) and q∗ k := q4k/ gcd(p4k, q4k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Definitely, they are both integers and (25) together with (9) imply |q∗ k| ⩽ 4 � 2k π � c3 4c1 �4k =: 4 � 2k π · c4k 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' (30) For arbitrary δ > 0 and large enough k, one can use the inequality (11) to get |q∗ k| ≪ 16k � c3 4(c2 − δ) �4k ≪ � c3 4c2 + δ1 �4k =: (c∗ 6 + δ1)4k (31) where δ1 > 0 can be made arbitrarily close to zero for large enough k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Denote the upper bound for b∗ k by Q(k, t, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Next, we combine the last two inequalities with (29) and get ||q∗ kx|| ⩽ |t1|c4k 5 · 4 � 2k π � c3 4c1 �4k ⩽ 4|t1| √ k �c3c5 4c1 �4k =: 4|t1| √ kc−4k 7 (32) or ||q∗ kx|| ≪ � 4c2 c3c5 − δ2 �−4k =: (c∗ 7 − δ2)−4k (33) where δ2 can be made arbitrarily small and k is large enough, depending on δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Denote the upper bound of ||q∗ kx|| by R(k, t, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' 15 Consider an arbitrary q ⩾ 1 2R(1,t,a) = q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' We now impose the condition c7 > e1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' In this case, by examining the derivative of √ kc−4k 7 , one can check that it strictly decreases for k ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Therefore, there exists a unique k ⩾ 2 such that R(k, t, a) < 1 2q ⩽ R(k − 1, t, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Let p ∈ Z be such that ||qx|| = |qx − p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since two vectors (p∗ k, q∗ k) and (p∗ k+1, q∗ k+1) are linearly independent, at least one of them must be linearly independent with (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Suppose that is (p∗ k, q∗ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Then we estimate the absolute value of the following determinant: 1 ⩽ ���� q q∗ k p p∗ k ���� ⩽ ���� q q∗ k p − qx p∗ k − q∗ kx ���� ⩽ qR(k, t, a) + ||qx||Q(k, t, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since qR(k, t, a) < 1 2, we must have ||qx|| ⩾ (2Q(k, t, a))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Analogously, if (p, q) is linearly independent with (p∗ k+1, q∗ k+1), we have ||qx|| ⩾ (2Q(k + 1, t, a))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The latter lower bound is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Now, we need to rewrite the right hand side of the inequality in terms of q rather than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Since 1 2q ⩽ R(k − 1, t, a), we have that c4(k−1) 7 8|t1| √ k − 1 ⩽ q =⇒ k − 1 ⩽ log(8|t1|q) + log log(8|t1|q) 4 log c7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' The last implication can be justified by standard techniques on working with logarithms, see [1, (41)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Finally, substitute the last lower bound for k in ||qx|| ⩾ (2Q(k + 1, t, a))−1 and get ||qx|| ⩾ √π 8 � 2(k + 1)c4(k+1) 6 ⩾ 2√π(log c7)1/2 8 √ 6c8 6(log(8|t1|q) + log log(8|t1|q))1/2 · (8|t1|q) log c6 log c7 (log(8|t1|q)) log c6 log c7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' ⩾ (log c7)1/2 8c8 6(8|t1|) log c6 log c7 q− log c6 log c7 (log(8|t1|q))− log c6 log c7 − 1 2 = τ g2 q−λ(log(8|t1|q))−λ− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' To finish the proof of Theorem 1, we recall, that for convenience, we in fact worked with the number x/g2 rather than x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' the inequality above is for ||qx/g2||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Hence one needs to multiply both sides by g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Regarding theorem 2, we use inequalities (31) and (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' in this case the computations are much easier and we get for any δ3 > 0 and large enough integer q that ||qx|| ⩾ q − log c∗ 6 log c∗ 7 −δ3 or in other words λeff(x) ⩽ log c∗ 6 log c∗ 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' That completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' References [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Badziahin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Continued fractions of cubic Laurent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49E0T4oBgHgl3EQfegBh/content/2301.02391v1.pdf'} +page_content='org/pdf/2211.' metadata={'source': 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Our focus is mostly on the canonical two-message two-database case, for +which a coding scheme based on random codebook generation and the binning technique is proposed. +This coding scheme reveals a hidden connection between PIR and the classic multiple description source +coding problem. We first show that when the retrieval rate is kept optimal, the proposed non-linear +scheme can achieve better performance over any linear scheme. Moreover, a non-trivial storage-retrieval +rate tradeoff can be achieved beyond space-sharing between this extreme point and the other optimal +extreme point, achieved by the retrieve-everything strategy. We further show that with a method akin +to the expurgation technique, one can extract a zero-error PIR code from the random code. Outer +bounds are also studied and compared to establish the superiority of the non-linear codes over linear +codes. +1 +Introduction +Private information retrieval (PIR) addresses the situation of storing K messages of L-bits each in N +databases, with the requirement that the identity of any requested message must be kept private from any +one (or any small subset) of the databases. The early works were largely computer science theoretic [1], +where L = 1, and the main question is the scaling law of the retrieval rate in terms of (K, N). +The storage overhead in PIR systems has been studied in the coding and information theory com- +munity, from several perspectives using mainly two problem formulations. Shah et al. [2] considered the +problem when N is allowed to vary with L and K, and obtained some conclusive results. In a similar +vein, for L = 1, Fazeli et al. [3] proposed a technique to convert any linear PIR code to a new one with +low storage overhead by increasing N. Other notable results along this line can be found in [4–9]. +An information theoretic formulation of the PIR problem was considered in [10], where L is allowed to +increase, while (N, K) are kept fixed. Important properties on the tradeoff between the storage rate and +retrieval rate were identified in [10], and a linear code construction was proposed. In this formulation, +even without any storage overhead constraint, characterizing the minimum retrieval rate in the PIR +systems is nontrivial, and this capacity problem was settled in [11]. Tajeddine et al. [12] considered the +capacity problem when the message is coded across the databases with a maximum-distance separable +(MDS) code, which was later solved by Banawan and Ulukus [13]. Capacity-achieving code designs with +optimal message sizes were given in [14,15]. Systems where servers can collude were considered in [16]. +There have been various extensions and generalizations, and the recent survey article [17] provides a +comprehensive overview on efforts following this information theoretic formulation. +In many existing works, the storage component and the PIR component are largely designed sepa- +rately, usually by placing certain structural constraints on one of them; e.g., the MDS coding requirement +for the storage component [13], or the storage is uncoded [18]; moreover, the code constructions are al- +most all linear. The few exceptions we are aware of are [19–21]. In this work, we consider the information +theoretic formulation of the PIR problem, without placing any additional structural constraints on the +two components, and explicitly investigate the storage-retrieval tradeoff region. We mostly focus on the +1 +arXiv:2301.02155v1 [cs.IT] 5 Jan 2023 + +case N = K = 2 here since it provides the most important intuition; we refer to this as the (2, 2) PIR +system. Our approach naturally allows the joint design of the two components using either linear or +non-linear schemes. +The work in [19] is of significant relevance to our work, where the storage overhead was considered +in both single-round and multi-round PIR systems, when the retrieval rate must be optimal. Although +multi-round PIR has the same capacity as single-round PIR, it was shown that at the minimum retrieval +rate, a multi-round, ϵ-error, non-linear code can indeed break the storage performance barrier of an +optimal single-round, zero error, linear code. The question whether all the three differences are essential +to overcome this barrier was left as an open question. +In this work, we show that a non-linear code is able to achieve better performance than the optimal +linear code in the single-round zero-error (2, 2) PIR system, over a range of the storage rates. This is +accomplished by providing a Shannon-theoretic coding scheme based on random codebook generation +and the binning technique. The proposed scheme at the minimum retrieval rate is conceptually simpler, +and we present it as an explicit example. The general inner bound is then provided, and we show an +improved tradeoff can be achieved beyond space-sharing between the minimum retrieval rate code and +the other optimal extreme point. By leveraging a method akin to the expurgation technique, we further +show that one can extract a zero-error deterministic PIR code from the random ϵ-error PIR code. Outer +bounds are also studied for both general codes and linear codes, which allow us to establish conclusively +the superiority of non-linear codes over linear codes. Our work essentially answers the open question +in [19], and shows that in fact only non-linearity is essential in breaking the aforementioned barrier. +A preliminary version of this work was presented first in part in [22]. In this updated article, we provide +a more general random coding scheme, which reveals a hidden connection to the multiple description +source coding problem [23]. Intuitively, we can view the retrieved message as certain partial reconstruction +of the full set of messages, instead of a complete reconstruction of a single message. Therefore, the answers +from the servers can be viewed as descriptions of the full set of messages, which are either stored directly at +the servers or formed at the time of request, and the techniques seen in multiple description coding become +natural in the PIR setting. Since the publication of the preliminary version [22], several subsequent efforts +have been made in studying the storage-retrieval tradeoff in the PIR setting, which provided stronger and +more general information theoretic outer bounds and several new linear code constructions [20, 21, 24]. +However, the Shannon-theoretic random coding scheme given in [22] remains the best performance for +the (2, 2) case, which motivate us to provide the general coding scheme in this work and to make the +connection to multiple description source coding more explicit. It is our hope that this connection may +bring existing coding techniques for the multiple description problem to the study of the PIR problem. +2 +Preliminaries +The problem we consider is essentially the same as that in [11], with the additional consideration on +the storage overhead constraint at the databases. We provide a formal problem definition in the more +traditional Shannon-theoretic language, to facilitate subsequent treatment. Some relevant results on this +problem are also reviewed briefly in this section. +2.1 +Problem Definition +There are two independent messages, denoted as W1 and W2, in this system, each of which is generated +uniformly at random in the finite field FL +2 , i.e., each message is an L-bit sequence. There are two databases +to store the messages, which are produced by two encoding functions operating on (W1, W2) +φn : FL +2 × FL +2 → Fαn +2 , +n = 1, 2, +where αn is the number of storage symbols at database-n, n = 1, 2, which is a deterministic function of +L, i.e., we are using fixed length codes for storage. We write S1 = φ1(W1, W2) and S2 = φ2(W1, W2). +2 + +When a user requests message-k, it generates two queries (Q[k] +1 , Q[k] +2 ) to be sent to the two databases, +randomly in the alphabet Q × Q. Note the joint distribution satisfies the condition +PW1,W2,Q[k] +1 ,Q[k] +2 = PW1,W2PQ[k] +1 ,Q[k] +2 , +k = 1, 2, +(1) +i.e., +the messages and the queries are independent. The marginal distributions PW1,W2 and PQ[k] +1 ,Q[k] +2 , +k = 1, 2, thus fully specify the randomness in the system. +After receiving the queries, the databases produce the answers to the query via a set of deterministic +functions +ϕ(q) +n +: Fαn +2 +→ Fβ(q) +n +2 +, +q ∈ Q, n = 1, 2. +(2) +We also write the answers A[k] +n = ϕ(Q[k] +n ) +n +(Sn), n = 1, 2. The user, with the retrieved information, wishes +to reproduce the desired message through a set of decoding functions +ψ(k,q1,q2) : Fβ(q1) +1 +2 +× Fβ(q2) +2 +2 +→ FL +2 . +(3) +The outputs of the functions ˆWk = ψ(k,Q[k] +1 ,Q[k] +2 )(A[k] +1 , A[k] +2 ) are essentially the retrieved messages. We +require the system to retrieve the message correctly (zero-error), i.e., ˆWk = Wk for k = 1, 2. +Alternatively, we can require the system to have a small error probability. Denote the average prob- +ability of coding error of a PIR code as +Pe = 0.5 +� +k=1,2 +PW1,W2,Q[k] +1 ,Q[k] +2 (Wk ̸= ˆWk). +(4) +An (L, α1, α2, β1, β2) ϵ-error PIR code is defined similar as a (zero-error) PIR code, except that the +correctness condition is replaced by the condition that the probability of error Pe ≤ ϵ. +Finally, the privacy constraint stipulates that the identical distribution condition must be satisfied: +PQ[1] +n ,A[1] +n ,Sn = PQ[2] +n ,A[2] +n ,Sn, +n = 1, 2. +(5) +Note that one obvious consequence is that PQ[1] +n = PQ[2] +n ≜ PQn, for n = 1, 2. +We refer to the code, which is specified by two probability distributions PQ[k] +1 ,Q[k] +2 , k = 1, 2, and a +valid set of coding functions {φn, ϕ(q) +n , ψk,q1,q2} that satisfy both the correctness and privacy constraints, +as an (L, α1, α2, β1, β2) PIR code, where βn = EQn[β(Qn) +n +], for n = 1, 2. +Definition 1. A normalized storage-retrieval rate pair (¯α, ¯β) is achievable, if for any ϵ > 0 and sufficiently +large L, there exists an (L, α1, α2, β1, β2) PIR code, such that +L(¯α + ϵ) ≥ 1 +2(α1 + α2), L(¯β + ϵ) ≥ 1 +2 (β1 + β2) . +(6) +The collection of the achievable normalized storage-retrieval rate pair (¯α, ¯β) is the achievable storage- +retrieval rate region, denoted as R. +Unless explicitly stated, the rate region R is used for the zero-error PIR setting. In the definition +above, we have used the average rates (¯α, ¯β) across the databases instead of the individual rate vectors +1 +n(α1, α2, EQ1[β(Q1) +1 +], EQ2[β(Q2) +2 +]). This can be justified using the following lemma. +Lemma 1. If an (L, α1, α2, β1, β2) PIR code exists, then a (2L, α, α, β, β) PIR code exists, where +α = α1 + α2, +β = β1 + β2. +(7) +This lemma can essentially be proved by a space-sharing argument, the details of which can be found +in [19]. The following lemma is also immediate using a conventional space-sharing argument. +Lemma 2. The region R is convex. +3 + +database-1 +database-2 +Figure 1: A possible coding structure. +2.2 +Some Relevant Known Results +The capacity of a general PIR system with K messages and N databases is identified in [11] as +C = 1 − 1/N +1 − 1/N K , +(8) +which in our definition corresponds to the case when ¯β is minimized, and the proposed linear code achieves +(¯α, ¯β) = (K, (1 − 1/N K)/(N − 1)). The capacity of MDS-code PIR systems was established in [13]. In +the context of storage-retrieval tradeoff, this result can be viewed as providing the achievable tradeoff +pairs +(¯α, ¯β) = +� +t, 1 − tK/N K +N − t +� +, t = 1, 2, . . . , N. +(9) +However when specialized to the (2, 2) PIR problem, this does not provide any improvement over the +space-sharing strategy between the trivial code of retrieval-everything and the code in [11]. By specializing +the code in [11], it was shown in [19] that for the (2, 2) PIR problem, at the minimal retrieval value +¯β = 0.75, the storage rate ¯αl = 1.5 is achievable using a single-round, zero-error linear code, and in fact, +it is the optimal storage rate that any single-round, zero-error linear code can achieve. +One of the key observations in [19] is that a special coding structure appears to be the main difficulty +in the (2, 2) PIR setting, which is illustrated in Fig. 1. Here message W1 can be recovered from either +(X1, Y1) or (X2, Y2), and message W2 can be recovered from either (X1, Y2) or (X2, Y1); (X1, X2) is +essentially S1 and is stored at database-1, and (Y1, Y2) is essentially S2 and is stored at database-2. +It is clear that we can use the following strategy to satisfy the privacy constraint: when message W1 +is requested, with probability 1/2, the user queries for either (X1, Y1) or (X2, Y2); for message 2, with +probability 1/2, the user queries for either (X1, Y2) or (X2, Y1). More precisely, the following probability +distribution PQ[1] +1 ,Q[1] +2 +and PQ[2] +1 ,Q[2] +2 +can be used: +PQ[1] +1 ,Q[1] +2 = +� +0.5 +(Q[1] +1 , Q[1] +2 ) = (11) +0.5 +(Q[1] +1 , Q[1] +2 ) = (22) +, +(10) +and +PQ[2] +1 ,Q[2] +2 = +� +0.5 +(Q[2] +1 , Q[2] +2 ) = (12) +0.5 +(Q[2] +1 , Q[2] +2 ) = (21) +. +(11) +4 + +2.3 +Multiple Description Source Coding +The multiple description source coding problem [23] considers compressing a memoryless source S into +a total of M descriptions, i.e., M compressed bit sequences, such that the combinations of any subset +of these descriptions can be used to reconstruct the source S to guarantee certain quality requirements. +The motivation of this problem is mainly to address the case when packets can be dropped randomly on +a communication network. +Denote the coding rate for each description as Ri, i = 1, 2, . . . , M. A coding scheme was proposed +in [25], which leads to the following rate region. +Let U1, U2, . . . , UM be M random variables jointly +distributed with S, then the following rates (R1, R2, . . . , RM) and distortions (DA, A ⊆ {1, 2, . . . , M}) +are achievable: +� +i∈A +Ri ≥ +� +i∈A +H(Ui) − H({Ui, i ∈ A}|S), +A ⊆ {1, 2, . . . , M}, +(12) +DA ≥ E[d(S, fA(Ui, i ∈ A))], +A ⊆ {1, 2, . . . , M}. +(13) +Here fA is a reconstruction mapping from the random variables {Ui, i ∈ A} to the reconstruction domain, +d(·, ·) is a distortion metric that is used to measure the distortion, and DA is the distortion achievable using +the descriptions in the set A. Roughly speaking, the coding scheme requires generating approximately +2nRi length-n codewords in an i.i.d. manner using the marginal distribution Ui for each i = 1, 2, . . . , M, +and the rate constraints insure that when n is sufficiently large, with overwhelming probability there is a +tuple of M codewords (un +1, un +2, . . . , un +M), one in each codebook constructed earlier, that are jointly typical +with the source vector Sn. In this coding scheme, the descriptions are simply the codeword indices of +these codewords in these codebooks. For a given joint distribution (S, U1, U2, . . . , UM), we refer to the +rate region in (12) as the MD rate region RMD(S, U1, U2, . . . , UM), and the corresponding random code +construction the MD codebooks associated with (S, U1, U2, . . . , UM). +The binning technique [26] can be applied in the multiple description problem to provide further per- +formance improvements, particularly when not all the combinations of the descriptions are required +to satisfy certain performance constraints, but only a subset of them are; this technique has pre- +viously been used in [27] and [28] for this purpose. +Assume that only the subsets of descriptions +A1, A2, . . . , AT ⊆ {1, 2, . . . , M} have distortion requirements associated with the reconstructions using +these descriptions, which are denoted as DAi, i = 1, 2, . . . , T. Consider the MD codebooks associated with +(S, U1, U2, . . . , UM) at rates (R′ +1, R′ +2, . . . , R′ +M) ∈ RMD(S, U1, U2, . . . , UM), then assign the codewords in +the i-th codebook uniformly at random into 2nRi bins with 0 ≤ Ri ≤ R′ +i. The coding rates and distortions +that satisfy the following constraints simultaneously for all Ai, i = 1, 2, . . . , T are achievable: +� +j∈J +(R′ +j − Rj) ≤ +� +j∈J +H(Uj) − H +� +{Uj, j ∈ J } +���� +� +Uj′, j′ ∈ Ai \ J +�� +, +∀J ⊆ Ai, +(14) +DAi ≥ E[d(S, fAi(Uj, j ∈ Ai))]. +(15) +We denote the collection of such rate vectors (R1, R2, . . . , RM, R′ +1, R′ +2, . . . , R′ +M) as R∗ +MD((S, U1, U2, . . . , UM), ({Uj, j ∈ +Ai}, i = 1, 2, . . . , T)), and refer to the corresponding codebooks as the MD∗ codebooks associated with +the random variables (S, U1, U2, . . . , UM) and the reconstruction sets (A1, A2, . . . , AT ). +3 +A Special Case: Slepian-Wolf Coding for Minimum Retrieval Rate +In this section, we consider the minimum-retrieval-rate case, and show that non-linear and Shannon- +theoretic codes are beneficial. We will be rather cavalier here and ignore some details, in the hope of +better conveyance of the intuition. In particular, we ignore the asymptotic-zero probability of error that +is usually associated with a random coding argument, but this will be addressed more carefully in Section +4. +5 + +Let us rewrite the L-bit messages as +Wk = (Vk[1], . . . , Vk[L]) ≜ V L +k , +k = 1, 2. +(16) +The messages can be viewed as being produced from a discrete memoryless source PV1,V2 = PV1 · PV2, +where V1 and V2 are independent uniform-distributed Bernoulli random variables. +Consider the following auxiliary random variables +X1 ≜ V1 ∧ V2, +X2 ≜ (¬V1) ∧ (¬V2), +Y1 ≜ V1 ∧ (¬V2), +Y2 ≜ (¬V1) ∧ V2, +(17) +where ¬ is the binary negation, and ∧ is the binary “and” operation. This particular distribution satisfies +the coding structure depicted in Fig. 1, with (V1, V2) taking the role of (W1, W2), and the relation is +non-linear. The same distribution was used in [19] to construct a multiround PIR code. This non-linear +mapping appears to allow the resultant code to be more efficient than linear codes. +We wish to store (XL +1 , XL +2 ) at the first database in a lossless manner, however, store only certain +necessary information regarding Y L +1 and Y L +2 to facilitate the recovery of W1 or W2. For this purpose, we +will encode the message as follows: +• At database-1, compress and store (XL +1 , XL +2 ) losslessly; +• At database-2, encode Y L +1 using a Slepian-Wolf code (or more precisely Sgarro’s code with uncer- +tainty side information [29]), with either XL +1 or XL +2 at the decoder, whose resulting code index is +denoted as CY1; encode Y L +2 in the same manner, independent of Y L +1 , whose code index is denoted +as CY2. +It is clear that for database-1, we need roughly ¯α1 = H(X1, X2). At database-2, in order to guarantee +successful decoding of the Slepian-Wolf code, we can chose roughly +¯α2 = max(H(Y1|X1), H(Y1|X2)) + max(H(Y2|X1), H(Y2|X2)) += 2H(Y1|X1), +(18) +where the second equality is due to the symmetry in the probability distribution. Thus we find that this +code achieves +¯αnl = 0.5[H(X1, X2) + 2H(Y1|X1)] += 0.75 + 0.75H(1/3, 2/3) += 0.25 + 0.75 log2 3 ≈ 1.4387. +(19) +The retrieval strategy is immediate from the coding structure in Fig. 1, with (V L +1 , V L +2 , XL +1 , XL +2 , CY1, CY2) +serving the roles of (W1, W2, X1, X2, Y1, Y2), and thus indeed the privacy constraint is satisfied. The re- +trieval rates are roughly as follows +¯β(1) +1 += ¯β(2) +1 += H(X1) = H(X2), +(20) +¯β(1) +2 += ¯β(2) +2 += H(Y1|X1), +(21) +implying +¯β = 0.5[H(X1) + H(Y1|X1)] = 0.5H(Y1, X1) = 0.75. +Thus at the optimal retrieval rate ¯β = 0.75, we have +¯αl = 1.5 vs. ¯αnl ≈ 1.4387, +(22) +and clearly the proposed non-linear Shannon-theoretic code is able to perform better than the optimal +linear code. We note that it was shown in [19] by using a multround approach, the storage rate ¯α can be +further reduced, however this issue is beyond the scope of this work. In the rest of the paper, we build +on the intuition in this special case to generalize and strengthen the coding scheme. +6 + +4 +Main Result +4.1 +A General Inner Bound +We first present a general inner bound to the storage-retrieval tradeoff region. Let (V1, V2) be independent +random variables uniformly distributed on Ft +2 × Ft +2. Define the region R(t) +in to be the collection of (¯α, ¯β) +pairs for which there exist random variables (X0, X1, X2, Y1, Y2) jointly distributed with (V1, V2) such +that: +1. There exist deterministic functions f1,1, f1,2, f2,1, and f2,2 such that +V1 = f1,1(X0, X1, Y1) = f2,2(X0, X2, Y2), +V2 = f1,2(X0, X1, Y2) = f2,1(X0, X2, Y1); +(23) +2. There exist non-negative coding rates +(β(0) +1 , β(1) +1 , β(2) +1 , β(1) +2 , β(2) +2 , γ(0) +1 , γ(1) +1 , γ(2) +1 , γ(1) +2 , γ(2) +2 ) +∈ R∗ +MD (((V1, V2), X0, X1, X2, Y1, Y2), ({X0, X1, Y1}, {X0, X1, Y2}, {X0, X2, Y1}, {X0, X2, Y2})) ; +(24) +3. There exist non-negative storage rates (α(0) +1 , α(1) +1 , α(2) +1 , α(1) +2 , α(2) +2 ) such that +α(0) +1 +≤ β(0) +1 , α(1) +1 +≤ β(1) +1 , α(2) +1 +≤ β(2) +1 , α(1) +2 +≤ β(1) +2 , α(2) +2 +≤ β(2) +2 , +(25) +and if +γ(0) +1 +− β(0) +1 ++ γ(1) +1 +− β(1) +1 ++ γ(2) +1 +− β(2) +1 +< H(X1) + H(X2) + H(X3) − H(X0, X1, X2), +(26) +choose +(α(0) +1 , α(1) +1 , α(2) +1 , γ(0) +1 , γ(1) +1 , γ(2) +1 ) ∈ R∗ +MD (((V1, V2), X0, X1, X2), ({X0, X1, X2})) ; +(27) +otherwise, choose (α(0) +1 , α(1) +1 , α(2) +1 ) = (β(0) +1 , β(1) +1 , β(2) +1 ). Similarly, if +γ(1) +2 +− β(1) +2 ++ γ(2) +2 +− β(2) +2 +< I(Y1; Y2), +(28) +choose +(α(1) +2 , α(2) +2 , γ(1) +2 , γ(2) +2 ) ∈ R∗ +MD (((V1, V2), Y1, Y2), ({Y1, Y2})) , +(29) +otherwise (α(1) +2 , α(2) +2 ) = (β(1) +1 , β(2) +1 ); +4. The normalized average retrieval and storage rates +2t¯α ≥ α(0) +1 ++ α(1) +1 ++ α(2) +1 ++ α(1) +2 ++ α(2) +2 , +(30) +4t¯β ≥ 2β(0) +1 ++ β(1) +1 ++ β(2) +1 ++ β(1) +2 ++ β(2) +2 . +(31) +Then we have the following theorem. +Theorem 1. R(t) +in ⊆ R. +We can in fact potentially enlarge the achievable region by taking ∪∞ +t=1R(t) +in . However, unless R(t+1) +in +⊆ +R(t) +in for all t ≥ 1, the region ∪∞ +t=1R(t) +in is even more difficult to characterize. Nevertheless, for each fixed +t, we can identify inner bounds by specifying a feasible set of random variables X0, X1, X2, Y1, Y2. +Instead of directly establishing this theorem, we shall prove the following theorem which establishes +the existence of a PIR code with diminishing error probability, and then use an expurgation technique +to extract a zero-error PIR code. +7 + +Theorem 2. Consider any (¯α, ¯β) ∈ R(t) +in . +For any ϵ > 0 and sufficiently large L, there exists an +(L, L(¯α + ϵ), L(¯α + ϵ), L(¯β + ϵ), L(¯β + ϵ)) ϵ-error PIR code with the query distribution given in (10) and +(11). +The key observation to establish this theorem is that there are five descriptions in this setting, however, +the retrieval and storage place different constraints on different combination of descriptions, and some +descriptions can in fact be stored, recompressed, and then retrieved. Such compression and recompression +may lead to storage savings. The description based on X0 can be viewed as some common information +to X1 and X2, which allows us to tradeoff the storage and retrieval rates. +Proof of Theorem 2. Codebook generation: Codebooks are built using the MD codebooks based on the +distribution ((V1, V2), X0, X1, X2, Y1, Y2). +Storage codes: +The bin indices of the codebooks are stored in the two servers: those of X0, X1, and X2 +are stored at server-1 at rates α(0) +1 , α(1) +1 , and α(2) +1 , respectively; those of Y1 and Y2 are stored at server-2 +at rates α(1) +2 +and α(2) +2 . Note that at such rates, the codewords for X0, X1, and X2 can be recovered jointly +with overwhelming probability, while those for Y1 and Y2 can also be recovered jointly with overwhelming +probability. +Retrieval codes: +A different set of bin indices of the codebooks are retrieved during the retrieval process, +again based on the MD∗ codebooks: those of X0, X1, and X2 are retrieved at server-1 at rates β(0) +1 , β(1) +1 , +and β(2) +1 , respectively; those of Y1 and Y2 are retrieved at server-2 at rates β(1) +2 +and β(2) +2 . Note that at such +rates, the codewords of X0, X1, and Y1 can be jointly recovered such that using the three corresponding +codewords, the required V1 source vector can be recovered with overwhelming probability. Similarly, the +three retrieval patterns of (X0, X1, Y2) → V2, (X0, X2, Y1) → V2, and (X0, X2, Y2) → V2 will succeed with +overwhelming probabilities. +Storage and retrieval rates: +The rates can be computed straightforwardly, after normalization by the +parameter t. +Next we use it to prove Theorem 1. +Proof of Theorem 1. Given an ϵ > 0, according to Proposition 2, we can find an (L, L(¯α + ϵ), L(¯α + +ϵ), L(¯β + ϵ), L(¯β + ϵ)) ϵ-error PIR code for some sufficient large L. The probability of error of this code +can be rewritten as +Pe = 0.5 +� +k=1,2 +� +(w1,w2) +2−2LPQ[k] +1 ,Q[k] +2 |(w1,w2)(wk ̸= ˆWk). +For a fixed (w1, w2) pair, denote the event that there exists a (q1, q2) ∈ {(11), (22)}, i.e., when (Q[1] +1 , Q[1] +2 ) = +(q1, q2), such that ˆw1 ̸= w1 as E(1) +w1,w2, and there exists a (q1, q2) ∈ {(12), (21)} such that ˆw2 ̸= w2 as +E(2) +w1,w2. Since (Q[k] +1 , Q[k] +2 ) is independent of (W1, W2), if P(E(k) +w1,w2) ̸= 0, we must have P(E(k) +w1,w2) ≥ 0.5. +It follows that +Pe ≥ 0.25 +� +(w1,w2) +2−2L1(E[1] +w1,w2 ∪ E[2] +w1,w2), +(32) +where (·) is the indicator function. This implies that for any ϵ ≤ 0.125, there are at most 22L−1 pairs of +(w1, w2) that will induce any coding error. We can use any 22L−2 of the remaining 22L−1 pairs of L-bit +sequence pairs to instead store a pair of (L − 1)-bit messages, through an arbitrary but fixed one-to-one +mapping. This new code has a factor of 1 + 1/(L − 1) increase in the normalized coding rates, which is +negligible when L is large. Thus a zero-error PIR code has been found with the same normalized rates +as the ϵ-error code asymptotically, and this completes the proof. +8 + +4.2 +Outer bounds +We next turn our attention to the outer bounds for R, summarized in the following theorem. +Theorem 3. Any (¯α, ¯β) ∈ R must satisfy +¯β ≥ 0.75, +¯α + ¯β ≥ 2, +3¯α + 8¯β ≥ 10. +(33) +Moreover, if (¯α, ¯β) ∈ R can be achieved by a linear code, it must satisfy +¯α + 6¯β ≥ 6. +(34) +The inequality ¯β ≥ 0.75 follows from [11], while the two other bounds in (33) were proved in [24]. +Therefore we only need to prove (34). +Proof of Theorem 3. Following [19], we make the following simplifying assumptions that have no loss of +generality. Define Q = {Q[1] +1 , Q[2] +1 , Q[1] +2 , Q[2] +2 }. +1. Q[1] +1 = Q[2] +1 +⇒ A[1] +1 = A[2] +1 , +(35) +2. H(A[1] +1 |Q) = H(A[1] +2 |Q) = H(A[2] +2 |Q), +H(S1) = H(S2) +(36) +⇒ H(A[1] +1 |Q) ≤ β ≤ (¯β + ϵ)L, +H(S2) ≤ α ≤ (¯α + ϵ)L. +(37) +Assumption 1 states that the query to the first database is the same regardless of the desired message +index. This is justified by the privacy condition that the query to one database is independent of the +desired message index. +Assumption 2 states that the scheme is symmetric after the symmetrization +operation in Lemma 1 (the proof is referred to Theorem 3 in [19]). (37) follows from the fact that to +describe S2, A[1] +1 , the number of bits needed can not be less than the entropy value, and Definition 1. +In the following, we use (c) to refer to the correctness condition, (i) to refer to the constraint that +queries are independent of the messages, (a) to refer to the constraint that answers are deterministic +functions of the storage variables and corresponding queries, and (p) to refer to the privacy condition. +From A[1] +1 , A[1] +2 , Q, we can decode W1. +H(A[1] +1 , A[1] +2 |W1, Q) += +H(A[1] +1 , A[1] +2 , W1|Q) − H(W1|Q) +(38) +(c)(i) += +H(A[1] +1 , A[1] +2 |Q) − L +(39) +(36) +≤ +2H(A[1] +1 |Q) − L. +(40) +Next, consider Ingleton’s inequality. +I(A[1] +2 ; A[2] +2 |Q) +≤ +I(A[1] +2 ; A[2] +2 |W1, Q) + I(A[1] +2 ; A[2] +2 |W2, Q) +(41) += +2I(A[1] +2 ; A[2] +2 |W1, Q) +(42) += +2 +� +H(A[1] +2 |W1, Q) + H(A[2] +2 |W1, Q) − H(A[1] +2 , A[2] +2 |W1, Q) +� +(43) +(p) += +2 +� +2H(A[1] +2 |W1, Q) − H(A[1] +2 , A[2] +2 |W1, Q) +� +(44) +≤ +2 +� +2H(A[1] +2 |W1, Q) + H(A[1] +1 , A[1] +2 |W1, Q) +− H(A[1] +1 , A[1] +2 , A[2] +2 |W1, Q) − H(A[1] +2 |W1, Q) +� +(45) +(c)(35) += +2 +� +H(A[1] +2 |W1, Q) + H(A[1] +1 , A[1] +2 |W1, Q) +− H(A[1] +1 , A[1] +2 , A[2] +2 , W2|W1, Q) +� +(46) +9 + +(i) +≤ +2 +� +2H(A[1] +1 , A[1] +2 |W1, Q) − H(W2) +� +(47) +(40) +≤ +2 +� +2(2H(A[1] +1 |Q) − L) − L +� +(48) +where (42) follows from the observation that the second term can be bounded using the same method as +that bounds the first term by switching the message index. A more detailed derivation of (44) appears +in (79) of [19]. (45) is due to sub-modularity of entropy. +Note that +I(A[1] +2 ; A[2] +2 |Q) += +H(A[1] +2 |Q) + H(A[2] +2 |Q) − H(A[1] +2 , A[2] +2 |Q) +(49) +(36) +≥ +2H(A[1] +1 |Q) − (¯α + ϵ)L +(50) +where in (50), and the second term is bounded as follows : +H(A[1] +2 , A[2] +2 |Q) ≤ H(A[1] +2 , A[2] +2 , S2|Q) +(a) += H(S2|Q) +(37) +≤ (¯α + ϵ)L. +(51) +Combining (48) and (50), we have +2H(A[1] +1 |Q)/L − (¯α + ϵ) ≥ 2(4H(A[1] +1 |Q)/L − 3) +⇒ +¯α + ϵ + 6H(A[1] +1 |Q)/L ≥ 6 +(52) +(37) +⇒ +¯α + 6¯β ≥ 6. +(53) +The proof is complete. +4.3 +Specialization of the Inner Bound +The inner bound given in Theorem 1 is general but more involved, and we can specialize it in multiple +ways in order to simplify it. One particularly interesting approach is as follows. Define the region ˜R(t) +in +to be the collection of (¯α, ¯β) pairs such that there exists random variables (X0, X1, X2, Y1, Y2) jointly +distributed with (V1, V2) such that +1. The distribution factorizes as follows +PV1,V2,X0,X1,X2,Y1,Y2 = PV1,V2PX0|V1,V2PX1|V1,V2PX2|V1,V2PY1|V1,V2PY2|V1,V2; +2. There exist deterministic functions f1,1, f1,2, f2,1, and f2,2 such that +V1 = f1,1(X0, X1, Y1) = f2,2(X0, X2, Y2), +(54) +V2 = f1,2(X0, X1, Y2) = f2,1(X0, X2, Y1); +(55) +3. A set of rates +γ(0) +1 += I(V1, V2; X0), γ(1) +1 += I(V1, V2; X1), γ(2) +1 += I(V1, V2; X2), +(56) +γ(1) +2 += I(V1, V2; Y1), γ(2) +2 += I(V1, V2; Y2), +(57) +β(0) +1 += γ(0) +1 , β(1) +1 += I(V1, V2; X1|X0), β(2) +1 += I(V1, V2; X2|X0), +(58) +β(1) +2 += max(I(V1, V2; Y1|X0, X1), I(V1, V2; Y1|X0, X2)), +(59) +β(2) +2 += max(I(V1, V2; Y2|X0, X1), I(V1, V2; Y2|X0, X2)), +(60) +and (α(0) +1 += γ(0) +1 , α(1) +1 , α(2) +1 , α(1) +2 , α(2) +2 ) as defined in item 3 for the general region R(t); +10 + +1 +1.05 +1.1 +1.15 +1.2 +1.25 +1.3 +1.35 +1.4 +1.45 +1.5 +0.75 +0.8 +0.85 +0.9 +0.95 +1 +inner bound via a non-linear scheme +non-linear scheme: time-sharing +inner bound via a linear scheme +an outer bound on linear schemes +information theoretic outer bound +Figure 2: Illustration of inner bounds and outer bounds. +4. The normalized average retrieval and storage rates +2t¯α ≥ α(0) +1 ++ α(1) +1 ++ α(2) +1 ++ α(1) +2 ++ α(2) +2 , +(61) +4t¯β ≥ 2β(0) +1 ++ β(1) +1 ++ β(2) +1 ++ β(1) +2 ++ β(2) +2 . +(62) +Then we have the following corollary. +Corollary 1. ˜R(t) +in ⊆ R. +This inner bound is illustrated together with the outer bounds in Fig. 2. +Proof. The main difference from Theorem 1 is in the special dependence structure of (X0, X1, X2, Y1, Y2) +jointly distributed with (V1, V2), i.e., the Markov structure. We verify that the rate assignments satisfy +all the constraints in Theorem 1. Due to the special dependence structure of (X0, X1, X2, Y1, Y2) jointly +distributed with (V1, V2), it is straightforward to verify that +(γ(0) +1 , γ(1) +1 , γ(2) +1 , γ(1) +2 , γ(2) +2 ) ∈ RMD((V1, V2), X0, X1, X2, Y1, Y2). +We next verify (24) holds with the choice given above. Due to the symmetry in the structure, we only need +to confirm one subset of random variables, i.e., {X0, X1, Y1}, and the three other subsets {X0, X1, Y2}, +{X0, X2, Y1}, and {X0, X2, Y2} follow similarly. There are a total of 7 conditions in the form of (14) +associated with this subset {X0, X1, Y1}. Notice that +γ(0) +1 +− β(0) +1 += 0, γ(1) +1 +− β(1) +1 += I(X1; X0), γ(2) +2 +− β(2) +2 +≤ I(Y1; X0, X1), +which in fact confirm three of the seven conditions when J is a singleton. Next when J has two elements, +we verify that +γ(0) +1 +− β(0) +1 ++ γ(1) +1 +− β(1) +1 += I(X1; X0) = H(X0) + H(X1) − H(X0, X1) +≤ H(X0) + H(X1) − H(X0, X1|Y1), +(63) +γ(0) +1 +− β(0) +1 ++ γ(1) +2 +− β(1) +2 +≤ I(Y1; X0, X1) = H(Y1) + H(X0, X1) − H(X0, X1, Y1) +≤ H(Y1) + H(X0) + H(X1) − H(X0, X1, Y1) +11 + +Table 1: Conditional distribution PX0|W1,W2 used in Corollary 2. +(w1, w2) x0 = (00) x0 = (01) x0 = (10) x0 = (11) +(00) +1/2 +1/2 +(10) +(1 − p)/2 +p +(1 − p)/2 +(01) +(1 − p)/2 +p +(1 − p)/2 +(11) +1/2 +1/2 += H(X0) + H(Y1) − H(X0, Y1|X1), +(64) +γ(1) +1 +− β(1) +1 ++ γ(1) +2 +− β(1) +2 +≤ I(X1; X0) + I(Y1; X0, X1) = H(X1) + H(Y1) − H(X1, Y1|X0). +(65) +Finally when J has all the three elements, we have +γ(0) +1 +− β(0) +1 ++ γ(1) +1 +− β(1) +1 ++ γ(1) +2 +− β(1) +2 += I(X0; X1) + I(V1, V2; X1) − max(I(V1, V2; Y1|X0, X1), I(V1, V2; Y1|X0, X2)) +(66) +≤ I(X0; X1) + I(V1, V2; X1) − I(V1, V2; Y1|X0, X1) +(67) += H(X0) + H(X1) + H(Y1) − H(X0, X1, Y1). +(68) +Thus (24) is indeed true with the assignments (56)-(60). This in fact completes the proof. +We can use any explicit distribution (X0, X1, X2, Y1, Y2) to obtain an explicit inner bound to ˜R(t) +in , and +the next corollary provides one such non-trivial bound. For convenience, we write the entropy function +of a probability mass (p1, . . . , pt) as H(p1, . . . , pt). +Corollary 2. The following (¯α, ¯β) ∈ R for any p ∈ [0, 1]: +¯α =9 +4 − H(1 +4, 3 +4) + 1 +4H(1 − p +2 +, 1 − p +2 +, p +2, p +2) ++ 1 +2H(2 − p +4 +, 2 − p +4 +, p +2) − 3 +4H(3 − 2p +6 +, 3 − 2p +6 +, p +3, p +3), +¯β =5 +8 + 1 +4H(2 − p +4 +, 2 − p +4 +, p +2) − 1 +8H(1 − p +2 +, 1 − p +2 +, p). +Proof. These tradeoff pairs are obtained by applying Corollary 1, and choosing t = 1 and setting +(X1, X2, Y1, Y2) as given in (17), and letting X0 be defined as in Table 1. Note that the joint distri- +bution indeed satisfies the required Markov structure, and in this case α(1) +2 += β(1) +2 +and α(2) +2 += β(2) +2 . +5 +Conclusion +We consider the problem of private information retrieval using a Shannon-theoretic approach. A new +coding scheme based on random coding and binning is proposed, which reveals a hidden connection to the +multiple description problem. It is shown that for the (2, 2) PIR setting, this non-linear coding scheme is +able to provide the best known tradeoff between retrieval rate and storage rate, which is strictly better +than that achievable using linear codes. We further investigate the relation between zero-error PIR codes +and ϵ-error PIR codes in this setting, and shows that they do not causes any essential difference in this +problem setting. We hope that the hidden connection to multiple description coding can provide a new +revenue to design more efficient PIR codes. +12 + +References +[1] B. Chor, O. Goldreich, E. 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Sgarro, “Source coding with side information at several decoders,” IEEE Transactions on Infor- +mation Theory, vol. 23, no. 2, pp. 179–182, 1977. +14 + diff --git a/6NA0T4oBgHgl3EQfN_-e/content/tmp_files/load_file.txt b/6NA0T4oBgHgl3EQfN_-e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7b8d6c26d1294447c072a6d535c202b0ebd7428 --- /dev/null +++ b/6NA0T4oBgHgl3EQfN_-e/content/tmp_files/load_file.txt @@ -0,0 +1,654 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf,len=653 +page_content='A Shannon-Theoretic Approach to the Storage-Retrieval Tradeoff in PIR Systems Chao Tian, Hua Sun, and Jun Chen Abstract We consider the storage-retrieval rate tradeoff in private information retrieval (PIR) systems using a Shannon-theoretic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Our focus is mostly on the canonical two-message two-database case, for which a coding scheme based on random codebook generation and the binning technique is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This coding scheme reveals a hidden connection between PIR and the classic multiple description source coding problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We first show that when the retrieval rate is kept optimal, the proposed non-linear scheme can achieve better performance over any linear scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Moreover, a non-trivial storage-retrieval rate tradeoff can be achieved beyond space-sharing between this extreme point and the other optimal extreme point, achieved by the retrieve-everything strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We further show that with a method akin to the expurgation technique, one can extract a zero-error PIR code from the random code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Outer bounds are also studied and compared to establish the superiority of the non-linear codes over linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 1 Introduction Private information retrieval (PIR) addresses the situation of storing K messages of L-bits each in N databases, with the requirement that the identity of any requested message must be kept private from any one (or any small subset) of the databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The early works were largely computer science theoretic [1], where L = 1, and the main question is the scaling law of the retrieval rate in terms of (K, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The storage overhead in PIR systems has been studied in the coding and information theory com- munity, from several perspectives using mainly two problem formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Shah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' [2] considered the problem when N is allowed to vary with L and K, and obtained some conclusive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In a similar vein, for L = 1, Fazeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' [3] proposed a technique to convert any linear PIR code to a new one with low storage overhead by increasing N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Other notable results along this line can be found in [4–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' An information theoretic formulation of the PIR problem was considered in [10], where L is allowed to increase, while (N, K) are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Important properties on the tradeoff between the storage rate and retrieval rate were identified in [10], and a linear code construction was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In this formulation, even without any storage overhead constraint, characterizing the minimum retrieval rate in the PIR systems is nontrivial, and this capacity problem was settled in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Tajeddine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' [12] considered the capacity problem when the message is coded across the databases with a maximum-distance separable (MDS) code, which was later solved by Banawan and Ulukus [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Capacity-achieving code designs with optimal message sizes were given in [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Systems where servers can collude were considered in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There have been various extensions and generalizations, and the recent survey article [17] provides a comprehensive overview on efforts following this information theoretic formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In many existing works, the storage component and the PIR component are largely designed sepa- rately, usually by placing certain structural constraints on one of them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', the MDS coding requirement for the storage component [13], or the storage is uncoded [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' moreover, the code constructions are al- most all linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The few exceptions we are aware of are [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In this work, we consider the information theoretic formulation of the PIR problem, without placing any additional structural constraints on the two components, and explicitly investigate the storage-retrieval tradeoff region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We mostly focus on the 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='02155v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='IT] 5 Jan 2023 case N = K = 2 here since it provides the most important intuition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' we refer to this as the (2, 2) PIR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Our approach naturally allows the joint design of the two components using either linear or non-linear schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The work in [19] is of significant relevance to our work, where the storage overhead was considered in both single-round and multi-round PIR systems, when the retrieval rate must be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Although multi-round PIR has the same capacity as single-round PIR, it was shown that at the minimum retrieval rate, a multi-round, ϵ-error, non-linear code can indeed break the storage performance barrier of an optimal single-round, zero error, linear code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The question whether all the three differences are essential to overcome this barrier was left as an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In this work, we show that a non-linear code is able to achieve better performance than the optimal linear code in the single-round zero-error (2, 2) PIR system, over a range of the storage rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This is accomplished by providing a Shannon-theoretic coding scheme based on random codebook generation and the binning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The proposed scheme at the minimum retrieval rate is conceptually simpler, and we present it as an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The general inner bound is then provided, and we show an improved tradeoff can be achieved beyond space-sharing between the minimum retrieval rate code and the other optimal extreme point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' By leveraging a method akin to the expurgation technique, we further show that one can extract a zero-error deterministic PIR code from the random ϵ-error PIR code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Outer bounds are also studied for both general codes and linear codes, which allow us to establish conclusively the superiority of non-linear codes over linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Our work essentially answers the open question in [19], and shows that in fact only non-linearity is essential in breaking the aforementioned barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A preliminary version of this work was presented first in part in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In this updated article, we provide a more general random coding scheme, which reveals a hidden connection to the multiple description source coding problem [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Intuitively, we can view the retrieved message as certain partial reconstruction of the full set of messages, instead of a complete reconstruction of a single message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Therefore, the answers from the servers can be viewed as descriptions of the full set of messages, which are either stored directly at the servers or formed at the time of request, and the techniques seen in multiple description coding become natural in the PIR setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Since the publication of the preliminary version [22], several subsequent efforts have been made in studying the storage-retrieval tradeoff in the PIR setting, which provided stronger and more general information theoretic outer bounds and several new linear code constructions [20, 21, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' However, the Shannon-theoretic random coding scheme given in [22] remains the best performance for the (2, 2) case, which motivate us to provide the general coding scheme in this work and to make the connection to multiple description source coding more explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' It is our hope that this connection may bring existing coding techniques for the multiple description problem to the study of the PIR problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 2 Preliminaries The problem we consider is essentially the same as that in [11], with the additional consideration on the storage overhead constraint at the databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We provide a formal problem definition in the more traditional Shannon-theoretic language, to facilitate subsequent treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Some relevant results on this problem are also reviewed briefly in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='1 Problem Definition There are two independent messages, denoted as W1 and W2, in this system, each of which is generated uniformly at random in the finite field FL 2 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', each message is an L-bit sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There are two databases to store the messages, which are produced by two encoding functions operating on (W1, W2) φn : FL 2 × FL 2 → Fαn 2 , n = 1, 2, where αn is the number of storage symbols at database-n, n = 1, 2, which is a deterministic function of L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', we are using fixed length codes for storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We write S1 = φ1(W1, W2) and S2 = φ2(W1, W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 2 When a user requests message-k, it generates two queries (Q[k] 1 , Q[k] 2 ) to be sent to the two databases, randomly in the alphabet Q × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Note the joint distribution satisfies the condition PW1,W2,Q[k] 1 ,Q[k] 2 = PW1,W2PQ[k] 1 ,Q[k] 2 , k = 1, 2, (1) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', the messages and the queries are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The marginal distributions PW1,W2 and PQ[k] 1 ,Q[k] 2 , k = 1, 2, thus fully specify the randomness in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' After receiving the queries, the databases produce the answers to the query via a set of deterministic functions ϕ(q) n : Fαn 2 → Fβ(q) n 2 , q ∈ Q, n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (2) We also write the answers A[k] n = ϕ(Q[k] n ) n (Sn), n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The user, with the retrieved information, wishes to reproduce the desired message through a set of decoding functions ψ(k,q1,q2) : Fβ(q1) 1 2 × Fβ(q2) 2 2 → FL 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (3) The outputs of the functions ˆWk = ψ(k,Q[k] 1 ,Q[k] 2 )(A[k] 1 , A[k] 2 ) are essentially the retrieved messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We require the system to retrieve the message correctly (zero-error), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', ˆWk = Wk for k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Alternatively, we can require the system to have a small error probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Denote the average prob- ability of coding error of a PIR code as Pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 � k=1,2 PW1,W2,Q[k] 1 ,Q[k] 2 (Wk ̸= ˆWk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (4) An (L, α1, α2, β1, β2) ϵ-error PIR code is defined similar as a (zero-error) PIR code, except that the correctness condition is replaced by the condition that the probability of error Pe ≤ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Finally, the privacy constraint stipulates that the identical distribution condition must be satisfied: PQ[1] n ,A[1] n ,Sn = PQ[2] n ,A[2] n ,Sn, n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (5) Note that one obvious consequence is that PQ[1] n = PQ[2] n ≜ PQn, for n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We refer to the code, which is specified by two probability distributions PQ[k] 1 ,Q[k] 2 , k = 1, 2, and a valid set of coding functions {φn, ϕ(q) n , ψk,q1,q2} that satisfy both the correctness and privacy constraints, as an (L, α1, α2, β1, β2) PIR code, where βn = EQn[β(Qn) n ], for n = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A normalized storage-retrieval rate pair (¯α, ¯β) is achievable, if for any ϵ > 0 and sufficiently large L, there exists an (L, α1, α2, β1, β2) PIR code, such that L(¯α + ϵ) ≥ 1 2(α1 + α2), L(¯β + ϵ) ≥ 1 2 (β1 + β2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (6) The collection of the achievable normalized storage-retrieval rate pair (¯α, ¯β) is the achievable storage- retrieval rate region, denoted as R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Unless explicitly stated, the rate region R is used for the zero-error PIR setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In the definition above, we have used the average rates (¯α, ¯β) across the databases instead of the individual rate vectors 1 n(α1, α2, EQ1[β(Q1) 1 ], EQ2[β(Q2) 2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This can be justified using the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' If an (L, α1, α2, β1, β2) PIR code exists, then a (2L, α, α, β, β) PIR code exists, where α = α1 + α2, β = β1 + β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (7) This lemma can essentially be proved by a space-sharing argument, the details of which can be found in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The following lemma is also immediate using a conventional space-sharing argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The region R is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 3 database-1 database-2 Figure 1: A possible coding structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='2 Some Relevant Known Results The capacity of a general PIR system with K messages and N databases is identified in [11] as C = 1 − 1/N 1 − 1/N K , (8) which in our definition corresponds to the case when ¯β is minimized, and the proposed linear code achieves (¯α, ¯β) = (K, (1 − 1/N K)/(N − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The capacity of MDS-code PIR systems was established in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In the context of storage-retrieval tradeoff, this result can be viewed as providing the achievable tradeoff pairs (¯α, ¯β) = � t, 1 − tK/N K N − t � , t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (9) However when specialized to the (2, 2) PIR problem, this does not provide any improvement over the space-sharing strategy between the trivial code of retrieval-everything and the code in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' By specializing the code in [11], it was shown in [19] that for the (2, 2) PIR problem, at the minimal retrieval value ¯β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75, the storage rate ¯αl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 is achievable using a single-round, zero-error linear code, and in fact, it is the optimal storage rate that any single-round, zero-error linear code can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' One of the key observations in [19] is that a special coding structure appears to be the main difficulty in the (2, 2) PIR setting, which is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Here message W1 can be recovered from either (X1, Y1) or (X2, Y2), and message W2 can be recovered from either (X1, Y2) or (X2, Y1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (X1, X2) is essentially S1 and is stored at database-1, and (Y1, Y2) is essentially S2 and is stored at database-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' It is clear that we can use the following strategy to satisfy the privacy constraint: when message W1 is requested, with probability 1/2, the user queries for either (X1, Y1) or (X2, Y2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' for message 2, with probability 1/2, the user queries for either (X1, Y2) or (X2, Y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' More precisely, the following probability distribution PQ[1] 1 ,Q[1] 2 and PQ[2] 1 ,Q[2] 2 can be used: PQ[1] 1 ,Q[1] 2 = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 (Q[1] 1 , Q[1] 2 ) = (11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 (Q[1] 1 , Q[1] 2 ) = (22) , (10) and PQ[2] 1 ,Q[2] 2 = � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 (Q[2] 1 , Q[2] 2 ) = (12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 (Q[2] 1 , Q[2] 2 ) = (21) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (11) 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='3 Multiple Description Source Coding The multiple description source coding problem [23] considers compressing a memoryless source S into a total of M descriptions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', M compressed bit sequences, such that the combinations of any subset of these descriptions can be used to reconstruct the source S to guarantee certain quality requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The motivation of this problem is mainly to address the case when packets can be dropped randomly on a communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Denote the coding rate for each description as Ri, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A coding scheme was proposed in [25], which leads to the following rate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Let U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM be M random variables jointly distributed with S, then the following rates (R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , RM) and distortions (DA, A ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , M}) are achievable: � i∈A Ri ≥ � i∈A H(Ui) − H({Ui, i ∈ A}|S), A ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , M}, (12) DA ≥ E[d(S, fA(Ui, i ∈ A))], A ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (13) Here fA is a reconstruction mapping from the random variables {Ui, i ∈ A} to the reconstruction domain, d(·, ·) is a distortion metric that is used to measure the distortion, and DA is the distortion achievable using the descriptions in the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Roughly speaking, the coding scheme requires generating approximately 2nRi length-n codewords in an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' manner using the marginal distribution Ui for each i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , M, and the rate constraints insure that when n is sufficiently large, with overwhelming probability there is a tuple of M codewords (un 1, un 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , un M), one in each codebook constructed earlier, that are jointly typical with the source vector Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In this coding scheme, the descriptions are simply the codeword indices of these codewords in these codebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' For a given joint distribution (S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM), we refer to the rate region in (12) as the MD rate region RMD(S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM), and the corresponding random code construction the MD codebooks associated with (S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The binning technique [26] can be applied in the multiple description problem to provide further per- formance improvements, particularly when not all the combinations of the descriptions are required to satisfy certain performance constraints, but only a subset of them are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' this technique has pre- viously been used in [27] and [28] for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Assume that only the subsets of descriptions A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , AT ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , M} have distortion requirements associated with the reconstructions using these descriptions, which are denoted as DAi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Consider the MD codebooks associated with (S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM) at rates (R′ 1, R′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , R′ M) ∈ RMD(S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM), then assign the codewords in the i-th codebook uniformly at random into 2nRi bins with 0 ≤ Ri ≤ R′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The coding rates and distortions that satisfy the following constraints simultaneously for all Ai, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , T are achievable: � j∈J (R′ j − Rj) ≤ � j∈J H(Uj) − H � {Uj, j ∈ J } ���� � Uj′, j′ ∈ Ai \\ J �� , ∀J ⊆ Ai, (14) DAi ≥ E[d(S, fAi(Uj, j ∈ Ai))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (15) We denote the collection of such rate vectors (R1, R2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , RM, R′ 1, R′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , R′ M) as R∗ MD((S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM), ({Uj, j ∈ Ai}, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , T)), and refer to the corresponding codebooks as the MD∗ codebooks associated with the random variables (S, U1, U2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , UM) and the reconstruction sets (A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , AT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 3 A Special Case: Slepian-Wolf Coding for Minimum Retrieval Rate In this section, we consider the minimum-retrieval-rate case, and show that non-linear and Shannon- theoretic codes are beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We will be rather cavalier here and ignore some details, in the hope of better conveyance of the intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In particular, we ignore the asymptotic-zero probability of error that is usually associated with a random coding argument, but this will be addressed more carefully in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 5 Let us rewrite the L-bit messages as Wk = (Vk[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , Vk[L]) ≜ V L k , k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (16) The messages can be viewed as being produced from a discrete memoryless source PV1,V2 = PV1 · PV2, where V1 and V2 are independent uniform-distributed Bernoulli random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Consider the following auxiliary random variables X1 ≜ V1 ∧ V2, X2 ≜ (¬V1) ∧ (¬V2), Y1 ≜ V1 ∧ (¬V2), Y2 ≜ (¬V1) ∧ V2, (17) where ¬ is the binary negation, and ∧ is the binary “and” operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This particular distribution satisfies the coding structure depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 1, with (V1, V2) taking the role of (W1, W2), and the relation is non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The same distribution was used in [19] to construct a multiround PIR code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This non-linear mapping appears to allow the resultant code to be more efficient than linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We wish to store (XL 1 , XL 2 ) at the first database in a lossless manner, however, store only certain necessary information regarding Y L 1 and Y L 2 to facilitate the recovery of W1 or W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' For this purpose, we will encode the message as follows: At database-1, compress and store (XL 1 , XL 2 ) losslessly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' At database-2, encode Y L 1 using a Slepian-Wolf code (or more precisely Sgarro’s code with uncer- tainty side information [29]), with either XL 1 or XL 2 at the decoder, whose resulting code index is denoted as CY1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' encode Y L 2 in the same manner, independent of Y L 1 , whose code index is denoted as CY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' It is clear that for database-1, we need roughly ¯α1 = H(X1, X2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' At database-2, in order to guarantee successful decoding of the Slepian-Wolf code, we can chose roughly ¯α2 = max(H(Y1|X1), H(Y1|X2)) + max(H(Y2|X1), H(Y2|X2)) = 2H(Y1|X1), (18) where the second equality is due to the symmetry in the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Thus we find that this code achieves ¯αnl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5[H(X1, X2) + 2H(Y1|X1)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75H(1/3, 2/3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='25 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75 log2 3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='4387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (19) The retrieval strategy is immediate from the coding structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 1, with (V L 1 , V L 2 , XL 1 , XL 2 , CY1, CY2) serving the roles of (W1, W2, X1, X2, Y1, Y2), and thus indeed the privacy constraint is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The re- trieval rates are roughly as follows ¯β(1) 1 = ¯β(2) 1 = H(X1) = H(X2), (20) ¯β(1) 2 = ¯β(2) 2 = H(Y1|X1), (21) implying ¯β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5[H(X1) + H(Y1|X1)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5H(Y1, X1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Thus at the optimal retrieval rate ¯β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75, we have ¯αl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' ¯αnl ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='4387, (22) and clearly the proposed non-linear Shannon-theoretic code is able to perform better than the optimal linear code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We note that it was shown in [19] by using a multround approach, the storage rate ¯α can be further reduced, however this issue is beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In the rest of the paper, we build on the intuition in this special case to generalize and strengthen the coding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 6 4 Main Result 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='1 A General Inner Bound We first present a general inner bound to the storage-retrieval tradeoff region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Let (V1, V2) be independent random variables uniformly distributed on Ft 2 × Ft 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Define the region R(t) in to be the collection of (¯α, ¯β) pairs for which there exist random variables (X0, X1, X2, Y1, Y2) jointly distributed with (V1, V2) such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There exist deterministic functions f1,1, f1,2, f2,1, and f2,2 such that V1 = f1,1(X0, X1, Y1) = f2,2(X0, X2, Y2), V2 = f1,2(X0, X1, Y2) = f2,1(X0, X2, Y1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (23) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There exist non-negative coding rates (β(0) 1 , β(1) 1 , β(2) 1 , β(1) 2 , β(2) 2 , γ(0) 1 , γ(1) 1 , γ(2) 1 , γ(1) 2 , γ(2) 2 ) ∈ R∗ MD (((V1, V2), X0, X1, X2, Y1, Y2), ({X0, X1, Y1}, {X0, X1, Y2}, {X0, X2, Y1}, {X0, X2, Y2})) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (24) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There exist non-negative storage rates (α(0) 1 , α(1) 1 , α(2) 1 , α(1) 2 , α(2) 2 ) such that α(0) 1 ≤ β(0) 1 , α(1) 1 ≤ β(1) 1 , α(2) 1 ≤ β(2) 1 , α(1) 2 ≤ β(1) 2 , α(2) 2 ≤ β(2) 2 , (25) and if γ(0) 1 − β(0) 1 + γ(1) 1 − β(1) 1 + γ(2) 1 − β(2) 1 < H(X1) + H(X2) + H(X3) − H(X0, X1, X2), (26) choose (α(0) 1 , α(1) 1 , α(2) 1 , γ(0) 1 , γ(1) 1 , γ(2) 1 ) ∈ R∗ MD (((V1, V2), X0, X1, X2), ({X0, X1, X2})) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (27) otherwise, choose (α(0) 1 , α(1) 1 , α(2) 1 ) = (β(0) 1 , β(1) 1 , β(2) 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Similarly, if γ(1) 2 − β(1) 2 + γ(2) 2 − β(2) 2 < I(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y2), (28) choose (α(1) 2 , α(2) 2 , γ(1) 2 , γ(2) 2 ) ∈ R∗ MD (((V1, V2), Y1, Y2), ({Y1, Y2})) , (29) otherwise (α(1) 2 , α(2) 2 ) = (β(1) 1 , β(2) 1 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The normalized average retrieval and storage rates 2t¯α ≥ α(0) 1 + α(1) 1 + α(2) 1 + α(1) 2 + α(2) 2 , (30) 4t¯β ≥ 2β(0) 1 + β(1) 1 + β(2) 1 + β(1) 2 + β(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (31) Then we have the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' R(t) in ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We can in fact potentially enlarge the achievable region by taking ∪∞ t=1R(t) in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' However, unless R(t+1) in ⊆ R(t) in for all t ≥ 1, the region ∪∞ t=1R(t) in is even more difficult to characterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Nevertheless, for each fixed t, we can identify inner bounds by specifying a feasible set of random variables X0, X1, X2, Y1, Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Instead of directly establishing this theorem, we shall prove the following theorem which establishes the existence of a PIR code with diminishing error probability, and then use an expurgation technique to extract a zero-error PIR code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 7 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Consider any (¯α, ¯β) ∈ R(t) in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' For any ϵ > 0 and sufficiently large L, there exists an (L, L(¯α + ϵ), L(¯α + ϵ), L(¯β + ϵ), L(¯β + ϵ)) ϵ-error PIR code with the query distribution given in (10) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The key observation to establish this theorem is that there are five descriptions in this setting, however, the retrieval and storage place different constraints on different combination of descriptions, and some descriptions can in fact be stored, recompressed, and then retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Such compression and recompression may lead to storage savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The description based on X0 can be viewed as some common information to X1 and X2, which allows us to tradeoff the storage and retrieval rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Codebook generation: Codebooks are built using the MD codebooks based on the distribution ((V1, V2), X0, X1, X2, Y1, Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Storage codes: The bin indices of the codebooks are stored in the two servers: those of X0, X1, and X2 are stored at server-1 at rates α(0) 1 , α(1) 1 , and α(2) 1 , respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' those of Y1 and Y2 are stored at server-2 at rates α(1) 2 and α(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Note that at such rates, the codewords for X0, X1, and X2 can be recovered jointly with overwhelming probability, while those for Y1 and Y2 can also be recovered jointly with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Retrieval codes: A different set of bin indices of the codebooks are retrieved during the retrieval process, again based on the MD∗ codebooks: those of X0, X1, and X2 are retrieved at server-1 at rates β(0) 1 , β(1) 1 , and β(2) 1 , respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' those of Y1 and Y2 are retrieved at server-2 at rates β(1) 2 and β(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Note that at such rates, the codewords of X0, X1, and Y1 can be jointly recovered such that using the three corresponding codewords, the required V1 source vector can be recovered with overwhelming probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Similarly, the three retrieval patterns of (X0, X1, Y2) → V2, (X0, X2, Y1) → V2, and (X0, X2, Y2) → V2 will succeed with overwhelming probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Storage and retrieval rates: The rates can be computed straightforwardly, after normalization by the parameter t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Next we use it to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Given an ϵ > 0, according to Proposition 2, we can find an (L, L(¯α + ϵ), L(¯α + ϵ), L(¯β + ϵ), L(¯β + ϵ)) ϵ-error PIR code for some sufficient large L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The probability of error of this code can be rewritten as Pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 � k=1,2 � (w1,w2) 2−2LPQ[k] 1 ,Q[k] 2 |(w1,w2)(wk ̸= ˆWk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' For a fixed (w1, w2) pair, denote the event that there exists a (q1, q2) ∈ {(11), (22)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', when (Q[1] 1 , Q[1] 2 ) = (q1, q2), such that ˆw1 ̸= w1 as E(1) w1,w2, and there exists a (q1, q2) ∈ {(12), (21)} such that ˆw2 ̸= w2 as E(2) w1,w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Since (Q[k] 1 , Q[k] 2 ) is independent of (W1, W2), if P(E(k) w1,w2) ̸= 0, we must have P(E(k) w1,w2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' It follows that Pe ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='25 � (w1,w2) 2−2L1(E[1] w1,w2 ∪ E[2] w1,w2), (32) where (·) is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This implies that for any ϵ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='125, there are at most 22L−1 pairs of (w1, w2) that will induce any coding error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We can use any 22L−2 of the remaining 22L−1 pairs of L-bit sequence pairs to instead store a pair of (L − 1)-bit messages, through an arbitrary but fixed one-to-one mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This new code has a factor of 1 + 1/(L − 1) increase in the normalized coding rates, which is negligible when L is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Thus a zero-error PIR code has been found with the same normalized rates as the ϵ-error code asymptotically, and this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='2 Outer bounds We next turn our attention to the outer bounds for R, summarized in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Any (¯α, ¯β) ∈ R must satisfy ¯β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75, ¯α + ¯β ≥ 2, 3¯α + 8¯β ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (33) Moreover, if (¯α, ¯β) ∈ R can be achieved by a linear code, it must satisfy ¯α + 6¯β ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (34) The inequality ¯β ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75 follows from [11], while the two other bounds in (33) were proved in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Therefore we only need to prove (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Following [19], we make the following simplifying assumptions that have no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Define Q = {Q[1] 1 , Q[2] 1 , Q[1] 2 , Q[2] 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q[1] 1 = Q[2] 1 ⇒ A[1] 1 = A[2] 1 , (35) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' H(A[1] 1 |Q) = H(A[1] 2 |Q) = H(A[2] 2 |Q), H(S1) = H(S2) (36) ⇒ H(A[1] 1 |Q) ≤ β ≤ (¯β + ϵ)L, H(S2) ≤ α ≤ (¯α + ϵ)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (37) Assumption 1 states that the query to the first database is the same regardless of the desired message index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This is justified by the privacy condition that the query to one database is independent of the desired message index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Assumption 2 states that the scheme is symmetric after the symmetrization operation in Lemma 1 (the proof is referred to Theorem 3 in [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (37) follows from the fact that to describe S2, A[1] 1 , the number of bits needed can not be less than the entropy value, and Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' In the following, we use (c) to refer to the correctness condition, (i) to refer to the constraint that queries are independent of the messages, (a) to refer to the constraint that answers are deterministic functions of the storage variables and corresponding queries, and (p) to refer to the privacy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' From A[1] 1 , A[1] 2 , Q, we can decode W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' H(A[1] 1 , A[1] 2 |W1, Q) = H(A[1] 1 , A[1] 2 , W1|Q) − H(W1|Q) (38) (c)(i) = H(A[1] 1 , A[1] 2 |Q) − L (39) (36) ≤ 2H(A[1] 1 |Q) − L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (40) Next, consider Ingleton’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' I(A[1] 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |Q) ≤ I(A[1] 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |W1, Q) + I(A[1] 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |W2, Q) (41) = 2I(A[1] 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) (42) = 2 � H(A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) + H(A[2] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) − H(A[1] 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) � (43) (p) = 2 � 2H(A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) − H(A[1] 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) � (44) ≤ 2 � 2H(A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) + H(A[1] 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) − H(A[1] 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[1] 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) − H(A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) � (45) (c)(35) = 2 � H(A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) + H(A[1] 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) − H(A[1] 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[1] 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' W2|W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) � (46) 9 (i) ≤ 2 � 2H(A[1] 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[1] 2 |W1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Q) − H(W2) � (47) (40) ≤ 2 � 2(2H(A[1] 1 |Q) − L) − L � (48) where (42) follows from the observation that the second term can be bounded using the same method as that bounds the first term by switching the message index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A more detailed derivation of (44) appears in (79) of [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (45) is due to sub-modularity of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Note that I(A[1] 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A[2] 2 |Q) = H(A[1] 2 |Q) + H(A[2] 2 |Q) − H(A[1] 2 , A[2] 2 |Q) (49) (36) ≥ 2H(A[1] 1 |Q) − (¯α + ϵ)L (50) where in (50), and the second term is bounded as follows : H(A[1] 2 , A[2] 2 |Q) ≤ H(A[1] 2 , A[2] 2 , S2|Q) (a) = H(S2|Q) (37) ≤ (¯α + ϵ)L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (51) Combining (48) and (50), we have 2H(A[1] 1 |Q)/L − (¯α + ϵ) ≥ 2(4H(A[1] 1 |Q)/L − 3) ⇒ ¯α + ϵ + 6H(A[1] 1 |Q)/L ≥ 6 (52) (37) ⇒ ¯α + 6¯β ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (53) The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='3 Specialization of the Inner Bound The inner bound given in Theorem 1 is general but more involved, and we can specialize it in multiple ways in order to simplify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' One particularly interesting approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Define the region ˜R(t) in to be the collection of (¯α, ¯β) pairs such that there exists random variables (X0, X1, X2, Y1, Y2) jointly distributed with (V1, V2) such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The distribution factorizes as follows PV1,V2,X0,X1,X2,Y1,Y2 = PV1,V2PX0|V1,V2PX1|V1,V2PX2|V1,V2PY1|V1,V2PY2|V1,V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There exist deterministic functions f1,1, f1,2, f2,1, and f2,2 such that V1 = f1,1(X0, X1, Y1) = f2,2(X0, X2, Y2), (54) V2 = f1,2(X0, X1, Y2) = f2,1(X0, X2, Y1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (55) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A set of rates γ(0) 1 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0), γ(1) 1 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X1), γ(2) 1 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X2), (56) γ(1) 2 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y1), γ(2) 2 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y2), (57) β(0) 1 = γ(0) 1 , β(1) 1 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X1|X0), β(2) 1 = I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X2|X0), (58) β(1) 2 = max(I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y1|X0, X1), I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y1|X0, X2)), (59) β(2) 2 = max(I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y2|X0, X1), I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y2|X0, X2)), (60) and (α(0) 1 = γ(0) 1 , α(1) 1 , α(2) 1 , α(1) 2 , α(2) 2 ) as defined in item 3 for the general region R(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 10 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='95 1 inner bound via a non-linear scheme non-linear scheme: time-sharing inner bound via a linear scheme an outer bound on linear schemes information theoretic outer bound Figure 2: Illustration of inner bounds and outer bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The normalized average retrieval and storage rates 2t¯α ≥ α(0) 1 + α(1) 1 + α(2) 1 + α(1) 2 + α(2) 2 , (61) 4t¯β ≥ 2β(0) 1 + β(1) 1 + β(2) 1 + β(1) 2 + β(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (62) Then we have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' ˜R(t) in ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This inner bound is illustrated together with the outer bounds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The main difference from Theorem 1 is in the special dependence structure of (X0, X1, X2, Y1, Y2) jointly distributed with (V1, V2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', the Markov structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We verify that the rate assignments satisfy all the constraints in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Due to the special dependence structure of (X0, X1, X2, Y1, Y2) jointly distributed with (V1, V2), it is straightforward to verify that (γ(0) 1 , γ(1) 1 , γ(2) 1 , γ(1) 2 , γ(2) 2 ) ∈ RMD((V1, V2), X0, X1, X2, Y1, Y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We next verify (24) holds with the choice given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Due to the symmetry in the structure, we only need to confirm one subset of random variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=', {X0, X1, Y1}, and the three other subsets {X0, X1, Y2}, {X0, X2, Y1}, and {X0, X2, Y2} follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' There are a total of 7 conditions in the form of (14) associated with this subset {X0, X1, Y1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Notice that γ(0) 1 − β(0) 1 = 0, γ(1) 1 − β(1) 1 = I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0), γ(2) 2 − β(2) 2 ≤ I(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0, X1), which in fact confirm three of the seven conditions when J is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Next when J has two elements, we verify that γ(0) 1 − β(0) 1 + γ(1) 1 − β(1) 1 = I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0) = H(X0) + H(X1) − H(X0, X1) ≤ H(X0) + H(X1) − H(X0, X1|Y1), (63) γ(0) 1 − β(0) 1 + γ(1) 2 − β(1) 2 ≤ I(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0, X1) = H(Y1) + H(X0, X1) − H(X0, X1, Y1) ≤ H(Y1) + H(X0) + H(X1) − H(X0, X1, Y1) 11 Table 1: Conditional distribution PX0|W1,W2 used in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (w1, w2) x0 = (00) x0 = (01) x0 = (10) x0 = (11) (00) 1/2 1/2 (10) (1 − p)/2 p (1 − p)/2 (01) (1 − p)/2 p (1 − p)/2 (11) 1/2 1/2 = H(X0) + H(Y1) − H(X0, Y1|X1), (64) γ(1) 1 − β(1) 1 + γ(1) 2 − β(1) 2 ≤ I(X1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0) + I(Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X0, X1) = H(X1) + H(Y1) − H(X1, Y1|X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (65) Finally when J has all the three elements, we have γ(0) 1 − β(0) 1 + γ(1) 1 − β(1) 1 + γ(1) 2 − β(1) 2 = I(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X1) + I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X1) − max(I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y1|X0, X1), I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y1|X0, X2)) (66) ≤ I(X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X1) + I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' X1) − I(V1, V2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Y1|X0, X1) (67) = H(X0) + H(X1) + H(Y1) − H(X0, X1, Y1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' (68) Thus (24) is indeed true with the assignments (56)-(60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' This in fact completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We can use any explicit distribution (X0, X1, X2, Y1, Y2) to obtain an explicit inner bound to ˜R(t) in , and the next corollary provides one such non-trivial bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' For convenience, we write the entropy function of a probability mass (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , pt) as H(p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' , pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' The following (¯α, ¯β) ∈ R for any p ∈ [0, 1]: ¯α =9 4 − H(1 4, 3 4) + 1 4H(1 − p 2 , 1 − p 2 , p 2, p 2) + 1 2H(2 − p 4 , 2 − p 4 , p 2) − 3 4H(3 − 2p 6 , 3 − 2p 6 , p 3, p 3), ¯β =5 8 + 1 4H(2 − p 4 , 2 − p 4 , p 2) − 1 8H(1 − p 2 , 1 − p 2 , p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' These tradeoff pairs are obtained by applying Corollary 1, and choosing t = 1 and setting (X1, X2, Y1, Y2) as given in (17), and letting X0 be defined as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Note that the joint distri- bution indeed satisfies the required Markov structure, and in this case α(1) 2 = β(1) 2 and α(2) 2 = β(2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 5 Conclusion We consider the problem of private information retrieval using a Shannon-theoretic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' A new coding scheme based on random coding and binning is proposed, which reveals a hidden connection to the multiple description problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' It is shown that for the (2, 2) PIR setting, this non-linear coding scheme is able to provide the best known tradeoff between retrieval rate and storage rate, which is strictly better than that achievable using linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We further investigate the relation between zero-error PIR codes and ϵ-error PIR codes in this setting, and shows that they do not causes any essential difference in this problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' We hope that the hidden connection to multiple description coding can provide a new revenue to design more efficient PIR codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' 12 References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Chor, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Goldreich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Kushilevitz, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NA0T4oBgHgl3EQfN_-e/content/2301.02155v1.pdf'} +page_content=' Sudan, “Private 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Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland +3Hybrid Photonics Laboratory, Skolkovo Institute of Science and Technology, +Territory of Innovation Center Skolkovo, Bolshoy Boulevard 30, building 1, 121205 Moscow, Russia +(Dated: January 12, 2023) +We report on experimental observation of next-nearest-neighbour coupling between ballistically +expanding spinor exciton-polariton condensates in a planar semiconductor microcavity. All-optical +control over the coupling strength between neighbouring condensates is demonstrated through +distance-periodic pseudospin screening of their ballistic particle outflow due to the inherent splitting +of the planar cavity transverse-electric (TE) and transverse-magnetic (TM) modes. By screening +the nearest-neighbour coupling we overcome the conventional spatial coupling hierarchy between +condensates. This offers a promising route towards creating unconventional non-planar many-body +Hamiltonians using networks of ballistically expanding spinor exciton-polariton condensates. +Strongly correlated quantum many-body systems have +attracted a lot of interest as a promising tool to engi- +neer and explore phases of matter in extreme settings +[1–3] and to simulate complex Hamiltonians [4, 5]. Such +systems include ultracold atomic ensembles [4], trapped +ions [6, 7], nuclear and electronic spins [8, 9], supercon- +ducting circuits [10, 11], and nonlinear photonic systems +[12]. Of interest, recent milestone achievements in pro- +grammable connectivity in condensed matter using cold +atomic gases [13] now permit construction of intriguing +networks of coupled elements. However, in general, many +lab systems are by their physical nature unable to form +unconventional graph topologies. +In the past decade, +driven-dissipative Bose-Einstein condensates of exciton- +polaritons (from here on, polaritons) in planar microcavi- +ties have substantially advanced in optical reprogramma- +bility [14–21]. There, each condensate is driven by a fo- +cused non-resonant optical excitation beam forming a lo- +calized macroscopically coherent wavefunction [22]. The +coupling strength between neighbouring condensates is +roughly given by their mutual overlap with an expo- +nential fall-off as a function of separation distance [23– +25]. This means that nearest-neighbour (NN) coupling +dominates over next-nearest-neighbour (NNN) coupling +making polariton networks inherently planar in a graph +topology sense. +Overcoming this spatial coupling hi- +erarchy can offer opportunities to observe spontaneous +ordering and emergent polariton effects in non-planar +graph topologies [26–31]. +However, this is extremely +challenging, requiring very fine control over the two- +dimensional polariton potential landscape with limita- +tions of its own [32]. +In this Letter, we demonstrate that spin-orbit coupled +(SOC) exciton-polariton condensates can overcome this +challenge. Polaritons are quasiparticles exhibiting inter- +mixed properties of excitons and photons, which appear +when light and matter are brought to the strong coupling +regime [33]. As a consequence, the photon polarisation is +explicitly connected to the polariton pseudospin (or just +"spin" for short) with ˆσz = ±1 spin-projections along +the cavity growth axis representing σ± circularly polar- +ized light. Their two-component integer spin structure +has led to deep exploration into nonequilibrium spinor +quantum fluids [34]. +Polaritons mostly decay through +photons leaking out of the cavity containing all the in- +formation on the condensate such as energy, momentum, +density, and spin. This salient feature allows direct, yet +non-destructive, measurement of the condensate spin dis- +tribution using polarization resolved photoluminescence +(PL) imaging. +Both the polariton condensate and the incoherent pho- +toexcited background of excitons sustaining it adopt the +circular polarisation of the nonresonant excitation [35, +36] due to the optical orientation effect of excitons [37, 38] +and spin-preserving stimulated scattering of excitons into +the condensate [39]. This permits excitation of a con- +densate of a well defined macroscopic Sz ∼ ⟨ˆσz⟩ spin +projection [40–43]. Subsequently, the inherent TE-TM +splitting of the microcavity [44] will start rotating the +spin of any condensate polaritons which obtain finite +wavevector and flow away from the pump spot [45, 46]. +This is also referred to as the optical spin Hall ef- +fect [47, 48]. +Namely, the splitting between TE and +TM polarized cavity photon modes acts as a direction- +ally dependent in-plane effective magnetic field [47, 49] +(i.e., effective SOC [50]) causing the spins of outflowing +condensate polaritons to start precessing [see Fig. 1(a) +and Fig. 1(b)]. The strength of this effective SOC scales +quadratically with the polariton momentum, ∝ k2 and +can even be electrically tuned [51, 52]. This makes so- +called ballistic condensates ideal for enhanced SOC ef- +fects [45, 46] due to their extremely high kinetic en- +ergies obtained through repulsive Coulomb interactions +with the locally pump-induced exciton reservoir. More- +over, because of their long-range coherent particle out- +flow, ballistic condensates can couple over macroscopic +distances much greater than their respective full-width- +at-half-maximum [24] while also preserving their spin in- +arXiv:2301.04210v1 [cond-mat.mes-hall] 10 Jan 2023 + +2 +formation [43, 45, 46]. +Recently, it was theoretically predicted that ballistic +condensates could invert their neighbour coupling hier- +archy, making NNN stronger than NN, through a spin- +screening effect made possible by the effective SOC stem- +ming from TE-TM splitting [53]. Here, we provide exper- +imental evidence of these recent predictions. We present +a study of a spinor polariton dyad (two coupled con- +densates) and a triad [three coupled condensates, see +schematic Fig. 1(c)] wherein each condensate ballistically +emits a coherent pseudospin current which rapidly pre- +cesses as it propagates [45, 46]. We demonstrate control +over the coupling strength between neighbouring conden- +sates by changing the spatial distance between them (de- +noted d) relative to the spatial precession period of the +condensate pseudospin (denoted ξ). +We briefly explain the idea of spin-screened polariton +coupling. The three peaks in Fig. 1(c) represent the con- +densate centers excited by three co-localized Gaussian +pump spots of equal intensity. The red-blue colour map +shows the precession of the polariton pseudospin as it +radially propagates in-plane away from each condensate +center, with red representing Sz = +1 (spin-up polari- +tons) and blue representing Sz = −1 (spin-down polari- +tons). The height of the peaks represents the intensity of +the condensate emission. The distance between the con- +densate centers relative to the spatial oscillations of the +pseudospin modifies the coupling between them. In the +non-screened state [Fig. 1(c)] NN condensates are excited +at a distance equal to integer number of periods of pseu- +dospin oscillations, d = nξ where n = 1, 2, 3, . . . . This +means that propagating condensate polaritons arrive at +NNs with unchanged spin projection. On the contrary, +in the screened state [Fig. 1(d)] NNs are separated by +d = (n−1/2)ξ and polaritons arrive at their NNs with op- +posite spin-projection which reduces the condensate cou- +pling, while coupling between NNNs is still maintained. +The microcavity used in this study consists of a 5λ/2 +AlGaAs cavity surrounded by two distributed Bragg mir- +rors (DBR) of 35 and 32 pairs of ALGaAs/AlAs for the +bottom and top DBR correspondingly with the 12 GaAs +QWs separated into four sets of three QWs placed at the +antinodes of electric field within the cavity. The cavity +quality factor is around Q ∼ 16000 with the correspond- +ing polariton lifetime τp ≈ 5 ps and Rabi splitting of 9 +meV. The measured TE-TM splitting is ≈ 0.2 meV at +k = 3 µm−1 in-plane wavevector. See section S1 in the +Supplemental Material [54] for further experimental de- +tails. +The normalized Stokes parameters of the cavity emis- +sion are written, +Sx,y,z(r) = IH,D,σ+(r) − IV,A,σ−(r) +IH,D,σ+(r) + IV,A,σ−(r), +(1) +where +r += +(x, y) +is +the +in-plane +coordinate +and +IH(V ),D(A),σ+(σ−)(r) +corresponds +to +horizon- +Figure 1. +(a) Schematic of the effective SOC magnetic field +distribution (dark olive arrows) from the TE-TM splitting +on a momentum-space circle. (b) Schematic of the Poincaré +sphere showing example pseudospin precession as polaritons +propagate (blue and red arrows). Schematic representing two +pump geometries where the distance between the central and +edge pump spots equals to (c) one full period of pseudospin +oscillation (NN is stronger than NNN) and (d) half oscilla- +tion period (NN is weaker than NNN). The height of the peaks +represents the intensity of the condensate emission, and the +red, white, and blue colour map shows the precession of the +polariton pseudospin propagating in the cavity plane, with +red representing Sz = +1 (spin-up polaritons) and blue repre- +senting Sz = −1 (spin-down polaritons). Red and blue arrows +show the pseudospin precession of the polaritons propagating +from the edge condensates along the triad axis +tally(vertically), diagonally(antidiagonally), and right- +circularly(left-circularly) +polarized +(RCP +and +LCP +for short) PL, respectively. +Formally, the Stokes pa- +rameters relate to the condensate pseudospin through +S = ⟨Ψ†|ˆσ|Ψ⟩/⟨Ψ†|Ψ⟩ where Ψ = (ψ+, ψ−)T is the +condensate spinor order parameter and ˆσ is the Pauli + +(b) +a +D +H +(c) NN > NNN +Microcavity +(d) NN < NNN +Microcavity3 +matrix-vector. The Sx(r) and Sy(r) components repre- +sent the degree of linear and diagonal polarisation but +are not important in this study (also due to the pre- +dominant circular polarisation of the condensates used +here). +Experimental measurements were reproduced +using a generalised two-dimensional Gross-Pitaevskii +equation (2DGPE) (see section S2 in the Supplemental +Material [54]). +In Fig. 2 we present results for two polariton con- +densates separated by d ≈ ξ/2. Data for a single iso- +lated condensate gives a Sz period around ξ ≈ 90 µm +(see section S1 in the Supplemental Material [54]). Fig- +ures 2(a) and 2(b) show the measured and simulated spa- +tial distribution of the Sz component with spatial pseu- +dospin oscillations clearly visible due to the SOC rotat- +ing the spin of the outflowing polaritons. Note that un- +avoidable dephasing of polaritons in experiment results +in lowered Sz values compared to simulations as indi- +cated on the colorbars. Smaller ripple-like modulations +are also visible due to the standing wave interference be- +tween the two phase-locked condensates as reported be- +fore [24, 43, 53]. +These ripples are characterized by a +small-scale period λ = 2π/⟨kc⟩ ≈ 3 µm, where ⟨kc⟩ is the +average outflow momentum of polaritons from their con- +densates. In contrast, the large-scale Sz period is given +by ξ = 2π/∆k ≫ λ where ℏ∆k/√2εc = |√mTE−√mTM| +and εc ≈ 3 meV is the condensate energy (measured from +k = 0 at the dispersion) and mTE,TM are the effective +masses of TE and TM polarized polaritons [44]. +The spin screening effect can be observed as periodic +extrema in the integrated PL intensity, which represents +the condensate occupation, as a function of separation +distance d in Fig. 2(c). At the maxima the coupling is +unscreened and NN coupling is strong. At the minima +the coupling is screened and NN coupling is weak. Black +dots and black solid curve denote experimental measure- +ments and calculations, respectively. In the absence of +SOC one would observe monotonically decreasing emis- +sion intensity with only short variations (order of λ) +corresponding to in-phase and anti-phase flip-flop tran- +sitions between the synchronized condensates [24]. In- +stead, we observe strong non-monotonic behaviour with +clearly visible maxima around 67 and 154 µm, and min- +ima around 56 and 135 µm. +Notice that the distance +between the two maxima and the two minima correlates +with the measured ξ ≈ 90 µm period of Sz oscillations. +The discrepancy between the absolute locations of the +minima and maxima with the predicted critical distances +for screened (ξ/2, 3ξ/2) and unscreened (ξ, 2ξ) coupling, +respectively, can be understood as follows. Firstly, when +two condensates are coupled their energy is redshifted +on average [24] leading to smaller εc and thus larger ξ in +the coupled system. Second, the finite width of the pump +spots modulates the phase of polaritons and causes a shift +in the Sz period. Third, the cavity here has higher levels +of disorder than strain-compensated cavities [55] which +Figure 2. +Two polariton condensates. (a) Experimentally +measured and (b) simulated numerically real space Sz compo- +nent of the Stokes vector of the polariton condensates emis- +sion. In panel (a) black circles show the position of pump +spots. (c) Total integrated emission intensity dependence on +the separation distance between two condensates pump spots. +In panel (c) black dots shows the experimentally measured +values with red region representing the error of the total inten- +sity value. Black curve shows the same dependence calculated +numerically +can affect the spatial coupling. That’s why the relative +distances between the extrema are more meaningful than +their absolute locations. This interpretation is verified in +2DGPE modeling which accurately reproduces the loca- +tions of the extrema. Note that the slight discrepancy +between modeling and experiment in Fig. 2(c) between +70 and 120 µm can be attributed to the large parame- +ter space of the 2DGPE making quantitative matching +somewhat challenging. +In order to demonstrate the NNN coupling using the +all-optical spin screening effect we investigated the sys- +tem containing a chain of three condensates similar to the +system depicted schematically in Fig. 1. As in the pre- +vious experiment with two condensates, all condensates +were excited using tightly focused RCP laser pump spots +of equal intensity above threshold. Figures 3(a) and 3(b) +show the measured and simulated spatial distribution of +the three condensate Sz component with NN distance of +d ≈ ξ/2. As in the previous case of two condensates, the +system forms a joint macroscopic coherent state result- +ing in an oscillating Sz pattern elongated along the hor- +izontal axis with three RCP condensate circles of equal + +X (um) +-100 +0 +100 +-100 +0 +100 +(b) +a +100 +(μm) +0 +-100 +0.6 +-0.6 +(c) +1.0 +0.8 +0.6 +0.4 +40 +80 +120 +160 +Pump spot separation distance (um)4 +degree of polarisation in the centre. Amazingly, the in- +tensity of the central condensate was suppressed relative +to the outer ones, evidencing reduced NN coupling due +to the spin screening effect, see in Fig. 3(c) measured +(red diamonds) and simulated (black solid curve) inten- +sity distribution along the triad axis. +To unambiguously demonstrate the spin screening ef- +fect in the triad, we measured (dots) and simulated (solid +curve) the dependence of the central condensate intensity +as a function of NN separation distance with results pre- +sented in Fig. 3(d). Both experiment and calculations +show a clear dip around d = 52 µm ≈ ξ/2, corresponding +to spin-screened NN coupling, followed by a small peak +around d = 80 µm ≈ ξ where the NN coupling is re- +stored. The observed suppression of the central conden- +sate intensity provides strong evidence of spin-screened +NN coupling mediated by the spin coherence of the sys- +tem. +Moreover, the experimentally measured pump power +dependence for each separation distance and the ex- +tracted polariton condensation threshold values are +shown in Fig. 3(e) (red circles). The horizontal dashed +line is the threshold value of the isolated condensate. In +the absence of the TE-TM splitting monotonic increase of +the threshold value converging to the isolated condensate +threshold is expected with the increase of the separation +distance between the condensates. In our system we ob- +serve maximum threshold at the separation distance of +52 µm, which precisely corresponded to the minimum of +the central condensate intensity in Figs. 3(c) and 3(d). It +confirms that the NN condensate interaction is effectively +screened at this separation distance due to the TE-TM +splitting. Around a separation distance close to the full +period of Sz oscillation (d ≈ ξ) a decrease in the thresh- +old power was observed, as expected with NN coupling +restored. A simple linear coupled oscillator model [solid +curve in Fig. 3(e)] is able to explain the behaviour of +the threshold power (see section S3 in the Supplemental +Material [54]). +In summary, we have experimentally demonstrated +that next-nearest-neighbours coupling can be made +stronger than nearest-neighbour coupling in ballistically +expanding spinor exciton-polariton condensates which +was recently proposed in Ref. [53]. This unconventional +near-inversion of the spatial coupling hierarchy between +condensates stems from the combination of TE-TM split- +ting and the ballistic polariton flow from each conden- +sate. +Outflowing polaritons experience effective spin- +orbit coupling which rotates their spin state as they prop- +agate from one condensate to the next. Depending on +distance, the overlap (coupling) between the condensates +can become spin-screened depending on the polariton +spin projection upon arrival at its neighbour. We believe +that the demonstrated alteration of the conventional con- +densate coupling hierarchy could pave the way towards +all-optical simulation of many-body ballistic systems be- +Figure 3. +Three polariton condensates. (a) Experimentally +measured and (b) simulated real space Sz component of the +Stokes vector of the polariton condensates emission. In panel +(a) black circles show the position of pump spots. (c) Mea- +sured experimentally (red diamonds) and calculated numeri- +cally (solid black curve) real space intensity distribution along +the triad axis. (d) Dependence of the central condensate PL +intensity on the separation distance between the condensates +pump spots measured experimentally (black dots) and calcu- +lated numerically (solid black curve); red region represents the +error of the total intensity value. The dashed curves are guides +to the eye. (e) The system threshold power dependence on +the separation distance between the condensates pump spots +measured experimentally (red circles) and calculated numer- +ically (solid black curve); red bars represent the error. Grey +dashed line in panel (e) shows the threshold power for single +isolated condensate. +longing to non-planar graph topologies using networks of +spinor polariton condensates. +The authors acknowledge the support of the European +Union’s Horizon 2020 program, through a FET Open re- +search and innovation action under the grant agreements +No. +899141 (PoLLoC) and no. +964770 (TopoLight). +H.S. acknowledges the Icelandic Research Fund (Rannis), +Grant No. 217631-051. + +X (μm) +-100 +0 +100 +-100 +0 +100 +(a) +100 +100 +Y +(um) +(μm) +0 +0 +Y +-100 +-100 +-0.7 +0.7 +SZ +(c) +1.0 +Int +1.0 +tegrated +0.8 +(a.u.) +0.8 +intensity +0.6 +Intensity +0.6 +0.4 +0.4 @ +0.2 +.u. +20 +40 +60 +80 +¥100 +-100 +0 +100 +(un) X +Pump spot separation (μm) +(e) +.0 +0.9 +0.8 +0 +20 +40 +60 +80 +100 +Pump spot separation (um)5 +∗ DovzhenkoDS@gmail.com +[1] F. Alet, A. M. Walczak, and M. P. Fisher, Exotic quan- +tum phases and phase transitions in correlated matter, +Phys. A Stat. 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Lett. 105, 191118 (2014). + +Supplemental Material: Next nearest neighbour coupling with spinor polariton +condensates +Dmitriy Dovzhenko,1, ∗ Denis Aristov,1 Lucy Pickup,1 Helgi Sigurdsson,1, 2 and Pavlos Lagoudakis1, 3 +1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK +2Science Institute, University of Iceland, Dunhagi 3, IS-107, Reykjavik, Iceland +3Hybrid Photonics Laboratory, Skolkovo Institute of Science and Technology, +Territory of Innovation Center Skolkovo, Bolshoy Boulevard 30, building 1, 121205 Moscow, Russia +(Dated: January 12, 2023) +S1. +EXPERIMENTAL DETAILS, TE-TM SPLITTING AND SINGLE ISOLATED CONDENSATE REAL +SPACE Sz COMPONENT OF THE STOKES VECTOR +In this supplemental section we present experimental details, experimentally measured TE-TM splitting and real +space distribution of Sz component of Stokes vector of the emission from a single isolated polariton condensate. +All measurements were performed at 6 K using a continuous flow cold finger cryostat. We used a right circularly +(σ+) polarized non-resonant continuous wave laser excitation tuned to the first Bragg minimum of the microcavity +reflection spectra at 754 nm. To reduce sample heating we used an acousto-optic modulator driven by rectangular +voltage pulse train at 10 kHz repetition rate with 5% duty cycle. A spatial light modulator was used to structure the +pump spatial profile into one, two, or three Gaussian spots focused on the sample using a microscope objective lens +with 0.4 numerical aperture. In order to obtain simultaneously real space, k-space, and spectrally resolved k-space +images of the time-averaged PL two separate CCD cameras and a 0.75 m monochromator with 1200g/mm diffraction +grating equipped with the CCD camera were used. A quarter wave plate and a Wollaston prism were introduced +in the optical path for real space imaging to simultaneously measure the right-circularly polarized and left-circularly +polarized components of the PL with the same CCD camera. +In Fig. S1(a) we show the Sz spatial oscillations due to the spin-orbit coupling (SOC) rotating the pseudospin of +the polaritons propagating away from the condensate excited using single Gaussian spot. The oscillation period ξ was +measured to be around 90 µm with the oscillations amplitude of ±0.6. +In order to experimentally estimate the value of TE-TM splitting we measured dispersion of the lower polariton +branch in a linear regime (i.e., below condensation threshold) along the in-plane k∥ momentum axis [see Fig. S1(b)]. +Splitting of the dispersion is clearly observed at the higher values of in-plane k vector with the higher energy branch +corresponding to the emission from the vertically polarized polaritons. The energy splitting between horizontally +and vertically polarized polaritons possesses parabolic dependence on the in-plane momentum and ≈ 0.2 meV at +k = 3 µm−1 in-plane wavevector. +S2. +TWO DIMENSIONAL SPINOR POLARITON MODEL +The experimental observations are reproduced by numerically solving a generalized Gross-Pitaevskii equation (S1) +for macroscopic spinor polariton wavefunction Ψ(r, t) = (ψ+, ψ−)T coupled to an active exciton reservoir with density +nA(r, t) = (nA+, nA−)T rate equation [1], +i∂ψ± +∂t += +� +− ℏ∇2 +2m + i +2 +� +RnA± − γ +� ++ α1|ψ±|2 + α2|ψ∓|2 + U±(r) + V (r) +� +ψ± + ∆LT +� ∂ +∂x +∓ i ∂ +∂y +�2 +ψ∓, +(S1) +U± = G1 +� +nA± + nI± +� ++ G2 +� +nA∓ + nI∓ +� +, +(S2) +∂nA± +∂t += − +� +ΓA + Γs + R|ψ±|2� +nA± + WnI± + ΓsnA∓. +(S3) +Here, ± represents the spin of polaritons and excitons along the cavity growth axis, m is the polariton effective +mass in parabolic dispersion approximation, γ is the polariton decay rate, G1 = 2g|χ|2 and α1 = g|χ|4 are the same +spin polariton-reservoir and polariton-polariton interaction strengths, respectively, g is the exciton-exciton Coulomb +∗ DovzhenkoDS@gmail.com +arXiv:2301.04210v1 [cond-mat.mes-hall] 10 Jan 2023 + +2 +Figure S1. +(a) Experimentally measured real space Sz component of the Stokes vector of the single isolated polariton +condensate emission. (b) Energy and in-plane wavevector resolved normalized PL intensity from the lower polariton branch +below the threshold +interaction strength, |χ|2 is the excitonic Hopfield fraction of the polariton, and ∆LT represents the strength of the +TE-TM splitting. Opposite spin interactions, usually much weaker, were chosen to be G2 = −0.2G1 and α2 = −0.2α1 +for completeness but we note that our results to not qualitatively depend on these terms. R is the scattering rate of +reservoir excitons into the condensate, ΓA is the active reservoir decay rate, and Γs represents exciton spin relaxation +rate [2]. +A so-called inactive reservoir of excitons nI,± also contributes to the blueshift of polaritons as depicted in Eq. (S2). +This reservoir corresponds to high-momentum excitons which do not scatter into the condensate but instead drive the +active low-momentum excitons (S3). In continuous wave experiments the inactive reservoir density can be written +WnI,+ = +P0(r) +W + 2Γs +(W cos2 (θ) + Γs), +WnI,− = +P0(r) +W + 2Γs +(W sin2 (θ) + Γs), +(S4) +where P0 is the total power density of the incident coherent light with degree of circular polarization expressed as S3 = +P0[cos2 (θ)−sin2 (θ)] = P0 cos (2θ). Since our experiment is performed with fully right hand circularly polarized light, +we set θ = 0 from here on. The phenomenological parameter W quantifies conversion rate between same-spin inactive +and active exciton reservoirs. The pump profile is written as a superposition of Gaussians P0(r) = p0 +� +n e−|r−rn|2/2w2. +To represent tight focusing of excitation beams we used Gaussians with 2 µm full-width-at-half-maximum. +Lastly, given the disorder present at the large spatial scales of the experiment we include a random potential +landscape in our simulation given by V (r) generated as a random Gaussian-correlated potential [3]. The simulation +parameters are based on previous GaAs microcavity experiments [4, 5]: m = 5×10−5 of free electron mass; γ−1 = 5.5 +ps; |χ|2 = 0.4; ℏg = 0.5 µeV µm2; R = 3.2g; W = ΓA = γ; Γs = γ/4; ∆LT = 0.036 ps−1 µm2. The disorder potential +was generated with 1.5 µm correlation length and 0.06 meV root mean squared amplitude. +We note that in order to compensate for additional background noise in experiment (i.e., additional light coming +from spontaneous emission of bottleneck excitons) we applied a global shift to the integrated densities of the condensate +|ψ±|2 by approximately 10 percent in order to match the experimental values in Figure 3(d) in the main text. This +difference between modeling and experiment is more evident in Figure 3(c) where the experimentally measured +photoluminescence (PL) intensity is more spread out than simulated condensate densities. This can also come from +the finite diffusion of excitons which we have neglected here for simplicity. +Nevertheless, the calculated relative + +X (μm) +k, (μm-l) +-200 +200 +0 +-3-2 +-1 +0 +1 +2 +[(b) +(a) +1.543 +200 +1.542 +(un) +0 +(eV) +1.540 +1.539 +-200 +1.538 +0.6 +Sz +-0.6 +PL intensity +03 +amplitude of the PL at the pump positions follows the experimental results quite precisely, which, therefore, justifies +the use of current model and provides a clear quantitative evidence of the spin-screening happening in the system. +S3. +THEORY OF THE THRESHOLD BEHAVIOUR IN A SPIN SCREENED CONDENSATE TRIAD +The behaviour of the pump threshold from experiment in the triad configuration can be reproduced by scrutinizing +the eigenenergies of an appropriate linear operator which couples the three condensates together. In other words, we +neglect polariton nonlinearities so close to the threshold. The threshold is reached when a single eigenvalue belonging +to the three coupled condensates crosses from the lower- to the upper-half of the complex plane. +We will start by defining the state vector of the system, +|Ψ⟩ = (ψ1,+, ψ1,−, ψ2,+, ψ2,−, ψ3,+, ψ3,−)T. +(S5) +Here, the index n ∈ {1, 2, 3} denotes the left, middle, and right condensate, respectively. The spectrum of the coupled +system in the linear regime (i.e., close to threshold |ψn,±|2 ≃ 0) can be described with the following non-Hermitian +operator separated into three parts for clarity, +ˆH = +� +� +� +� +� +� +� +ω+ +0 +0 +0 +0 +0 +0 +ω− +0 +0 +0 +0 +0 +0 +ω+ +0 +0 +0 +0 +0 +0 +ω− +0 +0 +0 +0 +0 +0 +ω+ +0 +0 +0 +0 +0 +0 +ω− +� +� +� +� +� +� +� ++ +� +� +� +� +� +� +� +0 +0 +J+ δJ +0 +0 +0 +0 +δJ J− +0 +0 +J+ δJ +0 +0 +J+ δJ +δJ J− +0 +0 +δJ J− +0 +0 +J+ δJ +0 +0 +0 +0 +δJ J− +0 +0 +� +� +� +� +� +� +� ++ +� +� +� +� +� +� +� +0 +0 +0 0 K+ δK +0 +0 +0 0 δK K− +0 +0 +0 0 +0 +0 +0 +0 +0 0 +0 +0 +K+ δK 0 0 +0 +0 +δK K− 0 0 +0 +0 +� +� +� +� +� +� +� +(S6) +The first matrix describes the complex self-energy of each oscillator (condensate) composed of the local pump blueshift +(G) and gain (R), and cavity losses (γ). This contribution from the pump can be parametrized in terms of the reservoir +spin populations, +ω± = +� +G1 + iR +2 +� +(NA,± + NI,±) − iγ +2 . +(S7) +where +� +nA(I),± dr = NA(I),± [i.e., spatially integrating (S3) and (S4)]. Here, we will neglect opposite spin interaction +G2 for simplicity. +Each condensate is coupled ballistically with its nearest neighbours with coupling strength J± and next-nearest +neighbours with strength K± determined by the overlap between different condensates over their respective pump +spots. Approximating the tightly focused pump spots as delta functions, we can write the coupling between the +ballistic condensates as [4], +J± = cos2 (ξd + Φ) +� +G1 + iR +2 +� +(NA,± + NI,±)H(1) +0 (kd + φ), +(S8) +K± = sin2 (2ξd + Φ) +� +G1 + iR +2 +� +(NA,± + NI,±)H(1) +0 (2kd + φ) +(S9) +The square cosine (sine) modulations in the coupling stem from a pseudospin screening effect coming from the strong +influence of TE-TM splitting on the ballistic condensates [6] as explained in the main manuscript. Here, ξ denotes +the period of the pseudospin precession for a single condensate in experiment. H(1) +0 (kd) is the zeroth order Hankel +function of the first kind. The coupling depends on the product kd where d is the separation distance between two +pump spots and k is the complex wavevector of the polaritons with mass m propagating outside the pump spot, +k ≈ kc + i Γm +2ℏkc +. +(S10) +Here, kc is the average real wavevector of the outflowing polaritons. The finite size of the Gaussian pump spots intro- +duces some lag into the pseudospin precession because outflowing polaritons need to gradually build up momentum +as they leave the pump spot. This is captured in the fitting parameter Φ. For the same reason, an overall phase-lag +fitting parameter φ is also needed in the coupling term between the condensates. + +4 +The presence of TE-TM splitting also introduces coupling between opposite spin components denoted δJ and δK +written in a similar fashion, +δJ = δ cos2 (ξd + Φ) +� +G1 + iR +2 +� +(NA,+ + NI,+ + NA,− + NI,−)H(1) +0 (kd + φ), +(S11) +δK = δ sin2 (2ξd + Φ) +� +G1 + iR +2 +� +(NA,+ + NI,+ + NA,− + NI,−)H(1) +0 (2kd + φ) +(S12) +Here, δ < 1 is a fitting parameter describing the amount of opposite spin coupling. Diagonalizing ˆH for increasing +pump power P0 we identify the threshold as the point in which a single eigenenergy crosses from the lower-half to +the upper-half of the complex plane. The results are plotted in Fig. 3(e) in the main text (solid curve) alongside the +experimental data, normalized in units of threshold power for the single isolated condensates Pthr,iso. +[1] H. Deng, H. Haug, and Y. Yamamoto, Exciton-polariton bose-einstein condensation, Rev. Mod. Phys. 82, 1489 (2010). +[2] M. Z. Maialle, E. A. de Andrada e Silva, and L. J. Sham, Exciton spin dynamics in quantum wells, Phys. Rev. B 47, 15776 +(1993). +[3] V. Savona, Effect of interface disorder on quantum well excitons and microcavity polaritons, J. Phys. Condens. Matter 19, +295208 (2007). +[4] J. D. T¨opfer, H. Sigurdsson, L. Pickup, and P. G. Lagoudakis, Time-delay polaritonics, Commun. Phys. 3, 2 (2020). +[5] L. Pickup, J. D. T¨opfer, H. Sigurdsson, and P. G. Lagoudakis, Polariton spin jets through optical control, Phys. Rev. B +103, 155302 (2021). +[6] D. Aristov, H. Sigurdsson, and P. G. Lagoudakis, Screening nearest-neighbor interactions in networks of exciton-polariton +condensates through spin-orbit coupling, Phys. Rev. B 105, 155306 (2022). + diff --git a/7NE2T4oBgHgl3EQf7giQ/content/tmp_files/load_file.txt b/7NE2T4oBgHgl3EQf7giQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6160570f361b3d2e640a67a4bcec7b014634942a --- /dev/null +++ b/7NE2T4oBgHgl3EQf7giQ/content/tmp_files/load_file.txt @@ -0,0 +1,1111 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf,len=1110 +page_content='Next nearest neighbour coupling with spinor polariton condensates Dmitriy Dovzhenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' ∗ Denis Aristov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1 Lucy Pickup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1 Helgi Sigurdsson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2 and Pavlos Lagoudakis3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1 1School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' University of Southampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Southampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' SO17 1BJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' UK 2Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' University of Iceland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Dunhagi 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' IS-107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Reykjavik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Iceland 3Hybrid Photonics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Skolkovo Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Territory of Innovation Center Skolkovo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Bolshoy Boulevard 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' building 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 121205 Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Russia (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2023) We report on experimental observation of next-nearest-neighbour coupling between ballistically expanding spinor exciton-polariton condensates in a planar semiconductor microcavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' All-optical control over the coupling strength between neighbouring condensates is demonstrated through distance-periodic pseudospin screening of their ballistic particle outflow due to the inherent splitting of the planar cavity transverse-electric (TE) and transverse-magnetic (TM) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' By screening the nearest-neighbour coupling we overcome the conventional spatial coupling hierarchy between condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This offers a promising route towards creating unconventional non-planar many-body Hamiltonians using networks of ballistically expanding spinor exciton-polariton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Strongly correlated quantum many-body systems have attracted a lot of interest as a promising tool to engi- neer and explore phases of matter in extreme settings [1–3] and to simulate complex Hamiltonians [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Such systems include ultracold atomic ensembles [4], trapped ions [6, 7], nuclear and electronic spins [8, 9], supercon- ducting circuits [10, 11], and nonlinear photonic systems [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Of interest, recent milestone achievements in pro- grammable connectivity in condensed matter using cold atomic gases [13] now permit construction of intriguing networks of coupled elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' However, in general, many lab systems are by their physical nature unable to form unconventional graph topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In the past decade, driven-dissipative Bose-Einstein condensates of exciton- polaritons (from here on, polaritons) in planar microcavi- ties have substantially advanced in optical reprogramma- bility [14–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' There, each condensate is driven by a fo- cused non-resonant optical excitation beam forming a lo- calized macroscopically coherent wavefunction [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The coupling strength between neighbouring condensates is roughly given by their mutual overlap with an expo- nential fall-off as a function of separation distance [23– 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This means that nearest-neighbour (NN) coupling dominates over next-nearest-neighbour (NNN) coupling making polariton networks inherently planar in a graph topology sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Overcoming this spatial coupling hi- erarchy can offer opportunities to observe spontaneous ordering and emergent polariton effects in non-planar graph topologies [26–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' However, this is extremely challenging, requiring very fine control over the two- dimensional polariton potential landscape with limita- tions of its own [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In this Letter, we demonstrate that spin-orbit coupled (SOC) exciton-polariton condensates can overcome this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Polaritons are quasiparticles exhibiting inter- mixed properties of excitons and photons, which appear when light and matter are brought to the strong coupling regime [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' As a consequence, the photon polarisation is explicitly connected to the polariton pseudospin (or just "spin" for short) with ˆσz = ±1 spin-projections along the cavity growth axis representing σ± circularly polar- ized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Their two-component integer spin structure has led to deep exploration into nonequilibrium spinor quantum fluids [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Polaritons mostly decay through photons leaking out of the cavity containing all the in- formation on the condensate such as energy, momentum, density, and spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This salient feature allows direct, yet non-destructive, measurement of the condensate spin dis- tribution using polarization resolved photoluminescence (PL) imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Both the polariton condensate and the incoherent pho- toexcited background of excitons sustaining it adopt the circular polarisation of the nonresonant excitation [35, 36] due to the optical orientation effect of excitons [37, 38] and spin-preserving stimulated scattering of excitons into the condensate [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This permits excitation of a con- densate of a well defined macroscopic Sz ∼ ⟨ˆσz⟩ spin projection [40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Subsequently, the inherent TE-TM splitting of the microcavity [44] will start rotating the spin of any condensate polaritons which obtain finite wavevector and flow away from the pump spot [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This is also referred to as the optical spin Hall ef- fect [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Namely, the splitting between TE and TM polarized cavity photon modes acts as a direction- ally dependent in-plane effective magnetic field [47, 49] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=', effective SOC [50]) causing the spins of outflowing condensate polaritons to start precessing [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The strength of this effective SOC scales quadratically with the polariton momentum, ∝ k2 and can even be electrically tuned [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This makes so- called ballistic condensates ideal for enhanced SOC ef- fects [45, 46] due to their extremely high kinetic en- ergies obtained through repulsive Coulomb interactions with the locally pump-induced exciton reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' More- over, because of their long-range coherent particle out- flow, ballistic condensates can couple over macroscopic distances much greater than their respective full-width- at-half-maximum [24] while also preserving their spin in- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='04210v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='mes-hall] 10 Jan 2023 2 formation [43, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Recently, it was theoretically predicted that ballistic condensates could invert their neighbour coupling hier- archy, making NNN stronger than NN, through a spin- screening effect made possible by the effective SOC stem- ming from TE-TM splitting [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Here, we provide exper- imental evidence of these recent predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We present a study of a spinor polariton dyad (two coupled con- densates) and a triad [three coupled condensates, see schematic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1(c)] wherein each condensate ballistically emits a coherent pseudospin current which rapidly pre- cesses as it propagates [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We demonstrate control over the coupling strength between neighbouring conden- sates by changing the spatial distance between them (de- noted d) relative to the spatial precession period of the condensate pseudospin (denoted ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We briefly explain the idea of spin-screened polariton coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The three peaks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1(c) represent the con- densate centers excited by three co-localized Gaussian pump spots of equal intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The red-blue colour map shows the precession of the polariton pseudospin as it radially propagates in-plane away from each condensate center, with red representing Sz = +1 (spin-up polari- tons) and blue representing Sz = −1 (spin-down polari- tons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The height of the peaks represents the intensity of the condensate emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The distance between the con- densate centers relative to the spatial oscillations of the pseudospin modifies the coupling between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In the non-screened state [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1(c)] NN condensates are excited at a distance equal to integer number of periods of pseu- dospin oscillations, d = nξ where n = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This means that propagating condensate polaritons arrive at NNs with unchanged spin projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' On the contrary, in the screened state [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1(d)] NNs are separated by d = (n−1/2)ξ and polaritons arrive at their NNs with op- posite spin-projection which reduces the condensate cou- pling, while coupling between NNNs is still maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The microcavity used in this study consists of a 5λ/2 AlGaAs cavity surrounded by two distributed Bragg mir- rors (DBR) of 35 and 32 pairs of ALGaAs/AlAs for the bottom and top DBR correspondingly with the 12 GaAs QWs separated into four sets of three QWs placed at the antinodes of electric field within the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The cavity quality factor is around Q ∼ 16000 with the correspond- ing polariton lifetime τp ≈ 5 ps and Rabi splitting of 9 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The measured TE-TM splitting is ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='2 meV at k = 3 µm−1 in-plane wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' See section S1 in the Supplemental Material [54] for further experimental de- tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The normalized Stokes parameters of the cavity emis- sion are written, Sx,y,z(r) = IH,D,σ+(r) − IV,A,σ−(r) IH,D,σ+(r) + IV,A,σ−(r), (1) where r = (x, y) is the in-plane coordinate and IH(V ),D(A),σ+(σ−)(r) corresponds to horizon- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (a) Schematic of the effective SOC magnetic field distribution (dark olive arrows) from the TE-TM splitting on a momentum-space circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (b) Schematic of the Poincaré sphere showing example pseudospin precession as polaritons propagate (blue and red arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Schematic representing two pump geometries where the distance between the central and edge pump spots equals to (c) one full period of pseudospin oscillation (NN is stronger than NNN) and (d) half oscilla- tion period (NN is weaker than NNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The height of the peaks represents the intensity of the condensate emission, and the red, white, and blue colour map shows the precession of the polariton pseudospin propagating in the cavity plane, with red representing Sz = +1 (spin-up polaritons) and blue repre- senting Sz = −1 (spin-down polaritons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Red and blue arrows show the pseudospin precession of the polaritons propagating from the edge condensates along the triad axis tally(vertically), diagonally(antidiagonally), and right- circularly(left-circularly) polarized (RCP and LCP for short) PL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Formally, the Stokes pa- rameters relate to the condensate pseudospin through S = ⟨Ψ†|ˆσ|Ψ⟩/⟨Ψ†|Ψ⟩ where Ψ = (ψ+, ψ−)T is the condensate spinor order parameter and ˆσ is the Pauli (b) a D H (c) NN > NNN Microcavity (d) NN < NNN Microcavity3 matrix-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The Sx(r) and Sy(r) components repre- sent the degree of linear and diagonal polarisation but are not important in this study (also due to the pre- dominant circular polarisation of the condensates used here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Experimental measurements were reproduced using a generalised two-dimensional Gross-Pitaevskii equation (2DGPE) (see section S2 in the Supplemental Material [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2 we present results for two polariton con- densates separated by d ≈ ξ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Data for a single iso- lated condensate gives a Sz period around ξ ≈ 90 µm (see section S1 in the Supplemental Material [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Fig- ures 2(a) and 2(b) show the measured and simulated spa- tial distribution of the Sz component with spatial pseu- dospin oscillations clearly visible due to the SOC rotat- ing the spin of the outflowing polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Note that un- avoidable dephasing of polaritons in experiment results in lowered Sz values compared to simulations as indi- cated on the colorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Smaller ripple-like modulations are also visible due to the standing wave interference be- tween the two phase-locked condensates as reported be- fore [24, 43, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' These ripples are characterized by a small-scale period λ = 2π/⟨kc⟩ ≈ 3 µm, where ⟨kc⟩ is the average outflow momentum of polaritons from their con- densates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In contrast, the large-scale Sz period is given by ξ = 2π/∆k ≫ λ where ℏ∆k/√2εc = |√mTE−√mTM| and εc ≈ 3 meV is the condensate energy (measured from k = 0 at the dispersion) and mTE,TM are the effective masses of TE and TM polarized polaritons [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The spin screening effect can be observed as periodic extrema in the integrated PL intensity, which represents the condensate occupation, as a function of separation distance d in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' At the maxima the coupling is unscreened and NN coupling is strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' At the minima the coupling is screened and NN coupling is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Black dots and black solid curve denote experimental measure- ments and calculations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In the absence of SOC one would observe monotonically decreasing emis- sion intensity with only short variations (order of λ) corresponding to in-phase and anti-phase flip-flop tran- sitions between the synchronized condensates [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In- stead, we observe strong non-monotonic behaviour with clearly visible maxima around 67 and 154 µm, and min- ima around 56 and 135 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Notice that the distance between the two maxima and the two minima correlates with the measured ξ ≈ 90 µm period of Sz oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The discrepancy between the absolute locations of the minima and maxima with the predicted critical distances for screened (ξ/2, 3ξ/2) and unscreened (ξ, 2ξ) coupling, respectively, can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Firstly, when two condensates are coupled their energy is redshifted on average [24] leading to smaller εc and thus larger ξ in the coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Second, the finite width of the pump spots modulates the phase of polaritons and causes a shift in the Sz period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Third, the cavity here has higher levels of disorder than strain-compensated cavities [55] which Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Two polariton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (a) Experimentally measured and (b) simulated numerically real space Sz compo- nent of the Stokes vector of the polariton condensates emis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In panel (a) black circles show the position of pump spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (c) Total integrated emission intensity dependence on the separation distance between two condensates pump spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In panel (c) black dots shows the experimentally measured values with red region representing the error of the total inten- sity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Black curve shows the same dependence calculated numerically can affect the spatial coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' That’s why the relative distances between the extrema are more meaningful than their absolute locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This interpretation is verified in 2DGPE modeling which accurately reproduces the loca- tions of the extrema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Note that the slight discrepancy between modeling and experiment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2(c) between 70 and 120 µm can be attributed to the large parame- ter space of the 2DGPE making quantitative matching somewhat challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In order to demonstrate the NNN coupling using the all-optical spin screening effect we investigated the sys- tem containing a chain of three condensates similar to the system depicted schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' As in the pre- vious experiment with two condensates, all condensates were excited using tightly focused RCP laser pump spots of equal intensity above threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Figures 3(a) and 3(b) show the measured and simulated spatial distribution of the three condensate Sz component with NN distance of d ≈ ξ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' As in the previous case of two condensates, the system forms a joint macroscopic coherent state result- ing in an oscillating Sz pattern elongated along the hor- izontal axis with three RCP condensate circles of equal X (um) 100 0 100 100 0 100 (b) a 100 (μm) 0 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='4 40 80 120 160 Pump spot separation distance (um)4 degree of polarisation in the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Amazingly, the in- tensity of the central condensate was suppressed relative to the outer ones, evidencing reduced NN coupling due to the spin screening effect, see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3(c) measured (red diamonds) and simulated (black solid curve) inten- sity distribution along the triad axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' To unambiguously demonstrate the spin screening ef- fect in the triad, we measured (dots) and simulated (solid curve) the dependence of the central condensate intensity as a function of NN separation distance with results pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Both experiment and calculations show a clear dip around d = 52 µm ≈ ξ/2, corresponding to spin-screened NN coupling, followed by a small peak around d = 80 µm ≈ ξ where the NN coupling is re- stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The observed suppression of the central conden- sate intensity provides strong evidence of spin-screened NN coupling mediated by the spin coherence of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Moreover, the experimentally measured pump power dependence for each separation distance and the ex- tracted polariton condensation threshold values are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3(e) (red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The horizontal dashed line is the threshold value of the isolated condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In the absence of the TE-TM splitting monotonic increase of the threshold value converging to the isolated condensate threshold is expected with the increase of the separation distance between the condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In our system we ob- serve maximum threshold at the separation distance of 52 µm, which precisely corresponded to the minimum of the central condensate intensity in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3(c) and 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' It confirms that the NN condensate interaction is effectively screened at this separation distance due to the TE-TM splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Around a separation distance close to the full period of Sz oscillation (d ≈ ξ) a decrease in the thresh- old power was observed, as expected with NN coupling restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' A simple linear coupled oscillator model [solid curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3(e)] is able to explain the behaviour of the threshold power (see section S3 in the Supplemental Material [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In summary, we have experimentally demonstrated that next-nearest-neighbours coupling can be made stronger than nearest-neighbour coupling in ballistically expanding spinor exciton-polariton condensates which was recently proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This unconventional near-inversion of the spatial coupling hierarchy between condensates stems from the combination of TE-TM split- ting and the ballistic polariton flow from each conden- sate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Outflowing polaritons experience effective spin- orbit coupling which rotates their spin state as they prop- agate from one condensate to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Depending on distance, the overlap (coupling) between the condensates can become spin-screened depending on the polariton spin projection upon arrival at its neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We believe that the demonstrated alteration of the conventional con- densate coupling hierarchy could pave the way towards all-optical simulation of many-body ballistic systems be- Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Three polariton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (a) Experimentally measured and (b) simulated real space Sz component of the Stokes vector of the polariton condensates emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In panel (a) black circles show the position of pump spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (c) Mea- sured experimentally (red diamonds) and calculated numeri- cally (solid black curve) real space intensity distribution along the triad axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (d) Dependence of the central condensate PL intensity on the separation distance between the condensates pump spots measured experimentally (black dots) and calcu- lated numerically (solid black curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' red region represents the error of the total intensity value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The dashed curves are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (e) The system threshold power dependence on the separation distance between the condensates pump spots measured experimentally (red circles) and calculated numer- ically (solid black curve);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' red bars represent the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Grey dashed line in panel (e) shows the threshold power for single isolated condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' longing to non-planar graph topologies using networks of spinor polariton condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The authors acknowledge the support of the European Union’s Horizon 2020 program, through a FET Open re- search and innovation action under the grant agreements No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 899141 (PoLLoC) and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 964770 (TopoLight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' acknowledges the Icelandic Research Fund (Rannis), Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 217631-051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' X (μm) 100 0 100 100 0 100 (a) 100 100 Y (um) (μm) 0 0 Y 100 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='7 SZ (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 Int 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 tegrated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='8 (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='8 intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 Intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='4 @ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 20 40 60 80 ¥100 100 0 100 (un) X Pump spot separation (μm) (e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='9 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Lagoudakis, Polariton condensation in a strain- compensated planar microcavity with ingaas quantum wells, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 105, 191118 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Supplemental Material: Next nearest neighbour coupling with spinor polariton condensates Dmitriy Dovzhenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' ∗ Denis Aristov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1 Lucy Pickup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1 Helgi Sigurdsson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2 and Pavlos Lagoudakis1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3 1School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' University of Southampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Southampton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' SO17 1BJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' UK 2Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' University of Iceland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Dunhagi 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' IS-107,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Reykjavik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Iceland 3Hybrid Photonics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Skolkovo Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Territory of Innovation Center Skolkovo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Bolshoy Boulevard 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' building 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 121205 Moscow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Russia (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 2023) S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' EXPERIMENTAL DETAILS, TE-TM SPLITTING AND SINGLE ISOLATED CONDENSATE REAL SPACE Sz COMPONENT OF THE STOKES VECTOR In this supplemental section we present experimental details, experimentally measured TE-TM splitting and real space distribution of Sz component of Stokes vector of the emission from a single isolated polariton condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' All measurements were performed at 6 K using a continuous flow cold finger cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We used a right circularly (σ+) polarized non-resonant continuous wave laser excitation tuned to the first Bragg minimum of the microcavity reflection spectra at 754 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' To reduce sample heating we used an acousto-optic modulator driven by rectangular voltage pulse train at 10 kHz repetition rate with 5% duty cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' A spatial light modulator was used to structure the pump spatial profile into one, two, or three Gaussian spots focused on the sample using a microscope objective lens with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='4 numerical aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In order to obtain simultaneously real space, k-space, and spectrally resolved k-space images of the time-averaged PL two separate CCD cameras and a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='75 m monochromator with 1200g/mm diffraction grating equipped with the CCD camera were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' A quarter wave plate and a Wollaston prism were introduced in the optical path for real space imaging to simultaneously measure the right-circularly polarized and left-circularly polarized components of the PL with the same CCD camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' S1(a) we show the Sz spatial oscillations due to the spin-orbit coupling (SOC) rotating the pseudospin of the polaritons propagating away from the condensate excited using single Gaussian spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The oscillation period ξ was measured to be around 90 µm with the oscillations amplitude of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In order to experimentally estimate the value of TE-TM splitting we measured dispersion of the lower polariton branch in a linear regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=', below condensation threshold) along the in-plane k∥ momentum axis [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' S1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Splitting of the dispersion is clearly observed at the higher values of in-plane k vector with the higher energy branch corresponding to the emission from the vertically polarized polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The energy splitting between horizontally and vertically polarized polaritons possesses parabolic dependence on the in-plane momentum and ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='2 meV at k = 3 µm−1 in-plane wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' TWO DIMENSIONAL SPINOR POLARITON MODEL The experimental observations are reproduced by numerically solving a generalized Gross-Pitaevskii equation (S1) for macroscopic spinor polariton wavefunction Ψ(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' t) = (ψ+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' ψ−)T coupled to an active exciton reservoir with density nA(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' t) = (nA+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' nA−)T rate equation [1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' i∂ψ± ∂t = � − ℏ∇2 2m + i 2 � RnA± − γ � + α1|ψ±|2 + α2|ψ∓|2 + U±(r) + V (r) � ψ± + ∆LT � ∂ ∂x ∓ i ∂ ∂y �2 ψ∓,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S1) U± = G1 � nA± + nI± � + G2 � nA∓ + nI∓ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S2) ∂nA± ∂t = − � ΓA + Γs + R|ψ±|2� nA± + WnI± + ΓsnA∓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S3) Here, ± represents the spin of polaritons and excitons along the cavity growth axis, m is the polariton effective mass in parabolic dispersion approximation, γ is the polariton decay rate, G1 = 2g|χ|2 and α1 = g|χ|4 are the same spin polariton-reservoir and polariton-polariton interaction strengths, respectively, g is the exciton-exciton Coulomb ∗ DovzhenkoDS@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='04210v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='mes-hall] 10 Jan 2023 2 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (a) Experimentally measured real space Sz component of the Stokes vector of the single isolated polariton condensate emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (b) Energy and in-plane wavevector resolved normalized PL intensity from the lower polariton branch below the threshold interaction strength, |χ|2 is the excitonic Hopfield fraction of the polariton, and ∆LT represents the strength of the TE-TM splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Opposite spin interactions, usually much weaker, were chosen to be G2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='2G1 and α2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='2α1 for completeness but we note that our results to not qualitatively depend on these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' R is the scattering rate of reservoir excitons into the condensate, ΓA is the active reservoir decay rate, and Γs represents exciton spin relaxation rate [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' A so-called inactive reservoir of excitons nI,± also contributes to the blueshift of polaritons as depicted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This reservoir corresponds to high-momentum excitons which do not scatter into the condensate but instead drive the active low-momentum excitons (S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In continuous wave experiments the inactive reservoir density can be written WnI,+ = P0(r) W + 2Γs (W cos2 (θ) + Γs), WnI,− = P0(r) W + 2Γs (W sin2 (θ) + Γs), (S4) where P0 is the total power density of the incident coherent light with degree of circular polarization expressed as S3 = P0[cos2 (θ)−sin2 (θ)] = P0 cos (2θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Since our experiment is performed with fully right hand circularly polarized light, we set θ = 0 from here on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The phenomenological parameter W quantifies conversion rate between same-spin inactive and active exciton reservoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The pump profile is written as a superposition of Gaussians P0(r) = p0 � n e−|r−rn|2/2w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' To represent tight focusing of excitation beams we used Gaussians with 2 µm full-width-at-half-maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Lastly, given the disorder present at the large spatial scales of the experiment we include a random potential landscape in our simulation given by V (r) generated as a random Gaussian-correlated potential [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The simulation parameters are based on previous GaAs microcavity experiments [4, 5]: m = 5×10−5 of free electron mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' γ−1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='5 ps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' |χ|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' ℏg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='5 µeV µm2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='2g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' W = ΓA = γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Γs = γ/4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' ∆LT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='036 ps−1 µm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The disorder potential was generated with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='5 µm correlation length and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='06 meV root mean squared amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We note that in order to compensate for additional background noise in experiment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=', additional light coming from spontaneous emission of bottleneck excitons) we applied a global shift to the integrated densities of the condensate |ψ±|2 by approximately 10 percent in order to match the experimental values in Figure 3(d) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This difference between modeling and experiment is more evident in Figure 3(c) where the experimentally measured photoluminescence (PL) intensity is more spread out than simulated condensate densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This can also come from the finite diffusion of excitons which we have neglected here for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Nevertheless, the calculated relative X (μm) k, (μm-l) 200 200 0 3-2 1 0 1 2 [(b) (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='543 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='542 (un) 0 (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='540 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='539 200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='538 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 Sz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='6 PL intensity 03 amplitude of the PL at the pump positions follows the experimental results quite precisely, which, therefore, justifies the use of current model and provides a clear quantitative evidence of the spin-screening happening in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' THEORY OF THE THRESHOLD BEHAVIOUR IN A SPIN SCREENED CONDENSATE TRIAD The behaviour of the pump threshold from experiment in the triad configuration can be reproduced by scrutinizing the eigenenergies of an appropriate linear operator which couples the three condensates together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' In other words, we neglect polariton nonlinearities so close to the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The threshold is reached when a single eigenvalue belonging to the three coupled condensates crosses from the lower- to the upper-half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' We will start by defining the state vector of the system, |Ψ⟩ = (ψ1,+, ψ1,−, ψ2,+, ψ2,−, ψ3,+, ψ3,−)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S5) Here, the index n ∈ {1, 2, 3} denotes the left, middle, and right condensate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The spectrum of the coupled system in the linear regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' close to threshold |ψn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='±|2 ≃ 0) can be described with the following non-Hermitian operator separated into three parts for clarity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='ˆH = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='K+ δK 0 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='δK K− 0 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='(S6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='The first matrix describes the complex self-energy of each oscillator (condensate) composed of the local pump blueshift ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='(G) and gain (R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' and cavity losses (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This contribution from the pump can be parametrized in terms of the reservoir spin populations, ω± = � G1 + iR 2 � (NA,± + NI,±) − iγ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S7) where � nA(I),± dr = NA(I),± [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=', spatially integrating (S3) and (S4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Here, we will neglect opposite spin interaction G2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Each condensate is coupled ballistically with its nearest neighbours with coupling strength J± and next-nearest neighbours with strength K± determined by the overlap between different condensates over their respective pump spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Approximating the tightly focused pump spots as delta functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' we can write the coupling between the ballistic condensates as [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' J± = cos2 (ξd + Φ) � G1 + iR 2 � (NA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='± + NI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='±)H(1) 0 (kd + φ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S8) K± = sin2 (2ξd + Φ) � G1 + iR 2 � (NA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='± + NI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content='±)H(1) 0 (2kd + φ) (S9) The square cosine (sine) modulations in the coupling stem from a pseudospin screening effect coming from the strong influence of TE-TM splitting on the ballistic condensates [6] as explained in the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Here, ξ denotes the period of the pseudospin precession for a single condensate in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' H(1) 0 (kd) is the zeroth order Hankel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The coupling depends on the product kd where d is the separation distance between two pump spots and k is the complex wavevector of the polaritons with mass m propagating outside the pump spot, k ≈ kc + i Γm 2ℏkc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' (S10) Here, kc is the average real wavevector of the outflowing polaritons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The finite size of the Gaussian pump spots intro- duces some lag into the pseudospin precession because outflowing polaritons need to gradually build up momentum as they leave the pump spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' This is captured in the fitting parameter Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' For the same reason, an overall phase-lag fitting parameter φ is also needed in the coupling term between the condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 4 The presence of TE-TM splitting also introduces coupling between opposite spin components denoted δJ and δK written in a similar fashion, δJ = δ cos2 (ξd + Φ) � G1 + iR 2 � (NA,+ + NI,+ + NA,− + NI,−)H(1) 0 (kd + φ), (S11) δK = δ sin2 (2ξd + Φ) � G1 + iR 2 � (NA,+ + NI,+ + NA,− + NI,−)H(1) 0 (2kd + φ) (S12) Here, δ < 1 is a fitting parameter describing the amount of opposite spin coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Diagonalizing ˆH for increasing pump power P0 we identify the threshold as the point in which a single eigenenergy crosses from the lower-half to the upper-half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' The results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' 3(e) in the main text (solid curve) alongside the experimental data, normalized in units of threshold power for the single isolated condensates Pthr,iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Deng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Haug, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Yamamoto, Exciton-polariton bose-einstein condensation, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQf7giQ/content/2301.04210v1.pdf'} +page_content=' Phys.' metadata={'source': 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Abratenko𝑎 A. Aduszkiewicz𝑏 F. Akbar𝑐 M. Artero Pons𝑑 J. Asaadi𝑒 M. Aslin 𝑓 ,1 +M. Babicz𝑔,2 W.F. Badgett 𝑓 L.F. Bagby 𝑓 B. Baibussinov𝑑 B. Beheraℎ V. Bellini𝑖 +O. Beltramello𝑔 R. Benocci 𝑗 J. Bergerℎ S. Berkman 𝑓 S. Bertolucci𝑘 R. Bertoni𝑗 +M. Betancourt 𝑓 M. Bettini𝑑 S. Biagi𝑙 K. Biery 𝑓 O. Bitter 𝑓 ,3 M. Bonesini𝑗 T. Booneℎ +B. Bottino𝑚 A. Braggiotti𝑑,4 D. Brailsford5 J. Bremer𝑔 S.J. Brice 𝑓 V. Brio𝑖 C. Brizzolari𝑗 +J. Brown 𝑓 H.S. Budd𝑐 F. Calaon𝑑 A. Campani𝑚 D. Carberℎ M. Carneiro𝑛 I. Caro Terrazasℎ +H. Carranza𝑒 D. Casazza𝑚 L. Castellani𝑑 A. Castro𝑜 S. Centro𝑑 G. Cerati 𝑓 M. Chalifour𝑔 +P. Chambouvet𝑔 A. Chatterjee𝑝 D. Cherdack𝑏 S. Cherubini𝑙 N. Chithirasreemadam𝑞 +M. Cicerchia𝑑 V. Cicero𝑘 T. Coan𝑟 A. Cocco𝑠 M.R. Convery𝑡 S. Copello𝑢 E. Cristaldo6 +A.A. Dange𝑒 I. de Icaza Astiz7 A. De Roeck𝑔 S. Di Domizio𝑚 L. Di Noto𝑚 C. Di Stefano𝑙 +D. Di Ferdinando𝑘 M. Diwan𝑛 S. Dolan𝑔 L. Domine𝑡 S. Donati𝑞 R. Doubnik 𝑓 F. Drielsma𝑡 +J. Dyerℎ S. Dytman𝑣 C. Fabre𝑔 F. Fabris𝑑 A. Falcone𝑗 C. Farnese𝑑 A. Fava 𝑓 H. Ferguson 𝑓 +A. Ferrari𝑤 F. Ferraro𝑚 N. Gallice𝑤 F.G. Garcia𝑡 M. Geynisman 𝑓 M. Giarin𝑑 D. Gibin𝑑 +S.G. Gigli𝑢 A. Gioiosa𝑞 W. Gu𝑛 M. Guerzoni𝑘 A. Guglielmi𝑑 G. Gurung𝑒 S. Hahn 𝑓 K. Hardin 𝑓 +H. Hausner 𝑓 A. Heggestuenℎ C. Hilgenbergℎ,8 M. Hoganℎ B. Howard 𝑓 R. Howell𝑐 +J. Hrivnak𝑔 M. Iliescu𝑘,9 I. Ingratta𝑘 C. James 𝑓 W. Jang𝑒 M. Jung𝑥,10 Y.-J. Jwa𝑡 L. Kashurℎ +W. Ketchum 𝑓 J.S. Kim𝑐 D.-H. Koh𝑡 U. Kose𝑔,11 J. Larkin𝑛 G. Laurenti𝑘 G. Lukhanin 𝑓 +S. Marchini𝑑 C.M. Marshall𝑐 S. Martynenko𝑛 N. Mauri𝑘 A. Mazzacane 𝑓 K.S. McFarland𝑐 +D.P. Méndez𝑛 A. Menegolli𝑢,12 G. Meng𝑑 O.G. Miranda𝑜 D. Mladenov𝑔 A. Moganℎ N. Moggi𝑘 +E. Montagna𝑘 C. Montanari 𝑓 ,13 A. Montanari𝑘 M. Mooneyℎ G. Moreno-Granados𝑜 J. Muellerℎ +D. Naples𝑣 M. Nebot-Guinot14 M. Nessi𝑔 T. Nichols 𝑓 M. Nicoletto𝑑 B. Norris 𝑓 S. Palestini𝑔 +M. Pallavicini𝑚 V. Paolone𝑣 R. Papaleo𝑙 L. Pasqualini𝑘 L. Patrizii𝑘 R. Peghin𝑑 G. Petrillo𝑡 +C. Petta𝑖 V. Pia𝑘 F. Pietropaolo𝑔,15 J. Poirot𝑔 F. Poppi𝑘 M. Pozzato𝑘 M.C. Prata𝑢 A. Prosser 𝑓 +G. Putnam𝑤 X. Qian𝑛 G. Rampazzo𝑑 A. Rappoldi𝑢 G.L. Raselli𝑢 R. Rechenmacher 𝑓 +F. Resnati𝑔 A.M. Ricci𝑞 G. Riccobene𝑙 L. Rice𝑣 E. Richards𝑣 A. Rigamonti𝑔 M. Rosenberg𝑎 +1Now at University of Wisconsin, Madison, USA +2Also at INP-Polish Acad. Sci, Krakow,Poland. Now at University of Zurich, Switzerland +3Now at Northwestern University, USA +4Also at Istituto di Neuroscienze, CNR, Padova, Italy +5SBND Collaboration, Lancaster University, UK +6SBND Collaboration, Universidad Nacional de Asuncion, San Lorenzo, Paraguay +7SBND Collaboration, University of Sussex, UK +8Now at University of Minnesota, USA +9Now at INFN-LNF +10SBND Collaboration +11Now at ETH Zurich, Switzerland +12Corresponding author. +13on leave of absence from INFN Pavia, Italy +14SBND Collaboration, University of Edinburgh, UK +15On leave of absence from INFN Padova, Italy +arXiv:2301.08634v1 [hep-ex] 20 Jan 2023 + +M. Rossella𝑢 C. Rubbia𝑦 P. Sala𝑤 P. Sapienza𝑙 G. Savage 𝑓 A. Scaramelli𝑢 A. Scarpelli𝑛 +D. Schmitz𝑥 A. Schukraft 𝑓 F. Sergiampietri𝑔,16 G. Sirri𝑘 J.S. Smedley𝑐 A.K. Soha 𝑓 +M. Spanu 𝑗 L. Stanco𝑑 J. Stewart𝑛 N.B. Suarez𝑣 C. Sutera𝑖 H.A. Tanaka𝑡 M. Tenti𝑘 K. Terao𝑡 +F. Terranova 𝑗 V. Togo𝑘 D. Torretta 𝑓 M. Torti 𝑗 F. Tortorici𝑖 N. Tosi𝑘 Y.-T. Tsai𝑡 S. Tufanli𝑔 +M. Turcato𝑑 T. Usher𝑡 F. Varanini𝑑 S. Ventura𝑑 F. Vercellati𝑢 M. Vicenzi𝑚 C. Vignoli𝑧 +B. Viren𝑛 D. Warnerℎ Z. Williams𝑒 R.J. Wilsonℎ P. Wilson 𝑓 J. Wolfs𝑐 T. Wongjirad𝑎 A. Wood𝑏 +E. Worcester𝑛 M. Worcester𝑛 M. Wospakrik 𝑓 H. Yu𝑛 J. Yu𝑒 A. Zani𝑤 P.G. Zatti𝑑 J. Zennamo 𝑓 +J.C. Zettlemoyer 𝑓 C. Zhang𝑛 S. Zucchelli𝑘 and M. Zuckerbrot 𝑓 +𝑎Tufts University, Medford, MA 02155, USA +𝑏University of Houston, Houston, TX 77204, USA +𝑐University of Rochester, Rochester, NY 14627, USA +𝑑INFN Sezione di Padova and University of Padova, Padova, Italy +𝑒University of Texas at Arlington, Arlington, TX 76019, USA +𝑓 Fermi National Accelerator Laboratory, Batavia, IL 60510, USA +𝑔CERN, European Organization for Nuclear Research 1211 Genève 23, Switzerland, CERN +ℎColorado State University, Fort Collins, CO 80523, USA +𝑖INFN Sezione di Catania and University of Catania, Catania, Italy +𝑗INFN Sezione di Milano Bicocca and University of Milano Bicocca, Milano, Italy +𝑘INFN Sezione di Bologna and University of Bologna, Bologna, Italy +𝑙INFN LNS, Catania, Italy +𝑚INFN Sezione di Genova and University of Genova, Genova, Italy +𝑛Brookhaven National Laboratory, Upton, NY 11973, USA +𝑜Centro de Investigacion y de Estudios Avanzados del IPN (Cinvestav), Mexico City +𝑝Physical Research Laboratory, Ahmedabad, India +𝑞INFN Sezione di Pisa, Pisa, Italy +𝑟Southern Methodist University, Dallas, TX 75275, USA +𝑠INFN Sezione di Napoli, Napoli, Italy +𝑡SLAC National Acceleratory Laboratory, Menlo Park, CA 94025, USA +𝑢INFN Sezione di Pavia and University of Pavia, Pavia, Italy +𝑣University of Pittsburgh, Pittsburgh, PA 15260, USA +𝑤INFN Sezione di Milano, Milano, Italy +𝑥University of Chicago, Chicago, IL 60637, USA +𝑦INFN GSSI, L’Aquila, Italy +𝑧INFN LNGS, Assergi, Italy +E-mail: alessandro.menegolli@unipv.it +16Now at IPSI-INAF Torino, Italy + +Abstract: The ICARUS collaboration employed the 760-ton T600 detector in a successful three- +year physics run at the underground LNGS laboratory studying neutrino oscillations with the +CERN Neutrino to Gran Sasso beam (CNGS) and searching for atmospheric neutrino interactions. +ICARUS performed a sensitive search for LSND-like anomalous 𝜈𝑒 appearance in the CNGS +beam, which contributed to the constraints on the allowed parameters to a narrow region around +1 eV2, where all the experimental results can be coherently accommodated at 90% C.L.. After a +significant overhaul at CERN, the T600 detector has been installed at Fermilab. In 2020, cryogenic +commissioning began with detector cool down, liquid argon filling and recirculation. ICARUS has +started operations and successfully completed its commissioning phase, collecting the first neutrino +events from the Booster Neutrino Beam (BNB) and the Neutrinos at the Main Injector (NuMI) +beam off-axis, which were used to test the ICARUS event selection, reconstruction and analysis +algorithms. The first goal of the ICARUS data taking will then be a study to either confirm or refute +the claim by Neutrino-4 short baseline reactor experiment both in the 𝜈𝜇 channel with the BNB and +in the 𝜈𝑒 with NuMI. ICARUS will also address other fundamental studies such as neutrino cross +sections with the NuMI beam and a number of Beyond Standard Model searches. After the first +year of operations, ICARUS will commence its search for evidence of a sterile neutrino jointly with +the Short Baseline Near Detector, within the Short-Baseline Neutrino program. +Keywords: Large detector systems for particle and astro-particle physics, Liquid Argon, Time +Projection Chambers (TPC) + +Contents +1 +Introduction +2 +2 +The ICARUS-T600 detector +3 +3 +The overhaul of ICARUS-T600 +4 +3.1 +The TPC electronics +4 +3.2 +The scintillation light detection system +5 +4 +The Cosmic Ray Tagger +6 +5 +First operations at FNAL +7 +5.1 +Cryogenic plant installation +7 +5.2 +TPC electronics installation +9 +5.3 +PMT system installation +9 +5.4 +Cosmic Ray Tagger installation +10 +6 +ICARUS-T600 commissioning +10 +6.1 +TPC commissioning +12 +6.2 +PMT commissioning +16 +6.3 +CRT commissioning +17 +6.4 +Triggering on the BNB and NuMI neutrinos +18 +6.5 +DAQ implementation +20 +6.6 +First operations with the BNB and NuMI +21 +7 +Observation and reconstruction of neutrino events +21 +7.1 +Wire signal reconstruction +21 +7.2 +PMT signal reconstruction +22 +7.3 +CRT reconstruction +23 +7.4 +Event display study +25 +7.5 +Event reconstruction +25 +– 1 – + +1 +Introduction +The Liquid Argon Time Projection Chamber +(LAr-TPC) is a continuously sensitive and self +triggering detector that can provide excellent +3D imaging and calorimetric reconstruction of +any ionizing event. First proposed by C. Rub- +bia in 1977 [1], this detection technique allows +a detailed study of neutrino interactions, span- +ning a wide energy spectrum (from a few keV +to several hundreds of GeV), as demonstrated +by the first large scale experiment performed by +the ICARUS Collaboration at the LNGS under- +ground laboratory. +Several experiments, in particular the Liq- +uid Scintillator Neutrino Detector (LSND) [2] +and MiniBooNE [3], have reported anomalous +signals that may imply the presence of additional +(mass-squared difference Δ𝑚2 +𝑛𝑒𝑤 ∼ 1 eV2) flavor +oscillations at small distances pointing toward +the possible existence of nonstandard heavier +sterile neutrino(s). A sensitive search for a possi- +ble 𝜈𝑒 excess related to the LSND anomaly in the +CNGS 𝜈𝜇 beam has already been performed us- +ing the neutrino events collected in the ICARUS- +T600 detector during the Gran Sasso run. A total +of 2,650 CNGS neutrino interactions, identified +in 7.9·1019 POT (Protons On Target) exposure, +have been studied to identify the 𝜈𝑒 interactions. +Globally, 7 electron-like events have been ob- +served to be compared to 8.5±1.1 expected from +the intrinsic beam contamination and standard +3-flavor oscillations. This study constrained the +LSND signal to a narrow parameter region at +sin22𝜃 ∼ 0.005, Δ𝑚2 < 1 eV2, which requires +further investigation [4]. +The primary goal of the Short-Baseline +Neutrino (SBN) program at Fermilab is to fur- +ther investigate the possibility of sterile neutri- +nos in the O(1 eV) mass range and provide the +required clarification of the LSND anomaly. It +is based on three LAr-TPC detectors (ICARUS- +T600, with 476 tons active mass, MicroBooNE +with 89 tons active mass and SBND with 112 +tons active mass) exposed at shallow depth to +the ∼ 0.8 GeV Booster Neutrino Beam (BNB) at +different distances from the target (600 m, 470 +m and 110 m respectively) [5, 6]. +The detection technique used will provide +an unambiguous identification of neutrino in- +teractions, measurement of their energy and a +strong mitigation of possible sources of back- +ground. Performing this study with almost iden- +tical detectors at various distances from the neu- +trino source allows identification of any variation +of the spectra, which is a clear signature of neu- +trino oscillations. +In particular, SBN will allow for a very sen- +sitive search for 𝜈𝜇 → 𝜈𝑒 appearance signals, +covering the LSND 99% C.L. allowed region at +∼ 5𝜎 C.L. [5, 6]. The high correlations between +the event samples of the three LAr-TPC’s and the +huge event statistics at the near detector will also +allow for a simultaneous sensitive search in the +𝜈𝜇 disappearance channel. +During data taking at Fermilab, the 760- +ton T600 detector is also exposed to the off-axis +neutrinos from the Neutrinos at the Main Injec- +tor (NuMI) beam, where most of events are in +the 0 – 3 GeV energy range, with an enriched +component of electron neutrinos (few %). The +analysis of these events will provide useful infor- +mation related to detection efficiencies and neu- +trino cross-sections at energies relevant to the +future long baseline experiment with the multi- +kiloton DUNE LAr-TPC detector. +In addition to the LSND anomaly, ICARUS +will test the oscillation signal reported by the +Neutrino-4 experiment [7] both in the 𝜈𝜇 and +𝜈𝑒 channels with the BNB and NuMI beams, +respectively. +This paper is organized as follows: in Sec- +tion 2 the ICARUS-T600 detector is described +with a particular emphasis on its achievements +during three years data taking at the INFN LNGS +underground laboratories in Italy; in Section 3, +– 2 – + +the ICARUS-T600 overhauling activities, most +of which were carried out at CERN in the Neu- +trino Platform framework [8], are shown; the +new Cosmic Ray Tagger (CRT) detector, used +to mitigate the cosmic ray background due to +operating ICARUS at shallow depth, is detailed +in Section 4. In Section 5, the first operations +of ICARUS at Fermilab, in particular the instal- +lation of the cryogenic plant, TPC electronics, +scintillation light detection system and CRT are +described. A successful commissioning phase +followed soon after as described in Section 6. +Finally, the procedure for the selection, recon- +struction, and analysis of the first collected BNB +and NuMI off-axis neutrino events is introduced +in Section 7. +2 +The ICARUS-T600 detector +The ICARUS-T600, with a total active mass of +476 ton, is the first large-scale operating LAr- +TPC detector [9]: it consists of two large and +identical adjacent modules with internal dimen- +sions 3.6 × 3.9 × 19.6 m3, filled with a total of +760 tons of ultra-pure liquid argon. Each mod- +ule houses two LAr-TPCs separated by a com- +mon cathode with a maximum drift distance of +1.5 m, equivalent to ∼ 1 ms drift time for the +nominal 500 V/cm electric drift field. The cath- +ode is built up by an array of nine panels made of +punched stainless-steel, allowing for a 58% op- +tical transparency between the two drift regions. +The anode is made of three parallel wire planes +positioned 3 mm apart, where the stainless-steel +100 µm wires are oriented on each plane at a +different angle with respect to the horizontal di- +rection: 0◦ (Induction 1), +60◦ (Induction 2) +and -60◦ (Collection). +In total, 53,248 wires +with a 3 mm pitch and length up to 9 m are in- +stalled in the detector. By appropriate voltage +biasing, the first two planes (Induction 1 and In- +duction 2) provide a nondestructive charge mea- +surement, whereas the ionization charge is fully +collected by the last Collection plane. Photo- +Multiplier Tubes (PMTs) are located behind the +wire planes to collect the scintillation light pro- +duced by charged particles in LAr and used for +the trigger of the detector. +In 2013, ICARUS concluded a very suc- +cessful 3-year long run in the Gran Sasso under- +ground laboratory [10], demonstrating the feasi- +bility of the LAr-TPC technology at the kiloton +scale in a deep underground environment and +paving the way to the construction of the next +generation of experiments dedicated to study +neutrino oscillation physics such as DUNE. Dur- +ing the data taking, the liquid argon was kept +at an exceptionally high purity level (< 50 ppt +of O2 equivalent contaminants) reaching in 2013 +a 16 ms lifetime corresponding to 20 ppt O2 +equivalent LAr contamination [11], demonstrat- +ing the possibility to build larger LAr-TPC de- +tectors with drift distances up to 5 m. +The detector has been exposed to the CNGS +neutrino beam and to cosmic rays, recording +events that demonstrate high-level performance +and the physical potential of this detection tech- +nique: the detector showed a remarkable 𝑒/𝛾 +separation and particle identification exploiting +the measurement of 𝑑𝐸/𝑑𝑥 versus range [12]. +The momentum of escaping muons has been +measured by studying the multiple Coulomb +scattering with ∼ 15% average resolution in the +0.4 – 4 GeV/c energy range, which is relevant for +the next generation neutrino experiments [13]. +Events related to cosmic rays have been +studied to identify atmospheric neutrino interac- +tions: 6 𝜈𝜇CC and 8 𝜈𝑒CC events in a 0.43 kton·y +exposure have been identified and reconstructed, +demonstrating that the automatic search for the +𝜈𝑒CC in the sub-GeV range of interest for the +future short and long baseline neutrino experi- +ments is feasible [14]. +– 3 – + +3 +The overhaul of ICARUS-T600 +The ICARUS-T600 detector at Fermilab takes +data at shallow depth, shielded by a ∼ 3-meter +concrete overburden: neutrino interactions must +be recognized among the ∼ 11 cosmic muons +that are expected to cross the detector randomly +in the 1 ms drift time during each triggered event. +High-energy photons produced by cosmic rays +can become a serious background source for the +𝜈𝑒 search since the electrons produced via Comp- +ton scattering and pair production can mimic +𝜈𝑒CC events. +In order to prepare the detector for SBN data +taking, the T600 underwent an intensive overhaul +at CERN in the Neutrino Platform framework +(WA104/NP01 project) before being shipped to +the USA in 2017, introducing several technology +developments while maintaining the achieved +performance at Gran Sasso. +The refurbishing +mainly consisted of: the realization of new cold +vessels (Fig. 1) with purely passive insulation; an +update of the cryogenics and of the LAr purifi- +cation equipment; flattening of the TPC cathode +(the punched hole stainless-steel panels under- +went a thermal treatment improving the planarity +to a few mm); the implementation of new, higher +performance TPC read-out electronics; the up- +grade of the LAr light detection system. +3.1 +The TPC electronics +The electronics used at LNGS was based on +flange modularity, each flange serving 576 TPC +wire-channels. +The analogue front-end was a +Radeka type amplifier, using a custom BiCMOS +chip to integrate the cascode stage with two dif- +ferent filtering, one for Collection and Induc- +tion 1, another for Induction 2 with the aim +to produce in all the cases a unipolar signal. +This solution, however, showed strong limita- +tions in the Induction 2 signals in the case of +dense showers. Analog signals were converted +to digital via multiplexers by 10-bit ADCs with +Figure 1. One of the two new ICARUS cryostats +during its assembly at a CERN workshop. +sampling rate of 400 ns. The analogue circuits +were housed in a custom crate, connected to the +flange by flat cables, with 18 boards (32 chan- +nels per board). Analogue boards had a digital +link to corresponding digital modules hosted in +VME crates that contained memory buffers and +performed lossless data compression and data +transmission through a VME bus. Both crates +were housed in a rack next to the flange. +One of the largest tasks of the overhauling +program was the design of new electronics for +the 53,248 wire-channels that would be compat- +ible with higher data rates foreseen at shallow +depth operation at FNAL. The new electronics +adopts the same modularity and architecture but +takes advantage of newer technology that allows +for integrating both the analogue and the digital +electronics on the same board on a custom crate +mounted onto the flange [15]. +New packaging for the BiCMOS custom +cascode allowed the design of a small piggyback +module with 8 amplifiers and to house 8 of these +modules on a single board serving 64 channels, +see Fig. 2 (top-left). The digital part is also com- +pletely contained in the same board. Moreover, +all the amplifiers now have the same filtering, +preserving the bipolar structure of Induction 2 +signals without distortion. Each amplifier is fol- +– 4 – + +lowed by a serial 12-bit ADC avoiding the cum- +bersome signal multiplexing. +The digital part +is based essentially on a large powerful FPGA +allowing the possibility to use different signal +treatments if required from running experience. +The VME standard was abandoned in favor of +a serial optical link, allowing for gigabit band- +width data transmission compatible with shallow +depth data rates. +Figure 2. A2795 custom board housing 64 amplifiers +(far end), AD converter, digital control, and optical +link (top-left). An assembled feed-through with nine +DBBs and the biasing cables (top-right). +A mini- +crate populated by the nine A2795 boards installed +on a feed-through flange (bottom). +TPC wire signals are fed into the front- +end amplifiers by means of Decoupling Biasing +Boards (DBBs). The DBB has two functions: +biasing of each wire and conveying, with block- +ing capacitors, the signals to the amplifiers. The +DBBs work in argon gas and can withstand up +to 400 V input biasing. The flange CF250 is re- +alized with a G10 multi-layer solid PCB, about +6 mm thick with three internal layers of copper +to guarantee the required stiffness. SMD exter- +nal connectors provide receptacles for the A2795 +boards, while another set of SMD connectors in +correspondence (inner side) provide receptacles +for DBBs, see Fig. 2 (top-right). Finally, nine +electronic A2795 boards are hosted by a mini- +crate which is installed on a feed-through CF250 +flange, see Fig. 2 (bottom). +3.2 +The scintillation light detection system +A new light detection system that is sensitive to +the photons produced by the LAr scintillation is +a fundamental feature for the T600 operation at +shallow depth (contributing to the rejection of the +cosmic background). The light detection system +complements the 3D track reconstruction, unam- +biguously providing the absolute timing for each +track and identifying the interactions occurring +in the BNB and NuMI spill gates. +The ICARUS-T600 light detection system +consists of 360 8" Hamamatsu R5912-MOD +PMTs deployed behind the 4 wire chambers, +90 PMTs per TPC [16, 17], see Fig. 3. Since +the PMT glass is not transparent to the 128 nm +wavelength scintillation light produced in liquid +argon, each unit is provided with a ≈ 200 µg/cm2 +coating of Tetra-Phenyl Butadiene (TPB), to con- +vert the VUV photons to visible light [18]. +All PMTs are mounted onto the wire cham- +ber mechanical frames using a supporting sys- +tem, that allows the PMT to be positioned about +5 mm behind the Collection planes wires. +A +stainless steel grid cage is mounted around each +PMT to mitigate the induction of fake signals +on the nearby wire planes by the relatively large +PMT signals. +The light detection setup, realized by INFN, +is complemented by a laser calibration system +allowing for gain equalization, timing and moni- +– 5 – + +Figure 3. The new ICARUS PMTs mounted behind +the wires of one TPC. +toring of all the PMTs. Laser pulses (𝜆 = 405 nm, +FWHM = 60 ps), generated by a laser diode head +(Hamamatsu PLP10), are sent to each PMT win- +dow by means of a light distribution system based +on optical fibers, light splitters and an optical +switch [19]. +4 +The Cosmic Ray Tagger +ICARUS-T600 based at FNAL faces more chal- +lenging experimental conditions than at LNGS: +due to its shallow depth operation, identifica- +tion of neutrino interactions among 11 kHz of +cosmic rays is required. A ∼ 3-meter concrete +overburden was designed to almost completely +remove the contribution from charged hadrons +and high energy photons [20]. However, ∼ 11 +muon tracks occur per triggered event in the 1 ms +TPC drift readout; photons associated with the +muons represent a serious background for identi- +fying 𝜈𝑒 candidates since electrons produced via +Compton scattering/pair production can mimic a +genuine 𝜈𝑒CC event. +Rejecting the cosmic background, i.e. re- +constructing the triggering event, requires to +know precisely the timing of each track in the +TPC image. Operating at FNAL, ICARUS ex- +ploits an improved light detection system with +high granularity and 𝑂(1 ns) time resolution, and +an external ∼ 4𝜋 high coverage Cosmic Ray Tag- +ger (CRT). The primary function of the CRT is to +tag muons passing through or near the cryostats. +Timestamps associated to a particle tagged +by the CRT are compared with timestamps from +PMT signals, both with a few nanosecond res- +olution, allow the determination of whether an +interaction in the TPC originated from an outside +cosmic ray or from an internal interaction. The +ICARUS CRT consists of a top, side and bottom +subsystem. +The ICARUS Top CRT system is divided in +123 detector modules covering a surface of about +426 m2: 84 horizontal and 39 vertical modules +along the perimeter of the cryostat top surface. +Its design is such that more than 80% of the cos- +mic muon flux is intercepted by the Top CRT. +Each module is a 1.86 × 1.86 m2 aluminum +box containing two orthogonal layers of eight +scintillator bars for position reconstruction. The +bars, coated with white paint, are 23 cm wide, +184 cm long and have different thickness de- +pending on the layer: 1 cm and 1.5 cm for the +top layer and the bottom layer, respectively. In +each scintillator, the light is collected by two +wave-length shifting (WLS) fibers Kuraray Y- +11(200) then read out from one end by a Silicon +Photo-Multiplier (SiPM), Hamamatsu S13360- +1350C model. The 32 SiPM signals of one mod- +ule are routed via 50 Ω micro-coaxial cables to +a patch panel connected to the CAEN DT5702 +Front End Board (FEB) which provides a bias +voltage adjustable for each channel. The FEB +triggers on the coincidence between two SiPM +signals of the same bar and provides a coinci- +dence logic between the two scintillator layers +in the module. In Fig. 4, a picture of a vertical +Top CRT module installed in the detector hall is +shown. The Top CRT was a brand new detector +– 6 – + +designed and built by INFN and CERN before +shipping to Fermilab in summer 2021. +Front End +Board +Aluminum Box +containing Top +CRT module +Figure 4. Picture of a vertical TOP CRT module +installed in the detector hall. +The ICARUS Side CRT makes use of scin- +tillator modules formerly used by the MINOS ex- +periment. Each module is composed of twenty +adjacent strips of 800 × 4 × 1 cm3 Polystyrene +(1.0% PPO, 0.03% POPOP) scintillator. +The +full Side CRT system consists of 2,710 readout +channels across 93 FEBs, with 136 full and 81 +cut modules in total. +The scintillator is con- +tained in a metal sheath and each strip has an +embedded WLS fiber running down the mid- +dle. These fibers are collected into “snouts” at +the ends of the modules, onto which the opti- +cal readout, consisting of an array of ten Hama- +matsu S14160-3050HS SiPMs, is mounted onto +a snout. Each SiPM reads out two fibers and cor- +responds to a single electronic readout channel +on CAEN A1702 Front-End Boards (FEBs). A +full MINOS module has two snouts, one on each +end. The ICARUS Side CRT System is double +layered, with an inner and outer layer of MINOS +modules to apply coincidence logic between the +two layers. To account for geometric constraints, +some MINOS modules were cut and sealed on +the cut end with mylar and tape to only have a +single snout for readout. The South Side CRT +wall consists of an inner and outer layer of cut +modules oriented orthogonally in an X-Y con- +figuration, with the added benefit of improved +position reconstruction on the southern side of +the TPCs, upstream along the BNB beam. The +East and West walls utilize full length MINOS +modules mounted horizontally, while the North +Wall use cut modules mounted horizontally. +The Bottom CRT consists of 14 modules di- +vided into two daisy chains of 7 modules each, +positioned underneath the warm vessel in a north +and south section. +These modules are refur- +bished veto modules from the Double Chooz re- +actor neutrino experiment. Each module consists +of 64 Polystyrene scintillator strips, running in +parallel and divided into two layers of 32 strips +offset 2.5 cm from each other. Scintillation light +is collected in a WLS optical fiber and read out at +one end of each strip by an Hamamatsu H7546B +M64 multi-anode PMT, while the other end is +mirrored to maximize light collection. +5 +First operations at FNAL +Following the overhauling activities at CERN, +ICARUS-T600 was shipped to Fermilab in July +2017 and the two cryostats hosting the TPCs were +finally deployed in their shallow depth position +in August 2018. Work began soon after to install +and test all main subsystems before the cryogenic +commissioning, see Fig. 5. +5.1 +Cryogenic plant installation +The ICARUS cryogenic plant was designed, +built and installed at Fermilab by a collabora- +tion of three international institutions, CERN, +INFN and Fermilab to support operations of the +ICARUS LAr-TPC. For the installation at Fer- +milab, the entire ICARUS-T600 cryogenic and +purification system was rebuilt anew. The new +design followed closely the original implementa- +tion at the LNGS with one important exception: +– 7 – + +Figure 5. Deployment of the ICARUS cryostats inside the pit of the SBN Far Detector experimental hall at +Fermilab in August 2018 (left). Installation of TPC, PMT and laser feed-through flanges in December 2018 +(center). Status of the ICARUS detector at the beginning of data taking for commissioning (right). +at Fermilab, the LN2 boiloff is vented to the +atmosphere (open loop cooling circuit), while +at LNGS the LN2 boiloff was re-condensed by +means of a set of cryocoolers (closed loop cool- +ing circuit). The main components of the cryo- +genic and purification system are the following: +• Main LAr containers (2× cold vessels): +273 m3 each, containing the TPC detectors +and the LAr scintillation light system. +• Cold shields: set of heat exchangers filled +with LN2, completely surrounding the +main LAr containers and designed to pre- +vent heat, coming from the thermal insu- +lation, to reach the LAr volumes. +• Thermal insulation: +polyurethane foam +panels, ∼ 600 mm thick, surrounding the +cold shields. +• Warm vessel: provides enclosure and me- +chanical support for the thermal insula- +tion. +• LN2 cooling circuits: piping, circulation +pumps, regulating valves, phase separa- +tors, etc., providing LN2 supply to heat +exchangers serving the cold shields and +the purifying units. +• Argon gas recirculation units (4×, two per +cold vessel): set of units that re-condense +and purify the argon flowing from the gas +phase on top of the main LAr containers. +• Liquid argon recirculation units (2×, one +per cold vessel): provide forced circula- +tion, with a cryogenic pump, of argon +coming from the cold vessels through a set +of purifiers before injecting it back into the +cold vessel. +• Cryogenic control system: +to provide +automation, data display, recording and +alarming. +• LN2 and LAr storage dewars and relative +transfer lines. +• A dedicated purification unit used for the +filling of the cold vessels, equipped with a +regeneration system and a set of gas ana- +lyzers. +The ICARUS cryogenic plant at the SBN Far +Detector Hall at Fermilab was fully designed, +delivered, and installed by July 2019, with the +commissioning phase started by January 2020. +The equipment included the ICARUS cryogenic +plant is schematically divided into the external +components supplied by Fermilab, the proximity +components supplied by Demaco Holland B.V. +under contract with CERN and components in- +ternal to the cryostats supplied by INFN. Fig. 6 +shows the ICARUS plant physical layout. +– 8 – + +lcarusTritFigure 6. ICARUS cryogenic plant physical layout. +5.2 +TPC electronics installation +Each mini-crate, housing nine A2795 boards, +was mounted onto the flange on top of the chim- +ney that contains flat cables connecting wires of +the chambers to DBBs and powered by a linear +power supply next to the chimney, see Fig. 7. +Each set of nine A2795 in a single crate are read +out through two fibers that implement a CAEN +proprietary protocol named CONET (Chain-able +Optical NETwork). The two sets of fibers are +read through an A3818 PCI Express board in- +stalled in dedicated PCs. +The full TPC electronics (96 mini-crates) is +synchronized by a serial link (one cable), named +TTLink, which sends clock, trigger, and com- +mands. The TTLinks are distributed to all mini- +crates by four fan-out modules with the same +cable lengths to guarantee equal time delay. The +TPC electronics system is fully installed and op- +erational. +5.3 +PMT system installation +Electrical connections between PMTs and elec- +tronics, located in a building alcove adjacent to +the detector, were realized by means of 360 sig- +nal cables and 360 high voltage cables. +Sig- +nal cables are RG316/U, 7 m of which are de- +Figure 7. Two Low Voltage Power Supply (LVPS) +modules powering the two adjacent mini-crates pop- +ulated with nine A2795 boards, serving 576 wires +each. +ployed inside the detector and 37 m outside, the +two parts connected by means of BNC-BNC +feedthrough flanges. +High voltage cables are +7-m long HTC-50-1-1 deployed inside the de- +tector and 37 m RG58/U outside; the two parts +connected by means of SHV-SHV feedthrough +flanges. Power supply voltages are generated and +distributed by 8 CAEN A7030 boards, each with +48 channels that can provide 3 kV, housed in two +CAEN SY4527 crates. +The PMT electronics are designed to allow +continuous read-out, digitization and indepen- +– 9 – + +Proximity cryogenics on detector top: +LN2 shields valve boxes +LAR +GAr re-condensers valve boxes +Transfer lines and gas collection piping +Fill Filter +External cryogenics: +LAr and LN2 dewars +Transfer and vent lines +Proximity cryogenics in pit and mezzanine: +Regeneration skids for filter media +LAr pumps valve boxes +Gas analyzers +LAr filters valve boxes + Process controls system +LN2 Phase separator and pumps valve boxes +← 23 m +Safety controls systemWE05/06 +TRIPP-LITE +CINENdent waveform recording of signals coming from +the 360 PMTs. This operation is performed by +24 CAEN V1730B digitizers installed in 8 VME +crates (3 digitizers per crate). Each module con- +sists of a 16-channel 14-bit 500-MSa/s FLASH +ADC with 2 Vpp input dynamic range. In each +board 15 channels are used for the acquisition of +PMT pulses, while one channel is used for the +acquisition of ancillary signals such as the beam +gates and the trigger pulses. +For each channel, an internal trigger-request +logic signal is generated every time the sam- +pled PMT pulse passes through a programmable +threshold. For each couple of adjacent channels, +trigger-requests are logically combined (OR, +AND, Ch0, Ch1) and the result is presented in a +low-voltage differential signaling (LVDS) logic +output with settable duration. For triggering pur- +poses, an OR logic between neighboring PMTs +is adopted. A total of 192 LVDS lines (8 lines +per digitizer) are connected to the ICARUS trig- +ger system for exploiting the scintillation light +information for trigger purposes. +The PMT electronics are complemented by +a common 62.5 MHz clock distribution system, +an external trigger network, an external time- +stamp reset network, and 24 optical link inter- +faces based on the CAEN CONET2 protocol. +5.4 +Cosmic Ray Tagger installation +The Side CRT system was installed over the pe- +riod from November 2019 to April 2021 (Fig. 8 +left). Following its shipping in summer 2021, the +installation of Top CRT modules was carried out +and completed in December 2021 (Fig. 8 right). +All Top and Side CRT modules were tested be- +fore and after their installation to check for elec- +tronic functionality of the channels. Data trans- +mission to the servers is performed via ethernet +cables connecting the modules in daisy chain. +The distribution of a Pulse Per Second (PPS) +signal (see Sec. 6.4) for absolute time reference +and trigger signal to the FEBs was performed +with lemo cables. A voltage of 5.5 V to be pro- +vided to the FEBs is distributed via power lines +assembled at FNAL during the installation. All +the information on modules to cables connec- +tions, SiPM bias voltages, module positions, etc. +are stored in a Fermilab SQL database. +The last ICARUS installation activity was +the deployment of the 2.85-meter concrete over- +burden above the Top CRT. The overburden is +composed of three layers of concrete blocks, +each approximately 1-meter tall, giving a total +mass of 5 million pounds. The installation of the +last concrete block was completed June 7, 2022, +marking the beginning of ICARUS data taking +for physics with both BNB and NuMI beams. +6 +ICARUS-T600 commissioning +After the placement of the two ICARUS modules +in the pit in August 2018, all the feed-through +flanges for the TPC and PMT signals and for +the injection of the laser flashes used to calibrate +the PMTs were installed in December 2018. The +gain and the dark rate for all 360 PMTs were mea- +sured as a function of the applied voltage at room +temperature. All the new TPC readout electron- +ics in the 96 mini-crates and the low noise power +supplies were installed and verified. In particular +the full readout chain has been tested by injecting +test pulses in wires at the far end of the cham- +ber and reading out the signals with the A2795 +boards on the other end to check the full system +for noise monitoring purposes. +In parallel all the cryogenic equipment were +installed, welded and the complete system has +been tested at 350 mbar over-pressure. The cold +vessels were then successfully brought to vac- +uum, with a 10−5 mbar residual pressure. +The +cryogenic +commissioning +of +the +ICARUS-T600 detector started on February 13, +2020 by breaking the vacuum in the two main +cold vessels with ultra-purified argon gas. Cool +down started on February 14 by injecting liq- +– 10 – + +Figure 8. +Left: picture of the Side CRT. Right: Top CRT horizontal modules whose installation was +completed in December 2021. +uid nitrogen in the cold shields. It took about +four days to bring the temperature on the wire +chamber below 100 K. The cooling process was +continuous and the maximum temperature gra- +dient on the wire chambers was about 35 K. On +February 19, the gas recirculation units were put +into operation to purify the argon gas before the +start of the liquid filling. +The continuous filling with ultra-purified +liquid argon started on February 24. The filling +was interrupted at around 50% to regenerate the +filling filter. The filling was stopped again when +the liquid reached the −6 cm LAr level probes +(6 cm below the nominal level) to perform the +final pressure test of the two cold vessels. After +the test, the gas recirculation units were put into +operation. +The filling was completed on April 19, see +Fig. 9. On April 21 the liquid recirculation was +started. The recirculation rates were 1.85 m3/h +in the West module and 2.25 m3/h in the East +module. +The cryogenic stabilization phase was com- +pleted around the end of May 2020. Pressures +and temperatures in the two modules were stable +and no cold spots were observed on the exter- +nal surface of the Warm Vessel. +At the start +of the cryogenic commissioning, all activities in +the detector building not related to cryogenics +were suspended and the building was put in a +high safety condition, with strong limitations to +the presence of people onsite. At the end of the +liquid argon filling, after the final pressure test, +the standard safety conditions were restored and +regular activities on top of the detector could be +restarted to complete the installation and test of +all sub-detectors. +During the cryogenic commissioning, there +were several activities both related to monitoring +the status of the detectors (wire chambers, wires +readout electronics, PMTs, CRT) and to develop- +ments for the following detector commissioning +phases. Noise data have been continuously taken +of wire readout electronics, PMTs and CRT. Ef- +fects on the noise from the activation of the cryo- +genic plant have been continuously monitored. +Functionality and stress tests of the DAQ were +conducted with several useful results. +The detector activation took place on Au- +gust 27, 2020 when the TPC wire planes and the +cathode high voltage (HV) were taken to nomi- +nal voltages. HV has remained stable at −75 kV. +Significant currents were found only on a few +wire bias and were addressed. All PMTs were +switched on and calibrated with the laser system. +Cosmic-ray interaction events were initially +collected with a random 5 Hz trigger and data +analyzed for calibration purposes (i.e. electron +– 11 – + +DUMMY +Genie29Figure 9. Trend of the liquid argon level inside the two ICARUS cryostats during the filling phase. +lifetime, space charge, drift velocity measure- +ments). +Dedicated runs were also carried out +for specific commissioning tasks, such as inves- +tigation of TPC noise, PMT calibration with the +laser system, DAQ upgrades/longevity tests, etc. +One of the first measurements carried out +was the free electron lifetime 𝜏𝑒𝑙𝑒. This parame- +ter is fundamental for the monitoring of the liquid +argon condition in the TPCs and to obtain the +precise measurement of the energy deposition +from the ionization charge signal in the collected +events. The LAr purity is continuously moni- +tored by measuring the charge attenuation along +the drift path of the electron ionization signals +generated by cosmic ray tracks crossing the de- +tector. A fast procedure has been setup starting +from the method developed and used during the +Gran Sasso run [11]; it has been applied to the +recorded data since the detector activation. +The 𝜏𝑒𝑙𝑒 measurement is based on a simpli- +fied identification of the wire signals in the Col- +lection plane and of the anode to cathode cross- +ing muon tracks that have no indication of asso- +ciated 𝛿-rays or electromagnetic showers along +the track. It is used to provide a fast, real time, +measurement within 5-10% precision dominated +mostly by effects related to space charge and to +the electron diffusion, see Fig. 10. +The steady state values of 𝜏𝑒𝑙𝑒, exceeding +3 ms in both cryostats, are high enough to al- +low for efficient detection and reconstruction of +ionizing events inside the active volume. +6.1 +TPC commissioning +After the TPC wires were biased and the cath- +ode HV was raised to nominal operating condi- +tions, the TPC commissioning began. With the +liquid argon at a sufficient level of purity, cos- +mogenic activity in the detector can be used to +study the detector response to ionization signals +in the TPC. To characterize the performance of +the ICARUS TPC, a variety of measurements +were taken between August 2020 and May 2022 +as summarized below. +Noise levels in the TPC can be measured us- +ing the RMS of waveforms from the TPC read- +out, with an equivalent noise charge (ENC) of +roughly 550 e−/ADC [21]. Measured TPC noise +levels at ICARUS are shown in Fig. 11, both +before and after the filtering of coherent noise, +which was performed across sets of 64 channels +associated with the same front-end electronics +board. +– 12 – + +4 +Start Gas and Liquid re-circulation +3,5 +3 +2, 5 +EAST Module +2 +WESTModule +1,5 +1 +0,5 +0 +0,00 +200,00 +400,00 +600,00 +800,00 +1000,00 +1200,00 +[400,00 +1600,00 +Feb 19 : Filling Start +Elapsed Time (hours) +Apr 19 : Filling CompleteFigure 10. Trend of the drift electron lifetime in the two ICARUS cryostats during the commissioning phase. +The sharp decreases of the lifetime are due to programmed interventions on the LAr recirculation pumps or +on the cryogenic system. The lifetime is quickly recovered after the end of the interventions. +Waveforms containing ionization signals are +identified by simply applying a threshold and re- +moving from consideration to ensure there is no +bias to the noise measurements. The measure- +ments were repeated with the cathode HV off +and consistent results were obtained, validating +the ionization signal identification methodology +and indicating that a negligible amount of TPC +noise is caused by interference from the cathode +HV system. +The noise levels after coherent noise filter- +ing shown in Fig. 11 are consistent with previous +noise measurements of the TPC electronics in a +test setup [21]. +Fast Fourier transforms (FFTs) of the same +noise waveforms used in the results shown in +Fig. 11 are calculated for each of the three +wire planes and averaged across the entire de- +tector; +these results are shown in Fig. 12. +FFTs are shown both before and after coherent +noise removal, showing the expected approxi- +mate Rayleigh distribution of the intrinsic noise +spectrum [22] on all three planes after coherent +noise is removed. This provides strong evidence +of extrinsic noise being almost completely re- +moved from the TPC waveform data by the noise +filtering algorithm. +The Induction 2 plane and Collection plane +spectra show a similar normalization, which is +expected given the same length of the wires of +these two planes. The Induction 1 plane spec- +trum has instead a larger normalization given the +longer wires and thus a higher capacitance, in- +creasing the intrinsic noise levels. Further work +is being carried out to understand the source of +the coherent noise. +In runs with sufficiently high electron life- +time (most runs after the very beginning of +commissioning in 2020), +ionization signals +from anode-cathode-crossing cosmic muons are +used to evaluate the peak signal-to-noise ratio +(PSNR) for minimum-ionizing particles (MIPs) +in the TPC. Anode-cathode-crossing cosmic +muon tracks traverse the full drift length of the +detector and therefore allow for knowledge of the +drift coordinate of each ionization signal along +the track. Fig. 13 shows the PSNR of ioniza- +tion signals for each plane using a large sample +of cosmic muons in ICARUS data with coherent +noise removed. +In this study, the peak signal +(numerator in the ratio) is defined as the maxi- +mum signal ADC value minus the baseline ADC +value for the unipolar signals of the Collection +plane and the absolute value of the maximum sig- +– 13 – + +Lifetime [ms] +West +L +East + +30/Sep +31/Dec +01/Apr +01/Jul +01/Oct +31/Dec +01/Apr +2020 +2020 +2021 +2021 +2021 +2021 +2022 +DateFigure 11. TPC noise levels at ICARUS before and after filtering of coherent noise, as measured by waveform +RMS in ADC counts (with ENC of roughly 550 e−/ADC [21]). Results are shown separately for the Induction +1 plane (left), Induction 2 plane (center), and Collection plane (right). Mean values of the shown distributions +are presented at the bottom of each figure. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Induction 1 +Raw Spectra +Noise-Filtered Spectra +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Induction 2 +0 +100 +200 +300 +400 +500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Collection +Frequency [kHz] +Power [ADC2/kHz] +Figure 12. Fast Fourier transforms (FFTs) of noise +waveform data collected by the ICARUS TPCs, be- +fore and after filtering of coherent noise. Results are +shown separately for the Induction 1 plane (top), In- +duction 2 plane (middle), and Collection plane (bot- +tom). +nal ADC value minus the minimum signal ADC +value for the bipolar signals of the two induc- +tion planes. The noise level (denominator in the +ratio) is the RMS of signal-removed waveforms +from the same TPC channel in units of ADCs, +as shown in Fig. 11. Cosmic muon tracks used +in the PSNR measurement are required to be +oriented at an angle of 20 degrees or less with +respect to the anode plane, and have a "3D pitch" +(track segment length corresponding to the ion- +ization signal from a single wire) of 4 mm or less +for the wire plane of interest. These selection +criteria probe the phase space most relevant for +beam neutrinos interacting in the detector, which +have interaction products that travel mainly in the +forward direction. Furthermore, only parts of the +track within 2 cm to 10 cm of the anode are used +in order to minimize impact from charge attenua- +tion due to impurities in the liquid argon. Fig. 13 +illustrates the performance of the TPC. +The detector enables robust identification of +ionization signals embedded within electronics +noise background, with more than 99% of the +MIP ionization signals having a PSNR greater +than four. +Anode-cathode-crossing +cosmic +muon +tracks are also used to make a measurement +of ionization drift velocity in the detector. +The distance between the anode and cathode, +148.2 cm, is divided by the maximum ionization +drift time, or the difference in time between +the first and last ionization signals associated +with the cosmic muon tracks. +The latter +measurement should yield the time it takes for +ionization to drift from the cathode (one end of +– 14 – + +Average Noise by Plane +Full Noise +Noise-Filtered +Induction 2 +Collection +Induction 1 +2.5 +2.0 +Units +1.5 +Arbitrary l +1.0 +0.5 +0.0 +0.0 +2.5 +5.0 +7.5 +10.0 +0.0 +2.5 +5.0 +7.5 +10.0 +0.0 +2.5 +5.0 +7.5 +10.0 +RMS[ADC] +RMS [ADC] +RMS [ADC] +μ: 3.80 ADC +6.02 ADC +μ: 2.51 ADC +3.65 ADC +μ: 2.53 ADC +3.47 ADC0 +5 +10 +15 +20 +25 +30 +35 +40 +Peak Signal-to-Noise Ratio (Noise-Filtered) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Arbitrary Units +Induction 1 (Peak: 6.74) +Induction 2 (Peak: 8.64) +Collection (Peak: 9.05) +Figure 13. Peak signal-to-noise ratio (PSNR) of ion- +ization signals for each of the three TPC wire planes +using cosmic muons in ICARUS data. Coherent noise +is removed from the TPC waveforms prior to iden- +tification and measurement of the ionization signal +amplitude. See text for details on the cosmic muon +data selection. +the track) to the anode (other end of the track), +so the ratio should provide the drift velocity of +the ionization electrons in liquid argon at the +nominal drift electric field of roughly 500 V/cm +and temperature of roughly 87.5 K. +A correction is made to account for a small +bias in precisely reconstructing the drift times as- +sociated with the track end points, derived from +Monte Carlo simulation. A Crystal Ball func- +tion1 is then fit to the maximum ionization drift +time distribution associated with cosmic muon +tracks in each TPC volume (two per cryostat), +with the peak value of each fit used in the ion- +ization drift velocity calculation. The results of +the ionization drift velocity measurements in the +west cryostat are shown in Fig. 14. The results +of the measurements, roughly 0.1572 cm/µs for +both TPC volumes in the west cryostat, agree +with the predicted value of 0.1576 cm/µs to +within 0.3% [23, 24]. +1The Crystal Ball function, named after the Crystal +Ball Collaboration, is a probability density function com- +monly used to model various lossy processes in high-energy +physics. It consists of a Gaussian core portion and a power- +law low-end tail, below a certain threshold. +880 +900 +920 +940 +960 +980 +1000 +Maximum Ionization Drift Time [ s] +0 +5000 +10000 +15000 +Tracks +TPC E Drift V: +0.1572 cm/ s +TPC W Drift V: +0.1572 cm/ s +Ionization Drift Velocity: West Cryostat +TPC E Fit +TPC E Data +TPC W Fit +TPC W Data +Figure 14. +Results of the ionization drift veloc- +ity measurement using ICARUS cosmic muon data. +Shown are Crystal Ball fits to the maximum ioniza- +tion drift time distributions associated with anode- +cathode-crossing cosmic muons in the two TPCs in +the west cryostat. +Electric field distortions in near-surface +LAr-TPCs can arise due to the accumulation +of space charge, i.e. +slow-moving positively- +charged argon ions originating from cosmic +muon ionization within the detector [25]. These +argon ions, which drift slowly toward the cath- +ode at a drift velocity of several millimeters per +second at a drift electric field of 500 V/cm [24], +linger around long enough to create substantial +electric field distortions that pull ionization elec- +trons toward the middle of the TPC volume as +they drift toward the anode. These electric field +distortions lead to biases in reconstructing the +point of origin of ionization within the detector, +a secondary effect referred to as "spatial distor- +tions" in LAr-TPC detectors; collectively, these +two related distortions are referred to as space +charge effects (SCE). +Using +anode-cathode-crossing +cosmic +muon tracks, the magnitude of SCE in the +ICARUS detector is estimated by utilizing +methodology developed to measure SCE in +previous near-surface running of the ICARUS +detector [26]. The results of measurements in +the two TPC volumes of the west cryostat are +shown in Fig. 15, where they are compared +– 15 – + +to a calculation of SCE [24] used in ICARUS +Monte Carlo simulations prior to measuring +the magnitude of SCE in ICARUS data. +The +magnitude of SCE is observed to be very similar +in the two TPC volumes, though underestimated +by roughly 30% in simulation. +0 +20 +40 +60 +80 +100 +120 +140 +Drift Coordinate X [cm] +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +X [cm] +∆ +Drift Direction Spatial Offset +WE TPC Data +WW TPC Data +Calculation for Simulation +X vs. X +∆ +Data SCE Comparison: +Figure 15. Measured spatial offsets in the drift di- +rection as a function of ionization drift distance for +the two TPCs in the west cryostat, evaluated us- +ing anode-cathode-crossing cosmic muon tracks in +ICARUS data. The results are compared with pre- +dictions of spatial distortions from a calculation of +space charge effects (SCE) presently used in ICARUS +Monte Carlo simulations (to be updated with data- +driven SCE measurement). +The energy scale of MIPs can be probed +with cosmic muons that stop in the ICARUS de- +tector, as done in similar calibrations performed +at other LAr-TPC neutrino experiments [27]. +The known profile of muon energy loss per unit +length (𝑑𝐸/𝑑𝑥) in liquid argon as a function of +kinetic energy [28] can be used to predict the +value of 𝑑𝐸/𝑑𝑥 versus residual range, the dis- +tance from the end of a stopped muon track +in reconstructed TPC data. +After accounting +for prompt electron-ion recombination [29] and +charge attenuation during ionization drift due to +electro-negative impurities in the detector, one +can compare the most-probable value (MPV) of +𝑑𝐸/𝑑𝑥 versus residual range from a sample of +stopping muons in ICARUS data (evaluated by +fitting the data with a Landau distribution con- +volved with a Gaussian, performed in bins of +residual range) to the MPV 𝑑𝐸/𝑑𝑥 curve ex- +pected from theory. +The result of the Collection plane energy +scale calibration for the east TPC of the west +cryostat is shown in Fig. 16 (left). Good agree- +ment between calibrated data and predictions +from theory is found for all values of stopping +muon residual range after this calibration has +been performed, with sub-percent agreement for +values of 𝑑𝐸/𝑑𝑥 < 4 MeV/cm; similar levels of +agreement are observed for the other three TPCs +as well. Additionally, the energy scale calibra- +tion is further scrutinized by comparing two dif- +ferent methods of stopping muon kinetic energy +reconstruction: one by calorimetry (summing up +charge associated with energy deposition along +the track), 𝐸calo, and another by range (convert- +ing distance from end of stopping muon track +to kinetic energy by use of a look-up table [28]), +𝐸range. The result of this cross-check is presented +in Fig. 16 (right), showing little bias between the +two methods for stopping muons in ICARUS cos- +mic muon data after the energy scale calibration +is applied. +Future measurements will include +protons from ICARUS data, allowing for probing +of the energy scale of highly-ionizing particles +in the detector. +6.2 +PMT commissioning +The whole light detection system was tested at +Fermilab before the cooling of the detector, once +the dark condition inside the cryostats was guar- +anteed. A total of 357 (out of 360) PMTs were +found to be working with performances consis- +tent with the tests performed at CERN [16]. The +same number of working PMTs were found af- +ter the filling of the detector with liquid argon, +demonstrating the ability of this PMT model to +withstand low temperatures. +A PMT signal, recorded by the light detec- +tion system electronics, is shown in Fig. 17. A +gain calibration/equalization campaign was car- +– 16 – + +0 +20 +40 +60 +80 +100 +Residual Range [cm] +1 +2 +3 +4 +5 +6 +Calibrated dE/dx [MeV/cm] +Predicted MPV dE/dx +0.4 +0.2 +0.0 +0.2 +0.4 +(Ecalo +Erange) / Erange +0 +5000 +10000 +15000 +20000 +25000 +30000 +Tracks +1: 4.7% +1: 0.3% +2: 14.5% +2: 0.8% +Figure 16. Calibrated Collection plane 𝑑𝐸/𝑑𝑥 as a function of residual range for a selection of stopping muons +in ICARUS cosmic muon data, including a comparison to the most-probable value (MPV) of 𝑑𝐸/𝑑𝑥 from +stopping muons predicted from theory [28] (left); comparison of cosmic muon kinetic energy reconstruction +by calorimetry, 𝐸calo, and by range, 𝐸range, showing little bias between the two methods for stopping muons +in ICARUS cosmic muon data after the energy scale calibration is applied (right). +ried out during the PMT commissioning. At first, +external fast laser pulses focused on each PMT +window by means of dedicated optical fibers +were used to obtain a coarse gain curve for each +PMT as a function of the applied voltage around +the expected values. Laser pulses were also used +to characterize, to within 1 ns precision, the delay +response of each PMT channel, which can dif- +fer due to different PMT and cable transit times. +Voltages were set to values corresponding to a +gain of 5 · 106, resulting in an equalization within +16%, as a first approximation. +Fine tuning was carried out to improve the +gain equalization by means of an automatic pro- +cedure. +To this purpose the response of each +PMT to background single photons (≈ 250 kHz) +was measured, and the voltages were adjusted +according to the gain curves. This procedure led +to a final equalization with a spread less than 1%, +as shown in Fig. 18. +6.3 +CRT commissioning +The side and top CRT modules were tested before +the installation at ICARUS using a test stand. Af- +ter the installation of all CRT modules, the cos- +mic rate over time was obtained. The event rates +Figure 17. PMT signal as recorded by the light de- +tection system electronics. +for each wall of the side CRT as a function of time +are constant, as shown in Fig. 19. The higher +rates on north wall (black) are due to the proxim- +ity with the cryogenic pumps, with these mod- +ules experiencing higher electrical noise rates in +addition to cosmic rates on the surface. In addi- +tion, the rates from the west north and east north +walls are slightly higher from being closer to the +cryogenics. Following work to characterize and +mitigate the noise, electrical chokes (inductors) +were installed along all Side CRT FEB power +cables to reduce noise rates. +Top CRT cosmic event rates before and after +the installation of concrete overburden are shown +– 17 – + +15000 +Amplitude (ADC Counts +14900 +14800 +14700 +14600 +14500 +0 +2 +4 +6 +8 +10 +Time (us)Figure 18. Gain distribution for 354 PMTs after the +fine tuning equalization. The automatic procedure +was not applied on 6 PMTs (not present in the plot) +that were manually calibrated. +in Fig. 20 for horizontal (left) and vertical (right) +modules. +Before the installation of the over- +burden the mean rate was ∼ 610 Hz and 260 Hz +for horizontal and vertical modules, respectively. +After the installation of the overburden the rates +reduced to 330 Hz and 180 Hz for horizontal and +vertical modules, respectively. Except for varia- +tion due to concrete blocks placement above the +detector, the rates are stable on a time scale of +months. +12/23/20 +12/30/20 +01/06/21 +Date +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Rate [kHz] +North Wall +West North Wall +West Central Wall +West South Wall +East North Wall +East Central Wall +East South Wall +Figure 19. Side CRT cosmic event rates as a function +of time. The black points corresponds to the rates +from the north side CRT wall, the pink and blue +points corresponds to East and West north walls, and +the remaining walls are at 1 kHz rate. +6.4 +Triggering on the BNB and NuMI neu- +trinos +The initial ICARUS trigger system exploits the +coincidence of the BNB and NuMI beams spills, +1.6 µs and 9.6 µs respectively, with the prompt +scintillation light detected by the PMT system in- +stalled behind the wire planes of each TPC [30]. +The generation of the beam spill gates is +based on receiving the “Early Warning” (EW) +signals for BNB and NuMI beams, 35 and 730 ms +in advance of protons on target, respectively. +LVDS signals from the PMT digitizers, in terms +of the OR signal of adjacent PMTs, are pro- +cessed by programmable FPGA logic boards to +implement trigger logic for the activation of the +ICARUS read-out. Additional trigger signals are +generated for calibration purposes in correspon- +dence with a subset of the beam spills without +any requirement on the scintillation light (Min- +Bias trigger) and outside of the beam spills to +detect cosmic ray interactions (Off-Beam trig- +ger). +To synchronize all detector subsystems’ +read-outs with the proton beam spill extraction +at the level of few nanosecond accuracy, a White +Rabbit (WR) network [31] has been deployed for +distributing the beam extraction signals. An ab- +solute GPS timing signal, in the form of PPS, is +used as a reference for generating phase locked +digitization clocks (62.5 MHz for the PMT and +2.5 MHz for the TPC) and for time-stamping +the beam gates and trigger signals. In addition, +the signals of Resistive Wall Monitor detectors +(RWM) at 2 GHz sampling frequency are also +recorded to precisely measure the timing and +the bunched structure of protons on target, see +Fig. 21. +In the presence of a global trigger signal, +1.5 ms and 30 µs acquisition windows are acti- +vated for the TPC and PMT signal recording, +respectively. In addition, PMT waveforms are +collected inside a 2 ms time window around the +– 18 – + +PMTs +90 +Entries +354 +Constant +94.57 +# +80 +Mean +0.4967 +70 +Sigma +0.003574 +60 +50 +40 +30 +20 +10 +0.45 +0.46 +0.47 +0.48 +0.49 +0.5 +0.510.520.530.540.55 +Gain [10' electrons]02/26/22 +03/28/22 +04/27/22 +05/27/22 +Date +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +Rate [kHz] +FEB 172 +FEB 114 +FEB 100 +FEB 150 +FEB 238 +FEB 234 +FEB 238 +FEB 170 +FEB 101 +FEB 142 +FEB 6 +FEB 232 +FEB 237 +FEB 239 +02/26/22 +03/28/22 +04/27/22 +05/27/22 +Date +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Rate [kHz] +FEB 81 +FEB 119 +FEB 87 +FEB 92 +FEB 180 +FEB 97 +FEB 174 +FEB 189 +FEB 190 +Figure 20. Cosmic ray rates as a function of time for a set of Top CRT horizontal (left) and vertical (right) +modules. Numbers in the legend indicate the module’s Front End Board and the black dot lines indicate the +beginning and the end of 3 m overburden installation over the displayed modules: the rates reduced from +∼ 610 (260) Hz before to 330 (180) Hz after the installation of the overburden for the horizontal (vertical) +modules. +Figure 21. +Layout of the trigger system. +SPEXI +board: synchronizes the whole ICARUS detector, +generates clocks and readout signals, handles beam +extraction messages; 7820 FPGA boards: generate a +Global Trigger in coincidence with beam extraction +(Early Warning) on the basis of selected PMT sig- +nal majorities to recognize an event interaction in the +LAr, to start the PMT activity recording; RT Con- +troller implements all the features for communication +with DAQ. +beam spill to record all cosmic muons crossing +the ICARUS TPCs during the electron drift time. +The timing of the beam spills was first ap- +proximately determined by measuring with an +oscilloscope the difference between the EW sig- +nals arrival time and the actual proton extraction +signal by RWM counters at the target. Then neu- +trino interactions were identified and associated +with the muons of the beam spill in excess to +cosmic rays that were clearly identified inside +the time profile of the scintillation light signals +(flashes) by requiring at least 5 fired PMT pairs +in the left and right TPC (Fig. 22). +Due to the energy range of BNB and NuMI +neutrino beams, neutrino interactions are ex- +pected to be contained in an ∼ 4 m section of +ICARUS along the beam direction, suggesting +the implementation of a trigger logic based on +the recognition of fired PMTs inside a limited +TPC region. The logic for processing the PMT +LVDS signals has been initially determined with +Monte Carlo calculations, and then it has been +refined by analyzing a sample of events collected +with a beam spill signal only (Min-Bias trigger), +i.e. without any requirement on the scintillation +light. The 18-m long TPC walls have been sub- +divided in 3 consecutive longitudinal slices of +6-m length including 30 PMTs each. In each of +opposite facing slices a majority of 5 LVDS sig- +– 19 – + +WhiteRabbitnetwork +PMT +PMT +CPU W RT +TRIG +GLOBAL +TRIG +controller +TRIG +SPEXI +EAST +WEST +PXle8135 +7820R +7820R +7820R +TPC +TPC +A2795's +A2795's +PMT +PMT +TPC WIRES +TPC WIRES +V1730B's +V1730B's +T300 E +T300 E +PMT DAQ +TPC DAQ +PMT's +PMT's +T300 E +T300 W +ENABLEGATE(2mS +Trigger +BEAMGATE[1.6uS,9.8ys] +laptop +GlobalTriggerOutput cryostat1-2 +Central +PMT Trigger cryostat 1-2 +DAQ +ALLBus LinesFigure 22. Time distribution of the recorded PMT light flashes (≥ 5 fired PMT pairs in the left and right +TPCs within 150 ns): the beam event excess is observed for BNB (left) and NuMI beam (right). The 1.6 µs +and 9.6 µs spills duration of the beams are well recognized. +nals, with 8 photo-electron (phe) discrimination +threshold and an OR of two adjacent PMTs, has +been required to produce a PMT trigger primi- +tive signal. The same logic with a majority of +10 LVDS PMT signals is applied to generate a +PMT trigger primitive in time period prior to and +after a beam spill. This trigger provides collec- +tion of data sampling the 15 kHz of cosmic rays +crossing the detector during the drift time. +With trigger gates of duration 4 ms and 14 +ms for BNB and NuMI, respectively, a trigger +rate of ∼ 0.7 Hz has been obtained (0.3 and 0.15 +Hz from the BNB and NuMI components, re- +spectively, and 0.25 Hz for the Off-Beam). This +is in a manageable data read-out bandwidth with +good operational stability. The trigger efficiency +for neutrino interactions is under study with data; +expectations based on the Monte Carlo simula- +tions indicate a > 90% efficiency for neutrino CC +interactions with >100 MeV energy deposition. +6.5 +DAQ implementation +The ICARUS data acquisition (DAQ) system uti- +lizes the general artdaq data acquisition software +development toolkit [32], providing customiz- +able applications for reading data from detector +elements (BoardReaders), and configurable ap- +plications for performing event-building, data- +logging, and data-dispatch to downstream online +data quality monitoring processes. +Customized BoardReaders acquire data +fragments from the TPC, PMT, and CRT read- +out electronics, and from the trigger and White +Rabbit timing systems. +They then assign ap- +propriate event counters and timestamps to each +fragment and then queue that data for transfer +to a configurable number of EventBuilder appli- +cations. For each triggered event, the ICARUS +trigger BoardReader sends its data fragment to +an EventBuilder, triggering a request for data +from all other configured BoardReaders in the +DAQ system. Events are written using the art +event-processing framework [33]. Data are writ- +ten on separate file streams using simple filters +on trigger type. Each event in ICARUS, after +lossless data compression, is approximately 160 +MB, with the majority of data corresponding to +the TPCs. The DAQ system is capable of stably +supporting trigger rates in excess of 5 Hz, though +typical operational trigger rates are of roughly 1 +Hz or below. +The BoardReader for the trigger system +sends a single fragment containing the trigger +and beam-gate timing, the type of beam gate, +a global trigger counter, and a counter for the +number of beam gates of each type in that DAQ +– 20 – + +BnB +NuM +4 +Background +.50 Ms +2140 +. +425 +Beam gate:1.6 μs +400 +Data +375 +lashes +lashes +100 +329 +Background +DL +Beamgate:9.5μs +275 +Data +Pmt flash start tinmne [us] +PMT flash start tirme Lus]run. +The global trigger counter and time are +used for collection of data from other subsys- +tems; the latter derives from the common White +Rabbit timing system, and is checked for validity +against the network protocol time of the trigger +BoardReader server. The number of beam gates +of each type in the run is used offline for proper +accounting of the total number of POT and de- +tector exposure within a run. +In order to handle large data volumes stored +on tape, the Fermilab based SAM (Serial Ac- +cess to Metadata) system is exploited. For this +purpose, a set of metadata is associated to each +data file using Python scripts. The metadata al- +low users to create large data sets for the analysis +by requiring matching with data’s relevant infor- +mation such as run number, data type (raw or +reconstructed), run configuration, date, etc. +6.6 +First operations with the BNB and +NuMI +The ICARUS-T600 detector was first fully op- +erational in June 2021 before the summer shut- +down. It restarted data collection when beam +returned November 5, 2021. Figure 23 shows +the amounts of POT delivered by the accelerator +and collected by the detector during its commis- +sioning phase, concluded in June 2022, for a +total of 296 · 1018 and 503 · 1018 POT collected +for BNB and NuMI, respectively. Beam utiliza- +tion - defined as the amount of POT collected +divided by the delivered - of 89% for BNB and +88% for NuMI. In Fig. 23, daily variations of the +beam utilization are also visible: periods with +low utilization (less than 60%) correspond to +days where the data acquisition was suspended +in order to proceed with detector commission- +ing activities. +Apart from this, the utilization +is an average over 91% per day for both beams, +which corresponds to a downtime of less than +two hours per day. The most frequent causes of +operation downtime are data acquisition issues +and less commonly hardware problems. +The +detector and data collection status are continu- +ously supervised with fully-remote shifts staffed +by collaborators and with the support of on-call +experts for each of the main detector subsystems. +7 +Observation and reconstruction of +neutrino events +The data collected by the detector are processed +by offline software to obtain information neces- +sary for reconstruction and analysis of events. +The procedure to reconstruct the TPC wire and +PMT signals is briefly described in the following +Sec. 7.1, 7.2 and 7.3. +The detector behavior was first investigated +by a visual selection of neutrino interactions in +the active liquid argon, as described in Sec. 7.4. +These sample were an important component of +the development and validation of an automated +event selection scheme. +7.1 +Wire signal reconstruction +The ICARUS wire signal processing chain fol- +lows a logic similar to other LAr-TPC experi- +ments, based on the deconvolution of the wire +signal waveform. This procedure, explained in +more detail in [34], has the goal to recover the +original time structure of the current of drift +electrons generating the signal on each wire, up- +stream of the distortions produced by the electric +field in the wire region and the shaping by the +front-end electronics. +Mathematically, this is +obtained by inverting the response functions de- +scribing both the electric field and the electronics +effects; the resulting deconvolved signal shape is +approximately Gaussian for all wire planes. +After the removal of the coherent noise (de- +scribed in 6.1), the deconvolution is performed +on each wire waveform. +Segments of wave- +forms corresponding to physical signals (hits) are +searched for in the deconvolved waveform with a +threshold-based hit finding algorithm. Each hit +– 21 – + +06-01 +2021 +11-19 +2021 +12-29 +2021 +02-14 +2022 +03-27 +2022 +05-06 +2022 +20 +40 +60 +80 +100 +Beam utilization [%] +06-01 +2021 +11-19 +2021 +12-29 +2021 +02-14 +2022 +03-27 +2022 +05-06 +2022 +20 +40 +60 +80 +100 +Beam utilization [%] +0 +50 +100 +150 +200 +250 +300 +POT (1018) +BNB +Delivered: 334.2 1018 POT +Collected: 296.1 1018 POT +0 +100 +200 +300 +400 +500 +POT (1018) +NuMI +Delivered: 573.6 1018 POT +Collected: 503.1 1018 POT +Figure 23. Cumulative sum of POT delivered by the accelerator and collected by the detector and daily beam +utilization coefficient as a function of the operation time for BNB (NuMI) on the left (right). The dotted +black line marks the separation between the two operation periods of the detector: the full month of June +2021 and between November 5, 2021 and June 1, 2022 (the long break between the two periods is hidden in +the plot). +is then fit with a Gaussian, whose area is propor- +tional to the number of drift electrons generating +the signal. +Globally, the efficiency for identifying a +wire signal and associating it with the corre- +sponding track that generated is exceeding 90% +for all three wire planes when the 3D track seg- +ment length contributing to each hit (pitch) is +larger than 3.4 mm (Fig. 24). +Figure 24. +Hit efficiency as a function of wire +"pitch": blue, red and green points correspond to +Induction 1, Induction 2 and Collection wires respec- +tively. Measurement made by means of a sample of +cosmic muon tracks crossing the cathode. +7.2 +PMT signal reconstruction +The reconstruction of the scintillation light as- +sociated with the event of interest is based on +the recorded PMTs signals in the event, sam- +pled at 500 MHz. +For any event triggered in +coincidence with the beam spill, all 360 PMTs +digitized signals are recorded in 30 µs long time +intervals. In addition, for cosmic rays crossing +the detector in ±1 ms around the beam gate and +identified by the trigger logic, all 180 PMTs be- +longing to the ICARUS module containing the +event are recorded in 10 µs long time intervals. +A threshold-based algorithm is applied to +each recorded signal, to identify fired PMTs and +to reconstruct the characteristics of the detected +light to be used in the event analysis. Whenever +a PMT signal exceeds the baseline by 0.5 phe, +a new OpHit object is created, characterized by +a start time, a time interval for the signal to re- +turn back to baseline, a maximal amplitude, and +an integral of the signal over the baseline. As +a second stage all OpHits in coincidence within +100 ns are clustered together into an OpFlash +object. The Opflash is then expanded to include +also OpHits within 1 µs after the first OpHit time. +Nominally, an OpFlash should correspond to the +– 22 – + +Efficiency +0.9 +0.8 +0.7 +efficiency profile +0.6 +0.5 +0.4 +0.3 +0.4 +0.5 +0.6 +0.7 +pitch [cm] +0.8total detected light associated to each interac- +tion, either due to cosmic rays or to a neutrino +interaction. The distribution of the PMT signals +in an OpFlash (time, amplitudes, integrals and +geometrical positions) is clearly determined by +the associated interaction in the TPC (Fig. 25). +Figure 25. The PMTs associated with a cosmic ray +muon crossing the cathode. +Initially, a very simple association between +the event in the TPC and the corresponding de- +tected light that is based on the comparison of +the track and the light barycentre along the lon- +gitudinal z axis (zTPC, zPMT) has been adopted. +A correlation within few tens of centimeters +was observed for the TPC and light barycen- +tre (Δz = zTPC − zPMT) for both cosmic muons +crossing the cathode (Fig. 26) and for a sample +of BNB neutrino interactions (Fig. 27) selected +by visual scanning. +By requiring |Δz| < 100 cm it is possible +to restrict the analysis of the event to a detector +slide that is approximately 5% of the total active +LAr, with a corresponding reduction of randomly +overlapping cosmic rays. +7.3 +CRT reconstruction +The CRT hit reconstruction algorithm was vali- +dated during the commissioning phase [35]. The +first step in the reconstruction chain is to con- +struct CRT hits defined as points in space and +time corresponding to a muon track crossing the +CRT volume. CRT data coming from Front End +Board (FEB) read-outs in a given event are or- +dered in time and grouped by CRT region. Due +Figure 26. Distribution of Δz = zTPC − zPMT for a +sample of cosmic ray muons crossing the cathode. +Figure 27. Distribution of Δz = zTPC − zPMT for a +sample BNB 𝜈 interactions identified by visual scan- +ning. +to the differences in design of the side and top +CRT systems, the Side and Top CRT Hits have +to be handled differently. +The coincidence logic in the Side CRTs is +performed offline in the reconstruction stage due +to the inner and outer CRT modules being con- +nected to FEBs in adjacent layers, whereas each +top CRT module is a self-contained coincidence +unit. In order to identify a coincident grouping of +CRT data objects, a software-based coincidence +gate is performed (the hardware-based coinci- +dence gate width is 150 ns and this value is the +minimum for the software gate). The reason for +not making the coincidence window too large +is to avoid introducing fake coincidences from +– 23 – + +PMTs (behindthe +Fired +wires) +PMTs +Central cathode +PMTs(behind thewires +50 +z axis24000 +Cosmic +Entries +22000 +282361 +Mean +0.7743 +20000 +muons +RMS +51.16 +18000 +16000 +14000 +12000 +10000 +8000 +6000 +4000 +2000 +0 +400 +-200 +0 +200 +400#events +BNB vuCC +12 +candidates +10 +8 +rms=41 cm +6 +-50 +0 +50 +100 +150 +200 +cmlow energy events. Studies are underway to es- +tablish a gate width that optimizes the tagging +efficiency while avoiding introducing fake coin- +cidences with low energy events if the gate is too +wide. +After the creation of coincident groupings +of CRT data, the spatial information is extracted +to reconstruct the position of the crossing track. +The channel with the largest amplitude is the +channel that generated the FEB trigger signal. +The channel position is identified and extracted +from the geometry based on the global coordi- +nates of the ICARUS building. The hit position +is taken as the mean strip position where a track +crosses multiple strips in each layer. +When the charge amplitude exceeds the dis- +criminator threshold, a CRT hit is acquired by +the front-end electronics recording the values of +two different time counters. The first counter, +T0, is reset every second by means of the PPS +signal (see Sec. 5.4) and it provides the global +timing of the recorded hit. The second counter, +T1, is reset by the event trigger signal and is used +to determine the hit relative timing with respect +to the event trigger. Each CRT hit timestamp is +corrected to account for cable delays and light +propagation in the scintillator and in the WLS +fiber. +The Top CRT hit is defined by the FEB inter- +nal triggering logic (see Sec. 4) where a signal +threshold of 1.5 phe is applied to each chan- +nel. The position within a module is determined +by selecting the four channels with the largest +amplitude and projected in the global detector +coordinates. +The CRT timing system has been cross- +calibrated with the PMT signals, using the com- +mon trigger pulse recorded by the CRT and PMT +systems. A preliminary evaluation of the Time- +Of-Flight (TOF) of cosmic muons has been per- +formed by selecting particles entering the detec- +tor from the Top CRT and generating a flash in +the active argon volume. The preliminary distri- +Figure 28. Time difference between matched CRT +hits and PMT flashes. The plot refers to Top CRT +data in time with the BNB spill. +bution of the time differences between Top CRT +hits and PMT signals is shown in Fig. 28: the +measured average TOF of 24±9 ns is in agree- +ment with the expected ∼ 26 ns evaluated from +the distance between the Top CRT plane and the +first PMT row. +10 +− +8 +− +6 +− +4 +− +2 +− +0 +2 +4 +6 +8 +10 +s) +µ +CRT Hit T0 - gate start time ( +0 +100 +200 +300 +400 +500 +600 +700 +800 +900 +Number of CRT Hits +BNB, Side, South +s +µ +bin size = 0.2 +Figure 29. CRT hit time relative to the neutrino gate +start time in the south wall (side CRT) for the BNB +beam. +Figure 29 shows the CRT hit time relative +to the neutrino gate start time in the south side +CRT wall for the BNB neutrino beam. Using +11 days of commissioning data, a clear peak can +be observed, showing activity in the 4 µs trigger +coincidence window. Additional activity due to +the beam appears inside the smaller BNB gate +– 24 – + +Bins +1400 +Events/100 I +Fit parameters: +1200 +mean= -24 ns +sigma= 9 ns +1000 +800 +600 +400 +200 +-80 +-60 +-40 +-20 +20 +40 +60 +0 +80 +100 +CRT Hit timestamp - PMT Flash [ns](1.6 µs within the 4 µs window), the rest of the +activity outside the 1.6 µs window is due to cos- +mic ray triggering. +7.4 +Event display study +As a first check of the general behavior of the de- +tector, a visual study campaign was performed +to select and identify neutrino interactions in the +active liquid argon using a graphical event dis- +play. +As a first step, all the events recorded in the +BNB and NuMI beam for some runs were studied +selecting the tracks in the cryostat where the trig- +ger signal has been produced. An interaction was +classified as a neutrino candidate if a clear vertex +with more than one track was visually identified: +electron neutrino CC candidate events require +the presence of a clear electromagnetic shower +connected to the primary vertex, while the muon +neutrino CC events are selected by requiring the +presence of a long track (at least 0.5 m) from +the primary vertex. In addition, only events with +the primary vertex at least 5 cm from top/bottom +TPC sides, 50 cm from the upstream/downstream +TPC wall, and 5 cm from the anode position have +been initially selected. +An example of a 𝜈𝜇CC candidate is shown in +Fig. 30, with an estimated total deposited energy +of ∼ 1.1 GeV. The CC muon candidate is 3.8 m +long, while the highly ionizing track from the pri- +mary vertex is identified as a 20 cm long proton. +The full wire signal calibration is in the finaliza- +tion stage, but by a very preliminary wire signal +conversion to estimate the deposited energy, it is +possible to reconstruct the dE/dx associated to +the individual hits of the muon candidate in the +same event, distributed as expected for a MIP +particle particle, as shown in Fig. 31. +Visual scanning also permitted identifica- +tion of 𝜈𝑒CC candidates in the NuMI beam: a +remarkable example is shown in Fig. 32 for an +event of ∼ 600 MeV deposited energy. +7.5 +Event reconstruction +For a given cryostat, hits identified and passing +a multi-plane matching algorithm are passed as +input to Pandora [36]: a pattern reconstruction +code that performs a 3D reconstruction of the +full image recorded in the collected event, in- +cluding the identification of interaction vertices +and of tracks and showers inside the TPC. These +are organized into a hierarchical structure (called +a slice) of particles generated starting from a pri- +mary interaction vertex or particle. +The analysis uses information reconstructed +in Pandora to tag and reject “clear cosmic” slices +by identifying straight tracks crossing the full ac- +tive liquid argon volume or that are clearly out +of time with respect to the beam gate. In Monte +Carlo studies, selection criteria require that the +reconstructed vertex is in the fiducial volume and +that PMT timing signals and the reconstructed +angle of the muon track are inconsistent with +that of a cosmic ray. +These requirements re- +ject 99.7% of cosmic rays, while accepting more +than 82% of true 𝜈𝜇CC events in the fiducial vol- +ume. Requiring that a particle identified as a pro- +ton be reconstructed in the event further reduces +background from cosmic rays. After all criteria +are applied, 0.8% of a selected 𝜈𝜇CC contained +sample is made up of background from cosmic +rays, with 0.6% coming from intime cosmic rays +and 0.2% coming from out-of-time cosmic rays. +Further tagging and rejection of cosmic rays out +of time with respect to the beam spill is possi- +ble with the CRT detector, which can provide a +few nanosecond absolute time measurement for +the TPC tracks when they are unambiguously +matched to signals on the CRT. This TPC track- +CRT hit matching algorithm is still being tuned +and validated with cosmic ray data collected off- +beam, but is expected to facilitate improved ef- +ficiency and allow further optimization of the +cosmic rejection criteria. +Pandora and a set of algorithms to iden- +– 25 – + +Figure 30. A visually selected 𝜈𝜇CC candidate from the BNB beam. +Figure 31. Distribution of the measured dE/dx of the +muon candidate in the event shown in Fig. 30. dE/dx +is reconstructed on each wire applying a preliminary +calibration constant. +tify, measure and reconstruct tracks and show- +ers can be exploited for the event reconstruction +and analysis. These reconstruction tools repre- +sent a legacy from past efforts and made avail- +able within the LArSoft framework [37], com- +plemented by new efforts carried out within the +joint SBN effort for a common near and far detec- +tor analysis. This set of algorithms is applied to +Figure 32. A visually selected 𝜈𝑒CC candidate from +the NuMI beam . +tracks and showers from any slice in the event to +perform particle identification and estimate the +momentum from range, calorimetry and multiple +Coulomb Scattering. +A dedicated visual study of events was per- +formed to select ∼ 600 𝜈𝜇CC interactions from +BNB in the active liquid argon. These events +have been used for validation of the Pandora +– 26 – + +Collection plane +p +Primary +vertex +Beam direction +Cathode90 +80 +Mean +2.235 +70 +RMS +1.211 +60 +50 +40 +30 +20 +10 +0 +9 +1 +2 +3 +4 +5 +6 +8 +dE/dx[MeV/cm]NuMI veCC +candidate +Track 1 +Track 2 +e-shower +(~600MeV) +COLL +1 m +Wiresreconstruction. In order to reduce the manual +effort, events to be visually studied are first se- +lected by requiring, offline, the absence of signals +in the CRT in coincidence with the trigger. In ad- +dition, full 3D reconstruction was performed for +the events and only reconstructed tracks longer +than 30 cm, fully contained in the detector, and +whose barycenter was in agreement within 1 m +with the barycenter of the light signal generating +the trigger, have been visually studied. For this +sample, the neutrino interaction vertex was iden- +tified and measured in 3D coordinates as well +as the final point associated with the muon can- +didate track. +Out of the full selected sample, +476 neutrino events present in the analysis files +showed a reasonable match with a reconstructed +object based on vertex location and were adopted +as a benchmark for the validation of the recon- +struction tools. +As an example, in ∼ 90% of +these events the reconstruction reasonably iden- +tifies the neutrino interaction vertex along the +beam direction, meaning the difference between +the two estimates is within 3 cm, as shown in +Fig. 33. +Comparison of the visual study to auto- +mated reconstruction, along with studies of +Monte Carlo simulation, will enable further un- +derstanding of where to focus efforts and im- +provements in the automatic reconstruction. For +example, in some cases inefficiencies in a wire +plane for a given event reconstruction leading to +loss of hits may impact some 3D steps and lead to +a track broken into one or more smaller pieces; +or algorithms may lead to improper clustering +or determination of particle types, etc. Further +tuning of the reconstruction is progressing, as +well as the complete calibration of the detec- +tor. However the first results are quite promis- +ing, demonstrating that the basic tools for the +event reconstruction and the event selection are +operational and allow an initial identification and +measurement of neutrino interactions. +10 +− +8 +− +6 +− +4 +− +2 +− +0 +2 +4 +6 +8 +10 + (cm) +vertex Z +∆ +0 +20 +40 +60 +80 +100 +120 +Slices +(scan-reco) +∆ +Figure 33. Difference Δ𝑍 between the automatic and +manual measured longitudinal (beam) coordinate of +the neutrino interaction vertex for a sample of 476 +𝜈𝜇CC candidates from the BNB beam. +Conclusions +After the successful three-year physics run at +the underground LNGS laboratories studying +neutrino oscillations with the CERN Neutrino +to Gran Sasso beam, the ICARUS T600 LAr- +TPC detector underwent a significant overhaul +at CERN and was then installed at Fermilab. +Detector activation began in 2020 with the cryo- +genic commissioning and, despite serious chal- +lenges and delays caused by prolonged restric- +tions related to the COVID-19 pandemic, it +started operations in 2021 and successfully com- +pleted its commissioning phase in 2022. It col- +lected neutrino events from both the Booster +Neutrino Beam (BNB) and the Main Injector +(NuMI) beam off-axis. +Data taking started in +June 2021 with the beam data acquisition, with +the detector commissioning activities being con- +ducted in parallel. An event sample correspond- +ing to ∼ 3 · 1020 and 5 · 1020 POT of the Booster +and NuMI beam respectively has been collected +with an efficiency exceeding 91% during the +normal operations. +This data set was used to +study the single detector subsystems calibration +and to test the ICARUS event selection and re- +construction procedure and analysis algorithms. +– 27 – + +ICARUS has already started the first year of reg- +ular data taking devoted to a sensitive study of the +claim by Neutrino-4 short baseline reactor exper- +iment both in the 𝜈𝜇 channel with the BNB and in +the 𝜈𝑒 channel with NuMI. ICARUS will also ad- +dress other fundamental studies such as neutrino +cross sections with the NuMI beam and a number +of Beyond Standard Model searches. The search +for evidence of a sterile neutrino jointly with the +Short-Baseline Near Detector, within the Short- +Baseline Neutrino program, will follow. +Acknowledgements +This document was prepared by the ICARUS +Collaboration using the resources of the Fermi +National Accelerator Laboratory (Fermilab), a +U.S. Department of Energy, Office of Science, +HEP User Facility. +Fermilab is managed by +Fermi Research Alliance, LLC (FRA), acting +under Contract No. +DE-AC02-07CH11359. +This work was supported by the US Depart- +ment of Energy, INFN, EU Horizon 2020 +Research and Innovation Program under the +Marie Sklodowska-Curie Grant Agreement No. +734303, 822185, 858199, and 101003460 and +Horizon Europe Program research and innova- +tion programme under the Marie Sklodowska- +Curie Grant Agreement No. 101081478. Part +of the work resulted from the implementation of +the research Project No. 2019/33/N/ST2/02874 +funded by the National Science Centre, Poland. +The ICARUS Collaboration would like to thank +the MINOS Collaboration for having provided +the side CRT panels as well as Double Chooz +(University of Chicago) for the bottom CRT pan- +els. We also acknowledge the contribution of +many SBND colleagues, in particular for the de- +velopment of a number of simulation, recon- +struction and analysis tools which are shared +within the SBN program. Finally, our experi- +ment could not have been carried out without +the major support of CERN in the detector over- +hauling within the Neutrino Platform framework +and of Fermilab in the detector installation and +commissioning, and in providing the BNB and +NuMI beams. +References +[1] C. Rubbia. The Liquid Argon Time Projection +Chamber: A New Concept for Neutrino +Detectors. CERN-EP, 77-08, 1977. +[2] A.A. 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PoS, ICHEP2016:182, 2016. +– 30 – + diff --git a/7tFAT4oBgHgl3EQfoR3-/content/tmp_files/load_file.txt b/7tFAT4oBgHgl3EQfoR3-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1a9b5086e90a526d3cb2c87d95791b9a24b134a5 --- /dev/null +++ b/7tFAT4oBgHgl3EQfoR3-/content/tmp_files/load_file.txt @@ -0,0 +1,1367 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf,len=1366 +page_content='ICARUS at the Fermilab Short-Baseline Neutrino Program Initial Operation P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Abratenko𝑎 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Aduszkiewicz𝑏 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Akbar𝑐 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Artero Pons𝑑 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Asaadi𝑒 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Aslin 𝑓 ,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Babicz𝑔,2 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Badgett 𝑓 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} 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Biagi𝑙 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Biery 𝑓 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Bitter 𝑓 ,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Bonesini𝑗 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Booneℎ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Bottino𝑚 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Braggiotti𝑑,4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Brailsford5 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Bremer𝑔 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Brice 𝑓 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Brio𝑖 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Brizzolari𝑗 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Brown 𝑓 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Budd𝑐 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Calaon𝑑 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Campani𝑚 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Carberℎ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Carneiro𝑛 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Caro Terrazasℎ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Carranza𝑒 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Casazza𝑚 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Castellani𝑑 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Castro𝑜 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Centro𝑑 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cerati 𝑓 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Chalifour𝑔 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Chambouvet𝑔 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Chatterjee𝑝 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cherdack𝑏 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cherubini𝑙 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Chithirasreemadam𝑞 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cicerchia𝑑 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cicero𝑘 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Coan𝑟 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cocco𝑠 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Convery𝑡 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Copello𝑢 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cristaldo6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Dange𝑒 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' de Icaza Astiz7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' De Roeck𝑔 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Di Domizio𝑚 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Di Noto𝑚 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Di Stefano𝑙 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Di Ferdinando𝑘 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Diwan𝑛 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Dolan𝑔 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Domine𝑡 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Donati𝑞 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Doubnik 𝑓 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Drielsma𝑡 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Dyerℎ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Dytman𝑣 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fabre𝑔 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fabris𝑑 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Falcone𝑗 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Farnese𝑑 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fava 𝑓 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ferguson 𝑓 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ferrari𝑤 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ferraro𝑚 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gallice𝑤 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Garcia𝑡 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Geynisman 𝑓 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Giarin𝑑 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gibin𝑑 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gigli𝑢 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gioiosa𝑞 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gu𝑛 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Guerzoni𝑘 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Guglielmi𝑑 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gurung𝑒 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hahn 𝑓 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hardin 𝑓 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hausner 𝑓 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Heggestuenℎ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hilgenbergℎ,8 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hoganℎ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Howard 𝑓 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Howell𝑐 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hrivnak𝑔 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Iliescu𝑘,9 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ingratta𝑘 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' James 𝑓 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Jang𝑒 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Jung𝑥,10 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Jwa𝑡 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Kashurℎ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ketchum 𝑓 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Kim𝑐 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Koh𝑡 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Kose𝑔,11 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Larkin𝑛 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Laurenti𝑘 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Lukhanin 𝑓 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Marchini𝑑 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Marshall𝑐 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Martynenko𝑛 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Mauri𝑘 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Mazzacane 𝑓 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' McFarland𝑐 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Méndez𝑛 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Menegolli𝑢,12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Meng𝑑 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Miranda𝑜 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Mladenov𝑔 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Moganℎ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Moggi𝑘 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Montagna𝑘 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Montanari 𝑓 ,13 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Montanari𝑘 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Mooneyℎ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Moreno-Granados𝑜 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Muellerℎ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Naples𝑣 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Nebot-Guinot14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Nessi𝑔 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Nichols 𝑓 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Nicoletto𝑑 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Norris 𝑓 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Palestini𝑔 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pallavicini𝑚 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Paolone𝑣 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Papaleo𝑙 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pasqualini𝑘 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Patrizii𝑘 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Peghin𝑑 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Petrillo𝑡 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Petta𝑖 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pia𝑘 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pietropaolo𝑔,15 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Poirot𝑔 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Poppi𝑘 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pozzato𝑘 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Prata𝑢 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Prosser 𝑓 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Putnam𝑤 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Qian𝑛 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rampazzo𝑑 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rappoldi𝑢 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Raselli𝑢 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rechenmacher 𝑓 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Resnati𝑔 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ricci𝑞 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Riccobene𝑙 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rice𝑣 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Richards𝑣 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rigamonti𝑔 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rosenberg𝑎 1Now at University of Wisconsin, Madison, USA 2Also at INP-Polish Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sci, Krakow,Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Now at University of Zurich, Switzerland 3Now at Northwestern University, USA 4Also at Istituto di Neuroscienze, CNR, Padova, Italy 5SBND Collaboration, Lancaster University, UK 6SBND Collaboration, Universidad Nacional de Asuncion, San Lorenzo, Paraguay 7SBND Collaboration, University of Sussex, UK 8Now at University of Minnesota, USA 9Now at INFN-LNF 10SBND Collaboration 11Now at ETH Zurich, Switzerland 12Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 13on leave of absence from INFN Pavia, Italy 14SBND Collaboration, University of Edinburgh, UK 15On leave of absence from INFN Padova, Italy arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='08634v1 [hep-ex] 20 Jan 2023 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rossella𝑢 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rubbia𝑦 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sala𝑤 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sapienza𝑙 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Savage 𝑓 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Scaramelli𝑢 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Scarpelli𝑛 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Schmitz𝑥 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Schukraft 𝑓 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sergiampietri𝑔,16 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sirri𝑘 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Smedley𝑐 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Soha 𝑓 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Spanu 𝑗 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Stanco𝑑 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Stewart𝑛 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Suarez𝑣 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sutera𝑖 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Tanaka𝑡 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Tenti𝑘 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Terao𝑡 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Terranova 𝑗 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Togo𝑘 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Torretta 𝑓 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Torti 𝑗 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Tortorici𝑖 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Tosi𝑘 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Tsai𝑡 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Tufanli𝑔 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Turcato𝑑 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Usher𝑡 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Varanini𝑑 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ventura𝑑 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Vercellati𝑢 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Vicenzi𝑚 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Vignoli𝑧 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Viren𝑛 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Warnerℎ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Williams𝑒 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Wilsonℎ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Wilson 𝑓 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Wolfs𝑐 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Wongjirad𝑎 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Wood𝑏 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Worcester𝑛 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Worcester𝑛 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Wospakrik 𝑓 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Yu𝑛 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Yu𝑒 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Zani𝑤 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Zatti𝑑 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Zennamo 𝑓 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Zettlemoyer 𝑓 C.' metadata={'source': 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+page_content=' Italy 𝑡SLAC National Acceleratory Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Menlo Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' CA 94025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' USA 𝑢INFN Sezione di Pavia and University of Pavia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pavia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Italy 𝑣University of Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pittsburgh,' 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+page_content='it 16Now at IPSI-INAF Torino, Italy Abstract: The ICARUS collaboration employed the 760-ton T600 detector in a successful three- year physics run at the underground LNGS laboratory studying neutrino oscillations with the CERN Neutrino to Gran Sasso beam (CNGS) and searching for atmospheric neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ICARUS performed a sensitive search for LSND-like anomalous 𝜈𝑒 appearance in the CNGS beam, which contributed to the constraints on the allowed parameters to a narrow region around 1 eV2, where all the experimental results can be coherently accommodated at 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='. After a significant overhaul at CERN, the T600 detector has been installed at Fermilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In 2020, cryogenic commissioning began with detector cool down, liquid argon filling and recirculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ICARUS has started operations and successfully completed its commissioning phase, collecting the first neutrino events from the Booster Neutrino Beam (BNB) and the Neutrinos at the Main Injector (NuMI) beam off-axis, which were used to test the ICARUS event selection, reconstruction and analysis algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The first goal of the ICARUS data taking will then be a study to either confirm or refute the claim by Neutrino-4 short baseline reactor experiment both in the 𝜈𝜇 channel with the BNB and in the 𝜈𝑒 with NuMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ICARUS will also address other fundamental studies such as neutrino cross sections with the NuMI beam and a number of Beyond Standard Model searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After the first year of operations, ICARUS will commence its search for evidence of a sterile neutrino jointly with the Short Baseline Near Detector, within the Short-Baseline Neutrino program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Keywords: Large detector systems for particle and astro-particle physics, Liquid Argon, Time Projection Chambers (TPC) Contents 1 Introduction 2 2 The ICARUS-T600 detector 3 3 The overhaul of ICARUS-T600 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 The TPC electronics 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 The scintillation light detection system 5 4 The Cosmic Ray Tagger 6 5 First operations at FNAL 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 Cryogenic plant installation 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 TPC electronics installation 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 PMT system installation 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 Cosmic Ray Tagger installation 10 6 ICARUS-T600 commissioning 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 TPC commissioning 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 PMT commissioning 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 CRT commissioning 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 Triggering on the BNB and NuMI neutrinos 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 DAQ implementation 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 First operations with the BNB and NuMI 21 7 Observation and reconstruction of neutrino events 21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 Wire signal reconstruction 21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 PMT signal reconstruction 22 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 CRT reconstruction 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 Event display study 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 Event reconstruction 25 – 1 – 1 Introduction The Liquid Argon Time Projection Chamber (LAr-TPC) is a continuously sensitive and self triggering detector that can provide excellent 3D imaging and calorimetric reconstruction of any ionizing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' First proposed by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rub- bia in 1977 [1], this detection technique allows a detailed study of neutrino interactions, span- ning a wide energy spectrum (from a few keV to several hundreds of GeV), as demonstrated by the first large scale experiment performed by the ICARUS Collaboration at the LNGS under- ground laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Several experiments, in particular the Liq- uid Scintillator Neutrino Detector (LSND) [2] and MiniBooNE [3], have reported anomalous signals that may imply the presence of additional (mass-squared difference Δ𝑚2 𝑛𝑒𝑤 ∼ 1 eV2) flavor oscillations at small distances pointing toward the possible existence of nonstandard heavier sterile neutrino(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A sensitive search for a possi- ble 𝜈𝑒 excess related to the LSND anomaly in the CNGS 𝜈𝜇 beam has already been performed us- ing the neutrino events collected in the ICARUS- T600 detector during the Gran Sasso run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A total of 2,650 CNGS neutrino interactions, identified in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='9·1019 POT (Protons On Target) exposure, have been studied to identify the 𝜈𝑒 interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Globally, 7 electron-like events have been ob- served to be compared to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 expected from the intrinsic beam contamination and standard 3-flavor oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This study constrained the LSND signal to a narrow parameter region at sin22𝜃 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='005, Δ𝑚2 < 1 eV2, which requires further investigation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The primary goal of the Short-Baseline Neutrino (SBN) program at Fermilab is to fur- ther investigate the possibility of sterile neutri- nos in the O(1 eV) mass range and provide the required clarification of the LSND anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' It is based on three LAr-TPC detectors (ICARUS- T600, with 476 tons active mass, MicroBooNE with 89 tons active mass and SBND with 112 tons active mass) exposed at shallow depth to the ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 GeV Booster Neutrino Beam (BNB) at different distances from the target (600 m, 470 m and 110 m respectively) [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The detection technique used will provide an unambiguous identification of neutrino in- teractions, measurement of their energy and a strong mitigation of possible sources of back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Performing this study with almost iden- tical detectors at various distances from the neu- trino source allows identification of any variation of the spectra, which is a clear signature of neu- trino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In particular, SBN will allow for a very sen- sitive search for 𝜈𝜇 → 𝜈𝑒 appearance signals, covering the LSND 99% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' allowed region at ∼ 5𝜎 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The high correlations between the event samples of the three LAr-TPC’s and the huge event statistics at the near detector will also allow for a simultaneous sensitive search in the 𝜈𝜇 disappearance channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' During data taking at Fermilab, the 760- ton T600 detector is also exposed to the off-axis neutrinos from the Neutrinos at the Main Injec- tor (NuMI) beam, where most of events are in the 0 – 3 GeV energy range, with an enriched component of electron neutrinos (few %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The analysis of these events will provide useful infor- mation related to detection efficiencies and neu- trino cross-sections at energies relevant to the future long baseline experiment with the multi- kiloton DUNE LAr-TPC detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In addition to the LSND anomaly, ICARUS will test the oscillation signal reported by the Neutrino-4 experiment [7] both in the 𝜈𝜇 and 𝜈𝑒 channels with the BNB and NuMI beams, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This paper is organized as follows: in Sec- tion 2 the ICARUS-T600 detector is described with a particular emphasis on its achievements during three years data taking at the INFN LNGS underground laboratories in Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' in Section 3, – 2 – the ICARUS-T600 overhauling activities, most of which were carried out at CERN in the Neu- trino Platform framework [8], are shown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the new Cosmic Ray Tagger (CRT) detector, used to mitigate the cosmic ray background due to operating ICARUS at shallow depth, is detailed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In Section 5, the first operations of ICARUS at Fermilab, in particular the instal- lation of the cryogenic plant, TPC electronics, scintillation light detection system and CRT are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A successful commissioning phase followed soon after as described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Finally, the procedure for the selection, recon- struction, and analysis of the first collected BNB and NuMI off-axis neutrino events is introduced in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 2 The ICARUS-T600 detector The ICARUS-T600, with a total active mass of 476 ton, is the first large-scale operating LAr- TPC detector [9]: it consists of two large and identical adjacent modules with internal dimen- sions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='9 × 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 m3, filled with a total of 760 tons of ultra-pure liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each mod- ule houses two LAr-TPCs separated by a com- mon cathode with a maximum drift distance of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 m, equivalent to ∼ 1 ms drift time for the nominal 500 V/cm electric drift field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The cath- ode is built up by an array of nine panels made of punched stainless-steel, allowing for a 58% op- tical transparency between the two drift regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The anode is made of three parallel wire planes positioned 3 mm apart, where the stainless-steel 100 µm wires are oriented on each plane at a different angle with respect to the horizontal di- rection: 0◦ (Induction 1), +60◦ (Induction 2) and -60◦ (Collection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In total, 53,248 wires with a 3 mm pitch and length up to 9 m are in- stalled in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' By appropriate voltage biasing, the first two planes (Induction 1 and In- duction 2) provide a nondestructive charge mea- surement, whereas the ionization charge is fully collected by the last Collection plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Photo- Multiplier Tubes (PMTs) are located behind the wire planes to collect the scintillation light pro- duced by charged particles in LAr and used for the trigger of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In 2013, ICARUS concluded a very suc- cessful 3-year long run in the Gran Sasso under- ground laboratory [10], demonstrating the feasi- bility of the LAr-TPC technology at the kiloton scale in a deep underground environment and paving the way to the construction of the next generation of experiments dedicated to study neutrino oscillation physics such as DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Dur- ing the data taking, the liquid argon was kept at an exceptionally high purity level (< 50 ppt of O2 equivalent contaminants) reaching in 2013 a 16 ms lifetime corresponding to 20 ppt O2 equivalent LAr contamination [11], demonstrat- ing the possibility to build larger LAr-TPC de- tectors with drift distances up to 5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The detector has been exposed to the CNGS neutrino beam and to cosmic rays, recording events that demonstrate high-level performance and the physical potential of this detection tech- nique: the detector showed a remarkable 𝑒/𝛾 separation and particle identification exploiting the measurement of 𝑑𝐸/𝑑𝑥 versus range [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The momentum of escaping muons has been measured by studying the multiple Coulomb scattering with ∼ 15% average resolution in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 – 4 GeV/c energy range, which is relevant for the next generation neutrino experiments [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Events related to cosmic rays have been studied to identify atmospheric neutrino interac- tions: 6 𝜈𝜇CC and 8 𝜈𝑒CC events in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='43 kton·y exposure have been identified and reconstructed, demonstrating that the automatic search for the 𝜈𝑒CC in the sub-GeV range of interest for the future short and long baseline neutrino experi- ments is feasible [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' – 3 – 3 The overhaul of ICARUS-T600 The ICARUS-T600 detector at Fermilab takes data at shallow depth, shielded by a ∼ 3-meter concrete overburden: neutrino interactions must be recognized among the ∼ 11 cosmic muons that are expected to cross the detector randomly in the 1 ms drift time during each triggered event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' High-energy photons produced by cosmic rays can become a serious background source for the 𝜈𝑒 search since the electrons produced via Comp- ton scattering and pair production can mimic 𝜈𝑒CC events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In order to prepare the detector for SBN data taking, the T600 underwent an intensive overhaul at CERN in the Neutrino Platform framework (WA104/NP01 project) before being shipped to the USA in 2017, introducing several technology developments while maintaining the achieved performance at Gran Sasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The refurbishing mainly consisted of: the realization of new cold vessels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 1) with purely passive insulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' an update of the cryogenics and of the LAr purifi- cation equipment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' flattening of the TPC cathode (the punched hole stainless-steel panels under- went a thermal treatment improving the planarity to a few mm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the implementation of new, higher performance TPC read-out electronics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the up- grade of the LAr light detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 The TPC electronics The electronics used at LNGS was based on flange modularity, each flange serving 576 TPC wire-channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The analogue front-end was a Radeka type amplifier, using a custom BiCMOS chip to integrate the cascode stage with two dif- ferent filtering, one for Collection and Induc- tion 1, another for Induction 2 with the aim to produce in all the cases a unipolar signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This solution, however, showed strong limita- tions in the Induction 2 signals in the case of dense showers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Analog signals were converted to digital via multiplexers by 10-bit ADCs with Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' One of the two new ICARUS cryostats during its assembly at a CERN workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' sampling rate of 400 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The analogue circuits were housed in a custom crate, connected to the flange by flat cables, with 18 boards (32 chan- nels per board).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Analogue boards had a digital link to corresponding digital modules hosted in VME crates that contained memory buffers and performed lossless data compression and data transmission through a VME bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Both crates were housed in a rack next to the flange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' One of the largest tasks of the overhauling program was the design of new electronics for the 53,248 wire-channels that would be compat- ible with higher data rates foreseen at shallow depth operation at FNAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The new electronics adopts the same modularity and architecture but takes advantage of newer technology that allows for integrating both the analogue and the digital electronics on the same board on a custom crate mounted onto the flange [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' New packaging for the BiCMOS custom cascode allowed the design of a small piggyback module with 8 amplifiers and to house 8 of these modules on a single board serving 64 channels, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 2 (top-left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The digital part is also com- pletely contained in the same board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Moreover, all the amplifiers now have the same filtering, preserving the bipolar structure of Induction 2 signals without distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each amplifier is fol- – 4 – lowed by a serial 12-bit ADC avoiding the cum- bersome signal multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The digital part is based essentially on a large powerful FPGA allowing the possibility to use different signal treatments if required from running experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The VME standard was abandoned in favor of a serial optical link, allowing for gigabit band- width data transmission compatible with shallow depth data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A2795 custom board housing 64 amplifiers (far end), AD converter, digital control, and optical link (top-left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' An assembled feed-through with nine DBBs and the biasing cables (top-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A mini- crate populated by the nine A2795 boards installed on a feed-through flange (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' TPC wire signals are fed into the front- end amplifiers by means of Decoupling Biasing Boards (DBBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The DBB has two functions: biasing of each wire and conveying, with block- ing capacitors, the signals to the amplifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The DBBs work in argon gas and can withstand up to 400 V input biasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The flange CF250 is re- alized with a G10 multi-layer solid PCB, about 6 mm thick with three internal layers of copper to guarantee the required stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' SMD exter- nal connectors provide receptacles for the A2795 boards, while another set of SMD connectors in correspondence (inner side) provide receptacles for DBBs, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 2 (top-right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Finally, nine electronic A2795 boards are hosted by a mini- crate which is installed on a feed-through CF250 flange, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 2 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 The scintillation light detection system A new light detection system that is sensitive to the photons produced by the LAr scintillation is a fundamental feature for the T600 operation at shallow depth (contributing to the rejection of the cosmic background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The light detection system complements the 3D track reconstruction, unam- biguously providing the absolute timing for each track and identifying the interactions occurring in the BNB and NuMI spill gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS-T600 light detection system consists of 360 8" Hamamatsu R5912-MOD PMTs deployed behind the 4 wire chambers, 90 PMTs per TPC [16, 17], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Since the PMT glass is not transparent to the 128 nm wavelength scintillation light produced in liquid argon, each unit is provided with a ≈ 200 µg/cm2 coating of Tetra-Phenyl Butadiene (TPB), to con- vert the VUV photons to visible light [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' All PMTs are mounted onto the wire cham- ber mechanical frames using a supporting sys- tem, that allows the PMT to be positioned about 5 mm behind the Collection planes wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A stainless steel grid cage is mounted around each PMT to mitigate the induction of fake signals on the nearby wire planes by the relatively large PMT signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The light detection setup, realized by INFN, is complemented by a laser calibration system allowing for gain equalization, timing and moni- – 5 – Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The new ICARUS PMTs mounted behind the wires of one TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' toring of all the PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Laser pulses (𝜆 = 405 nm, FWHM = 60 ps), generated by a laser diode head (Hamamatsu PLP10), are sent to each PMT win- dow by means of a light distribution system based on optical fibers, light splitters and an optical switch [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 4 The Cosmic Ray Tagger ICARUS-T600 based at FNAL faces more chal- lenging experimental conditions than at LNGS: due to its shallow depth operation, identifica- tion of neutrino interactions among 11 kHz of cosmic rays is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A ∼ 3-meter concrete overburden was designed to almost completely remove the contribution from charged hadrons and high energy photons [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' However, ∼ 11 muon tracks occur per triggered event in the 1 ms TPC drift readout;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' photons associated with the muons represent a serious background for identi- fying 𝜈𝑒 candidates since electrons produced via Compton scattering/pair production can mimic a genuine 𝜈𝑒CC event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Rejecting the cosmic background, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' re- constructing the triggering event, requires to know precisely the timing of each track in the TPC image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Operating at FNAL, ICARUS ex- ploits an improved light detection system with high granularity and 𝑂(1 ns) time resolution, and an external ∼ 4𝜋 high coverage Cosmic Ray Tag- ger (CRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The primary function of the CRT is to tag muons passing through or near the cryostats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Timestamps associated to a particle tagged by the CRT are compared with timestamps from PMT signals, both with a few nanosecond res- olution, allow the determination of whether an interaction in the TPC originated from an outside cosmic ray or from an internal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS CRT consists of a top, side and bottom subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS Top CRT system is divided in 123 detector modules covering a surface of about 426 m2: 84 horizontal and 39 vertical modules along the perimeter of the cryostat top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Its design is such that more than 80% of the cos- mic muon flux is intercepted by the Top CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each module is a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='86 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='86 m2 aluminum box containing two orthogonal layers of eight scintillator bars for position reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The bars, coated with white paint, are 23 cm wide, 184 cm long and have different thickness de- pending on the layer: 1 cm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 cm for the top layer and the bottom layer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In each scintillator, the light is collected by two wave-length shifting (WLS) fibers Kuraray Y- 11(200) then read out from one end by a Silicon Photo-Multiplier (SiPM), Hamamatsu S13360- 1350C model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The 32 SiPM signals of one mod- ule are routed via 50 Ω micro-coaxial cables to a patch panel connected to the CAEN DT5702 Front End Board (FEB) which provides a bias voltage adjustable for each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The FEB triggers on the coincidence between two SiPM signals of the same bar and provides a coinci- dence logic between the two scintillator layers in the module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 4, a picture of a vertical Top CRT module installed in the detector hall is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Top CRT was a brand new detector – 6 – designed and built by INFN and CERN before shipping to Fermilab in summer 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Front End Board Aluminum Box containing Top CRT module Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Picture of a vertical TOP CRT module installed in the detector hall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS Side CRT makes use of scin- tillator modules formerly used by the MINOS ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each module is composed of twenty adjacent strips of 800 × 4 × 1 cm3 Polystyrene (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0% PPO, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='03% POPOP) scintillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The full Side CRT system consists of 2,710 readout channels across 93 FEBs, with 136 full and 81 cut modules in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The scintillator is con- tained in a metal sheath and each strip has an embedded WLS fiber running down the mid- dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These fibers are collected into “snouts” at the ends of the modules, onto which the opti- cal readout, consisting of an array of ten Hama- matsu S14160-3050HS SiPMs, is mounted onto a snout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each SiPM reads out two fibers and cor- responds to a single electronic readout channel on CAEN A1702 Front-End Boards (FEBs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A full MINOS module has two snouts, one on each end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS Side CRT System is double layered, with an inner and outer layer of MINOS modules to apply coincidence logic between the two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' To account for geometric constraints, some MINOS modules were cut and sealed on the cut end with mylar and tape to only have a single snout for readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The South Side CRT wall consists of an inner and outer layer of cut modules oriented orthogonally in an X-Y con- figuration, with the added benefit of improved position reconstruction on the southern side of the TPCs, upstream along the BNB beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The East and West walls utilize full length MINOS modules mounted horizontally, while the North Wall use cut modules mounted horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Bottom CRT consists of 14 modules di- vided into two daisy chains of 7 modules each, positioned underneath the warm vessel in a north and south section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These modules are refur- bished veto modules from the Double Chooz re- actor neutrino experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each module consists of 64 Polystyrene scintillator strips, running in parallel and divided into two layers of 32 strips offset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 cm from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Scintillation light is collected in a WLS optical fiber and read out at one end of each strip by an Hamamatsu H7546B M64 multi-anode PMT, while the other end is mirrored to maximize light collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5 First operations at FNAL Following the overhauling activities at CERN, ICARUS-T600 was shipped to Fermilab in July 2017 and the two cryostats hosting the TPCs were finally deployed in their shallow depth position in August 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Work began soon after to install and test all main subsystems before the cryogenic commissioning, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 Cryogenic plant installation The ICARUS cryogenic plant was designed, built and installed at Fermilab by a collabora- tion of three international institutions, CERN, INFN and Fermilab to support operations of the ICARUS LAr-TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For the installation at Fer- milab, the entire ICARUS-T600 cryogenic and purification system was rebuilt anew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The new design followed closely the original implementa- tion at the LNGS with one important exception: – 7 – Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Deployment of the ICARUS cryostats inside the pit of the SBN Far Detector experimental hall at Fermilab in August 2018 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Installation of TPC, PMT and laser feed-through flanges in December 2018 (center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Status of the ICARUS detector at the beginning of data taking for commissioning (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' at Fermilab, the LN2 boiloff is vented to the atmosphere (open loop cooling circuit), while at LNGS the LN2 boiloff was re-condensed by means of a set of cryocoolers (closed loop cool- ing circuit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The main components of the cryo- genic and purification system are the following: Main LAr containers (2× cold vessels): 273 m3 each, containing the TPC detectors and the LAr scintillation light system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cold shields: set of heat exchangers filled with LN2, completely surrounding the main LAr containers and designed to pre- vent heat, coming from the thermal insu- lation, to reach the LAr volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Thermal insulation: polyurethane foam panels, ∼ 600 mm thick, surrounding the cold shields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Warm vessel: provides enclosure and me- chanical support for the thermal insula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' LN2 cooling circuits: piping, circulation pumps, regulating valves, phase separa- tors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=', providing LN2 supply to heat exchangers serving the cold shields and the purifying units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Argon gas recirculation units (4×, two per cold vessel): set of units that re-condense and purify the argon flowing from the gas phase on top of the main LAr containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Liquid argon recirculation units (2×, one per cold vessel): provide forced circula- tion, with a cryogenic pump, of argon coming from the cold vessels through a set of purifiers before injecting it back into the cold vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cryogenic control system: to provide automation, data display, recording and alarming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' LN2 and LAr storage dewars and relative transfer lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A dedicated purification unit used for the filling of the cold vessels, equipped with a regeneration system and a set of gas ana- lyzers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS cryogenic plant at the SBN Far Detector Hall at Fermilab was fully designed, delivered, and installed by July 2019, with the commissioning phase started by January 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The equipment included the ICARUS cryogenic plant is schematically divided into the external components supplied by Fermilab, the proximity components supplied by Demaco Holland B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' under contract with CERN and components in- ternal to the cryostats supplied by INFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6 shows the ICARUS plant physical layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' – 8 – lcarusTritFigure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ICARUS cryogenic plant physical layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 TPC electronics installation Each mini-crate, housing nine A2795 boards, was mounted onto the flange on top of the chim- ney that contains flat cables connecting wires of the chambers to DBBs and powered by a linear power supply next to the chimney, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each set of nine A2795 in a single crate are read out through two fibers that implement a CAEN proprietary protocol named CONET (Chain-able Optical NETwork).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The two sets of fibers are read through an A3818 PCI Express board in- stalled in dedicated PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The full TPC electronics (96 mini-crates) is synchronized by a serial link (one cable), named TTLink, which sends clock, trigger, and com- mands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The TTLinks are distributed to all mini- crates by four fan-out modules with the same cable lengths to guarantee equal time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The TPC electronics system is fully installed and op- erational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 PMT system installation Electrical connections between PMTs and elec- tronics, located in a building alcove adjacent to the detector, were realized by means of 360 sig- nal cables and 360 high voltage cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Sig- nal cables are RG316/U, 7 m of which are de- Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Two Low Voltage Power Supply (LVPS) modules powering the two adjacent mini-crates pop- ulated with nine A2795 boards, serving 576 wires each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ployed inside the detector and 37 m outside, the two parts connected by means of BNC-BNC feedthrough flanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' High voltage cables are 7-m long HTC-50-1-1 deployed inside the de- tector and 37 m RG58/U outside;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the two parts connected by means of SHV-SHV feedthrough flanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Power supply voltages are generated and distributed by 8 CAEN A7030 boards, each with 48 channels that can provide 3 kV, housed in two CAEN SY4527 crates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The PMT electronics are designed to allow continuous read-out,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' digitization and indepen- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='– 9 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Proximity cryogenics on detector top: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='LN2 shields valve boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='LAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='GAr re-condensers valve boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Transfer lines and gas collection piping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Fill Filter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='External cryogenics: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='LAr and LN2 dewars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Transfer and vent lines ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Proximity cryogenics in pit and mezzanine: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Regeneration skids for filter media ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='LAr pumps valve boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Gas analyzers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='LAr filters valve boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Process controls system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='LN2 Phase separator and pumps valve boxes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='← 23 m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='Safety controls systemWE05/06 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='TRIPP-LITE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='CINENdent waveform recording of signals coming from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='the 360 PMTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This operation is performed by 24 CAEN V1730B digitizers installed in 8 VME crates (3 digitizers per crate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each module con- sists of a 16-channel 14-bit 500-MSa/s FLASH ADC with 2 Vpp input dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In each board 15 channels are used for the acquisition of PMT pulses, while one channel is used for the acquisition of ancillary signals such as the beam gates and the trigger pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For each channel, an internal trigger-request logic signal is generated every time the sam- pled PMT pulse passes through a programmable threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For each couple of adjacent channels, trigger-requests are logically combined (OR, AND, Ch0, Ch1) and the result is presented in a low-voltage differential signaling (LVDS) logic output with settable duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For triggering pur- poses, an OR logic between neighboring PMTs is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A total of 192 LVDS lines (8 lines per digitizer) are connected to the ICARUS trig- ger system for exploiting the scintillation light information for trigger purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The PMT electronics are complemented by a common 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 MHz clock distribution system, an external trigger network, an external time- stamp reset network, and 24 optical link inter- faces based on the CAEN CONET2 protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 Cosmic Ray Tagger installation The Side CRT system was installed over the pe- riod from November 2019 to April 2021 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 8 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Following its shipping in summer 2021, the installation of Top CRT modules was carried out and completed in December 2021 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 8 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' All Top and Side CRT modules were tested be- fore and after their installation to check for elec- tronic functionality of the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Data trans- mission to the servers is performed via ethernet cables connecting the modules in daisy chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The distribution of a Pulse Per Second (PPS) signal (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4) for absolute time reference and trigger signal to the FEBs was performed with lemo cables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A voltage of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 V to be pro- vided to the FEBs is distributed via power lines assembled at FNAL during the installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' All the information on modules to cables connec- tions, SiPM bias voltages, module positions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' are stored in a Fermilab SQL database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The last ICARUS installation activity was the deployment of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='85-meter concrete over- burden above the Top CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The overburden is composed of three layers of concrete blocks, each approximately 1-meter tall, giving a total mass of 5 million pounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The installation of the last concrete block was completed June 7, 2022, marking the beginning of ICARUS data taking for physics with both BNB and NuMI beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6 ICARUS-T600 commissioning After the placement of the two ICARUS modules in the pit in August 2018, all the feed-through flanges for the TPC and PMT signals and for the injection of the laser flashes used to calibrate the PMTs were installed in December 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The gain and the dark rate for all 360 PMTs were mea- sured as a function of the applied voltage at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' All the new TPC readout electron- ics in the 96 mini-crates and the low noise power supplies were installed and verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In particular the full readout chain has been tested by injecting test pulses in wires at the far end of the cham- ber and reading out the signals with the A2795 boards on the other end to check the full system for noise monitoring purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In parallel all the cryogenic equipment were installed, welded and the complete system has been tested at 350 mbar over-pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The cold vessels were then successfully brought to vac- uum, with a 10−5 mbar residual pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The cryogenic commissioning of the ICARUS-T600 detector started on February 13, 2020 by breaking the vacuum in the two main cold vessels with ultra-purified argon gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cool down started on February 14 by injecting liq- – 10 – Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Left: picture of the Side CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Right: Top CRT horizontal modules whose installation was completed in December 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' uid nitrogen in the cold shields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' It took about four days to bring the temperature on the wire chamber below 100 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The cooling process was continuous and the maximum temperature gra- dient on the wire chambers was about 35 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' On February 19, the gas recirculation units were put into operation to purify the argon gas before the start of the liquid filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The continuous filling with ultra-purified liquid argon started on February 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The filling was interrupted at around 50% to regenerate the filling filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The filling was stopped again when the liquid reached the −6 cm LAr level probes (6 cm below the nominal level) to perform the final pressure test of the two cold vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After the test, the gas recirculation units were put into operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The filling was completed on April 19, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' On April 21 the liquid recirculation was started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The recirculation rates were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='85 m3/h in the West module and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='25 m3/h in the East module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The cryogenic stabilization phase was com- pleted around the end of May 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pressures and temperatures in the two modules were stable and no cold spots were observed on the exter- nal surface of the Warm Vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' At the start of the cryogenic commissioning, all activities in the detector building not related to cryogenics were suspended and the building was put in a high safety condition, with strong limitations to the presence of people onsite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' At the end of the liquid argon filling, after the final pressure test, the standard safety conditions were restored and regular activities on top of the detector could be restarted to complete the installation and test of all sub-detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' During the cryogenic commissioning, there were several activities both related to monitoring the status of the detectors (wire chambers, wires readout electronics, PMTs, CRT) and to develop- ments for the following detector commissioning phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Noise data have been continuously taken of wire readout electronics, PMTs and CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Ef- fects on the noise from the activation of the cryo- genic plant have been continuously monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Functionality and stress tests of the DAQ were conducted with several useful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The detector activation took place on Au- gust 27, 2020 when the TPC wire planes and the cathode high voltage (HV) were taken to nomi- nal voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' HV has remained stable at −75 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Significant currents were found only on a few wire bias and were addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' All PMTs were switched on and calibrated with the laser system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cosmic-ray interaction events were initially collected with a random 5 Hz trigger and data analyzed for calibration purposes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' electron – 11 – DUMMY Genie29Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Trend of the liquid argon level inside the two ICARUS cryostats during the filling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' lifetime, space charge, drift velocity measure- ments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Dedicated runs were also carried out for specific commissioning tasks, such as inves- tigation of TPC noise, PMT calibration with the laser system, DAQ upgrades/longevity tests, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' One of the first measurements carried out was the free electron lifetime 𝜏𝑒𝑙𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This parame- ter is fundamental for the monitoring of the liquid argon condition in the TPCs and to obtain the precise measurement of the energy deposition from the ionization charge signal in the collected events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The LAr purity is continuously moni- tored by measuring the charge attenuation along the drift path of the electron ionization signals generated by cosmic ray tracks crossing the de- tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A fast procedure has been setup starting from the method developed and used during the Gran Sasso run [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' it has been applied to the recorded data since the detector activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The 𝜏𝑒𝑙𝑒 measurement is based on a simpli- fied identification of the wire signals in the Col- lection plane and of the anode to cathode cross- ing muon tracks that have no indication of asso- ciated 𝛿-rays or electromagnetic showers along the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' It is used to provide a fast, real time, measurement within 5-10% precision dominated mostly by effects related to space charge and to the electron diffusion, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The steady state values of 𝜏𝑒𝑙𝑒, exceeding 3 ms in both cryostats, are high enough to al- low for efficient detection and reconstruction of ionizing events inside the active volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 TPC commissioning After the TPC wires were biased and the cath- ode HV was raised to nominal operating condi- tions, the TPC commissioning began.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' With the liquid argon at a sufficient level of purity, cos- mogenic activity in the detector can be used to study the detector response to ionization signals in the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' To characterize the performance of the ICARUS TPC, a variety of measurements were taken between August 2020 and May 2022 as summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Noise levels in the TPC can be measured us- ing the RMS of waveforms from the TPC read- out, with an equivalent noise charge (ENC) of roughly 550 e−/ADC [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Measured TPC noise levels at ICARUS are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 11, both before and after the filtering of coherent noise, which was performed across sets of 64 channels associated with the same front-end electronics board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' – 12 – 4 Start Gas and Liquid re-circulation 3,5 3 2, 5 EAST Module 2 WESTModule 1,5 1 0,5 0 0,00 200,00 400,00 600,00 800,00 1000,00 1200,00 [400,00 1600,00 Feb 19 : Filling Start Elapsed Time (hours) Apr 19 : Filling CompleteFigure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Trend of the drift electron lifetime in the two ICARUS cryostats during the commissioning phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The sharp decreases of the lifetime are due to programmed interventions on the LAr recirculation pumps or on the cryogenic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The lifetime is quickly recovered after the end of the interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Waveforms containing ionization signals are identified by simply applying a threshold and re- moving from consideration to ensure there is no bias to the noise measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The measure- ments were repeated with the cathode HV off and consistent results were obtained, validating the ionization signal identification methodology and indicating that a negligible amount of TPC noise is caused by interference from the cathode HV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The noise levels after coherent noise filter- ing shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 11 are consistent with previous noise measurements of the TPC electronics in a test setup [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fast Fourier transforms (FFTs) of the same noise waveforms used in the results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 11 are calculated for each of the three wire planes and averaged across the entire de- tector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' these results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' FFTs are shown both before and after coherent noise removal, showing the expected approxi- mate Rayleigh distribution of the intrinsic noise spectrum [22] on all three planes after coherent noise is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This provides strong evidence of extrinsic noise being almost completely re- moved from the TPC waveform data by the noise filtering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Induction 2 plane and Collection plane spectra show a similar normalization, which is expected given the same length of the wires of these two planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Induction 1 plane spec- trum has instead a larger normalization given the longer wires and thus a higher capacitance, in- creasing the intrinsic noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Further work is being carried out to understand the source of the coherent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In runs with sufficiently high electron life- time (most runs after the very beginning of commissioning in 2020), ionization signals from anode-cathode-crossing cosmic muons are used to evaluate the peak signal-to-noise ratio (PSNR) for minimum-ionizing particles (MIPs) in the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Anode-cathode-crossing cosmic muon tracks traverse the full drift length of the detector and therefore allow for knowledge of the drift coordinate of each ionization signal along the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 13 shows the PSNR of ioniza- tion signals for each plane using a large sample of cosmic muons in ICARUS data with coherent noise removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In this study, the peak signal (numerator in the ratio) is defined as the maxi- mum signal ADC value minus the baseline ADC value for the unipolar signals of the Collection plane and the absolute value of the maximum sig- – 13 – Lifetime [ms] West L East 30/Sep 31/Dec 01/Apr 01/Jul 01/Oct 31/Dec 01/Apr 2020 2020 2021 2021 2021 2021 2022 DateFigure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' TPC noise levels at ICARUS before and after filtering of coherent noise, as measured by waveform RMS in ADC counts (with ENC of roughly 550 e−/ADC [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Results are shown separately for the Induction 1 plane (left), Induction 2 plane (center), and Collection plane (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Mean values of the shown distributions are presented at the bottom of each figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 Induction 1 Raw Spectra Noise-Filtered Spectra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 Induction 2 0 100 200 300 400 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 Collection Frequency [kHz] Power [ADC2/kHz] Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fast Fourier transforms (FFTs) of noise waveform data collected by the ICARUS TPCs, be- fore and after filtering of coherent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Results are shown separately for the Induction 1 plane (top), In- duction 2 plane (middle), and Collection plane (bot- tom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' nal ADC value minus the minimum signal ADC value for the bipolar signals of the two induc- tion planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The noise level (denominator in the ratio) is the RMS of signal-removed waveforms from the same TPC channel in units of ADCs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cosmic muon tracks used in the PSNR measurement are required to be oriented at an angle of 20 degrees or less with respect to the anode plane, and have a "3D pitch" (track segment length corresponding to the ion- ization signal from a single wire) of 4 mm or less for the wire plane of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These selection criteria probe the phase space most relevant for beam neutrinos interacting in the detector, which have interaction products that travel mainly in the forward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Furthermore, only parts of the track within 2 cm to 10 cm of the anode are used in order to minimize impact from charge attenua- tion due to impurities in the liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 13 illustrates the performance of the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The detector enables robust identification of ionization signals embedded within electronics noise background, with more than 99% of the MIP ionization signals having a PSNR greater than four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Anode-cathode-crossing cosmic muon tracks are also used to make a measurement of ionization drift velocity in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The distance between the anode and cathode, 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 cm, is divided by the maximum ionization drift time, or the difference in time between the first and last ionization signals associated with the cosmic muon tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The latter measurement should yield the time it takes for ionization to drift from the cathode (one end of – 14 – Average Noise by Plane Full Noise Noise-Filtered Induction 2 Collection Induction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 Units 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 Arbitrary l 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 RMS[ADC] RMS [ADC] RMS [ADC] μ: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='80 ADC 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='02 ADC μ: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='51 ADC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='65 ADC μ: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='53 ADC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='47 ADC0 5 10 15 20 25 30 35 40 Peak Signal-to-Noise Ratio (Noise-Filtered) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='45 Arbitrary Units Induction 1 (Peak: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='74) Induction 2 (Peak: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='64) Collection (Peak: 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='05) Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Peak signal-to-noise ratio (PSNR) of ion- ization signals for each of the three TPC wire planes using cosmic muons in ICARUS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Coherent noise is removed from the TPC waveforms prior to iden- tification and measurement of the ionization signal amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' See text for details on the cosmic muon data selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the track) to the anode (other end of the track), so the ratio should provide the drift velocity of the ionization electrons in liquid argon at the nominal drift electric field of roughly 500 V/cm and temperature of roughly 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A correction is made to account for a small bias in precisely reconstructing the drift times as- sociated with the track end points, derived from Monte Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A Crystal Ball func- tion1 is then fit to the maximum ionization drift time distribution associated with cosmic muon tracks in each TPC volume (two per cryostat), with the peak value of each fit used in the ion- ization drift velocity calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The results of the ionization drift velocity measurements in the west cryostat are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The results of the measurements, roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1572 cm/µs for both TPC volumes in the west cryostat, agree with the predicted value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1576 cm/µs to within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3% [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 1The Crystal Ball function, named after the Crystal Ball Collaboration, is a probability density function com- monly used to model various lossy processes in high-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' It consists of a Gaussian core portion and a power- law low-end tail, below a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 880 900 920 940 960 980 1000 Maximum Ionization Drift Time [ s] 0 5000 10000 15000 Tracks TPC E Drift V: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1572 cm/ s TPC W Drift V: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1572 cm/ s Ionization Drift Velocity: West Cryostat TPC E Fit TPC E Data TPC W Fit TPC W Data Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Results of the ionization drift veloc- ity measurement using ICARUS cosmic muon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Shown are Crystal Ball fits to the maximum ioniza- tion drift time distributions associated with anode- cathode-crossing cosmic muons in the two TPCs in the west cryostat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Electric field distortions in near-surface LAr-TPCs can arise due to the accumulation of space charge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' slow-moving positively- charged argon ions originating from cosmic muon ionization within the detector [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These argon ions, which drift slowly toward the cath- ode at a drift velocity of several millimeters per second at a drift electric field of 500 V/cm [24], linger around long enough to create substantial electric field distortions that pull ionization elec- trons toward the middle of the TPC volume as they drift toward the anode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These electric field distortions lead to biases in reconstructing the point of origin of ionization within the detector, a secondary effect referred to as "spatial distor- tions" in LAr-TPC detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' collectively, these two related distortions are referred to as space charge effects (SCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Using anode-cathode-crossing cosmic muon tracks, the magnitude of SCE in the ICARUS detector is estimated by utilizing methodology developed to measure SCE in previous near-surface running of the ICARUS detector [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The results of measurements in the two TPC volumes of the west cryostat are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 15, where they are compared – 15 – to a calculation of SCE [24] used in ICARUS Monte Carlo simulations prior to measuring the magnitude of SCE in ICARUS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The magnitude of SCE is observed to be very similar in the two TPC volumes, though underestimated by roughly 30% in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 Drift Coordinate X [cm] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 X [cm] ∆ Drift Direction Spatial Offset WE TPC Data WW TPC Data Calculation for Simulation X vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' X ∆ Data SCE Comparison: Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Measured spatial offsets in the drift di- rection as a function of ionization drift distance for the two TPCs in the west cryostat, evaluated us- ing anode-cathode-crossing cosmic muon tracks in ICARUS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The results are compared with pre- dictions of spatial distortions from a calculation of space charge effects (SCE) presently used in ICARUS Monte Carlo simulations (to be updated with data- driven SCE measurement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The energy scale of MIPs can be probed with cosmic muons that stop in the ICARUS de- tector, as done in similar calibrations performed at other LAr-TPC neutrino experiments [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The known profile of muon energy loss per unit length (𝑑𝐸/𝑑𝑥) in liquid argon as a function of kinetic energy [28] can be used to predict the value of 𝑑𝐸/𝑑𝑥 versus residual range, the dis- tance from the end of a stopped muon track in reconstructed TPC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After accounting for prompt electron-ion recombination [29] and charge attenuation during ionization drift due to electro-negative impurities in the detector, one can compare the most-probable value (MPV) of 𝑑𝐸/𝑑𝑥 versus residual range from a sample of stopping muons in ICARUS data (evaluated by fitting the data with a Landau distribution con- volved with a Gaussian, performed in bins of residual range) to the MPV 𝑑𝐸/𝑑𝑥 curve ex- pected from theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The result of the Collection plane energy scale calibration for the east TPC of the west cryostat is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 16 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Good agree- ment between calibrated data and predictions from theory is found for all values of stopping muon residual range after this calibration has been performed, with sub-percent agreement for values of 𝑑𝐸/𝑑𝑥 < 4 MeV/cm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' similar levels of agreement are observed for the other three TPCs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Additionally, the energy scale calibra- tion is further scrutinized by comparing two dif- ferent methods of stopping muon kinetic energy reconstruction: one by calorimetry (summing up charge associated with energy deposition along the track), 𝐸calo, and another by range (convert- ing distance from end of stopping muon track to kinetic energy by use of a look-up table [28]), 𝐸range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The result of this cross-check is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 16 (right), showing little bias between the two methods for stopping muons in ICARUS cos- mic muon data after the energy scale calibration is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Future measurements will include protons from ICARUS data, allowing for probing of the energy scale of highly-ionizing particles in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 PMT commissioning The whole light detection system was tested at Fermilab before the cooling of the detector, once the dark condition inside the cryostats was guar- anteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A total of 357 (out of 360) PMTs were found to be working with performances consis- tent with the tests performed at CERN [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The same number of working PMTs were found af- ter the filling of the detector with liquid argon, demonstrating the ability of this PMT model to withstand low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A PMT signal, recorded by the light detec- tion system electronics, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A gain calibration/equalization campaign was car- – 16 – 0 20 40 60 80 100 Residual Range [cm] 1 2 3 4 5 6 Calibrated dE/dx [MeV/cm] Predicted MPV dE/dx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 (Ecalo Erange) / Erange 0 5000 10000 15000 20000 25000 30000 Tracks 1: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7% 1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3% 2: 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5% 2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8% Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Calibrated Collection plane 𝑑𝐸/𝑑𝑥 as a function of residual range for a selection of stopping muons in ICARUS cosmic muon data, including a comparison to the most-probable value (MPV) of 𝑑𝐸/𝑑𝑥 from stopping muons predicted from theory [28] (left);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' comparison of cosmic muon kinetic energy reconstruction by calorimetry, 𝐸calo, and by range, 𝐸range, showing little bias between the two methods for stopping muons in ICARUS cosmic muon data after the energy scale calibration is applied (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ried out during the PMT commissioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' At first, external fast laser pulses focused on each PMT window by means of dedicated optical fibers were used to obtain a coarse gain curve for each PMT as a function of the applied voltage around the expected values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Laser pulses were also used to characterize, to within 1 ns precision, the delay response of each PMT channel, which can dif- fer due to different PMT and cable transit times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Voltages were set to values corresponding to a gain of 5 · 106, resulting in an equalization within 16%, as a first approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fine tuning was carried out to improve the gain equalization by means of an automatic pro- cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' To this purpose the response of each PMT to background single photons (≈ 250 kHz) was measured, and the voltages were adjusted according to the gain curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This procedure led to a final equalization with a spread less than 1%, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 CRT commissioning The side and top CRT modules were tested before the installation at ICARUS using a test stand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Af- ter the installation of all CRT modules, the cos- mic rate over time was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The event rates Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' PMT signal as recorded by the light de- tection system electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' for each wall of the side CRT as a function of time are constant, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The higher rates on north wall (black) are due to the proxim- ity with the cryogenic pumps, with these mod- ules experiencing higher electrical noise rates in addition to cosmic rates on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In addi- tion, the rates from the west north and east north walls are slightly higher from being closer to the cryogenics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Following work to characterize and mitigate the noise, electrical chokes (inductors) were installed along all Side CRT FEB power cables to reduce noise rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Top CRT cosmic event rates before and after the installation of concrete overburden are shown – 17 – 15000 Amplitude (ADC Counts 14900 14800 14700 14600 14500 0 2 4 6 8 10 Time (us)Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Gain distribution for 354 PMTs after the fine tuning equalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The automatic procedure was not applied on 6 PMTs (not present in the plot) that were manually calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 20 for horizontal (left) and vertical (right) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Before the installation of the over- burden the mean rate was ∼ 610 Hz and 260 Hz for horizontal and vertical modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After the installation of the overburden the rates reduced to 330 Hz and 180 Hz for horizontal and vertical modules, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Except for varia- tion due to concrete blocks placement above the detector, the rates are stable on a time scale of months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 12/23/20 12/30/20 01/06/21 Date 0 2 4 6 8 10 12 14 16 18 20 Rate [kHz] North Wall West North Wall West Central Wall West South Wall East North Wall East Central Wall East South Wall Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Side CRT cosmic event rates as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The black points corresponds to the rates from the north side CRT wall, the pink and blue points corresponds to East and West north walls, and the remaining walls are at 1 kHz rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 Triggering on the BNB and NuMI neu- trinos The initial ICARUS trigger system exploits the coincidence of the BNB and NuMI beams spills, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 µs and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 µs respectively, with the prompt scintillation light detected by the PMT system in- stalled behind the wire planes of each TPC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The generation of the beam spill gates is based on receiving the “Early Warning” (EW) signals for BNB and NuMI beams, 35 and 730 ms in advance of protons on target, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' LVDS signals from the PMT digitizers, in terms of the OR signal of adjacent PMTs, are pro- cessed by programmable FPGA logic boards to implement trigger logic for the activation of the ICARUS read-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Additional trigger signals are generated for calibration purposes in correspon- dence with a subset of the beam spills without any requirement on the scintillation light (Min- Bias trigger) and outside of the beam spills to detect cosmic ray interactions (Off-Beam trig- ger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' To synchronize all detector subsystems’ read-outs with the proton beam spill extraction at the level of few nanosecond accuracy, a White Rabbit (WR) network [31] has been deployed for distributing the beam extraction signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' An ab- solute GPS timing signal, in the form of PPS, is used as a reference for generating phase locked digitization clocks (62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 MHz for the PMT and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 MHz for the TPC) and for time-stamping the beam gates and trigger signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In addition, the signals of Resistive Wall Monitor detectors (RWM) at 2 GHz sampling frequency are also recorded to precisely measure the timing and the bunched structure of protons on target, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In the presence of a global trigger signal, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 ms and 30 µs acquisition windows are acti- vated for the TPC and PMT signal recording, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In addition, PMT waveforms are collected inside a 2 ms time window around the – 18 – PMTs 90 Entries 354 Constant 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='57 # 80 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4967 70 Sigma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='003574 60 50 40 30 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='46 0.' metadata={'source': 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+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 Rate [kHz] FEB 81 FEB 119 FEB 87 FEB 92 FEB 180 FEB 97 FEB 174 FEB 189 FEB 190 Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cosmic ray rates as a function of time for a set of Top CRT horizontal (left) and vertical (right) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Numbers in the legend indicate the module’s Front End Board and the black dot lines indicate the beginning and the end of 3 m overburden installation over the displayed modules: the rates reduced from ∼ 610 (260) Hz before to 330 (180) Hz after the installation of the overburden for the horizontal (vertical) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Layout of the trigger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' SPEXI board: synchronizes the whole ICARUS detector, generates clocks and readout signals, handles beam extraction messages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7820 FPGA boards: generate a Global Trigger in coincidence with beam extraction (Early Warning) on the basis of selected PMT sig- nal majorities to recognize an event interaction in the LAr, to start the PMT activity recording;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' RT Con- troller implements all the features for communication with DAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' beam spill to record all cosmic muons crossing the ICARUS TPCs during the electron drift time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The timing of the beam spills was first ap- proximately determined by measuring with an oscilloscope the difference between the EW sig- nals arrival time and the actual proton extraction signal by RWM counters at the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Then neu- trino interactions were identified and associated with the muons of the beam spill in excess to cosmic rays that were clearly identified inside the time profile of the scintillation light signals (flashes) by requiring at least 5 fired PMT pairs in the left and right TPC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Due to the energy range of BNB and NuMI neutrino beams, neutrino interactions are ex- pected to be contained in an ∼ 4 m section of ICARUS along the beam direction, suggesting the implementation of a trigger logic based on the recognition of fired PMTs inside a limited TPC region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The logic for processing the PMT LVDS signals has been initially determined with Monte Carlo calculations, and then it has been refined by analyzing a sample of events collected with a beam spill signal only (Min-Bias trigger), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' without any requirement on the scintillation light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The 18-m long TPC walls have been sub- divided in 3 consecutive longitudinal slices of 6-m length including 30 PMTs each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=" In each of opposite facing slices a majority of 5 LVDS sig- – 19 – WhiteRabbitnetwork PMT PMT CPU W RT TRIG GLOBAL TRIG controller TRIG SPEXI EAST WEST PXle8135 7820R 7820R 7820R TPC TPC A2795's A2795's PMT PMT TPC WIRES TPC WIRES V1730B's V1730B's T300 E T300 E PMT DAQ TPC DAQ PMT's PMT's T300 E T300 W ENABLEGATE(2mS Trigger BEAMGATE[1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6uS,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8ys] laptop GlobalTriggerOutput cryostat1-2 Central PMT Trigger cryostat 1-2 DAQ ALLBus LinesFigure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Time distribution of the recorded PMT light flashes (≥ 5 fired PMT pairs in the left and right TPCs within 150 ns): the beam event excess is observed for BNB (left) and NuMI beam (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 µs and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 µs spills duration of the beams are well recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' nals, with 8 photo-electron (phe) discrimination threshold and an OR of two adjacent PMTs, has been required to produce a PMT trigger primi- tive signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The same logic with a majority of 10 LVDS PMT signals is applied to generate a PMT trigger primitive in time period prior to and after a beam spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This trigger provides collec- tion of data sampling the 15 kHz of cosmic rays crossing the detector during the drift time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' With trigger gates of duration 4 ms and 14 ms for BNB and NuMI, respectively, a trigger rate of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7 Hz has been obtained (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='15 Hz from the BNB and NuMI components, re- spectively, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='25 Hz for the Off-Beam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This is in a manageable data read-out bandwidth with good operational stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The trigger efficiency for neutrino interactions is under study with data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' expectations based on the Monte Carlo simula- tions indicate a > 90% efficiency for neutrino CC interactions with >100 MeV energy deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 DAQ implementation The ICARUS data acquisition (DAQ) system uti- lizes the general artdaq data acquisition software development toolkit [32], providing customiz- able applications for reading data from detector elements (BoardReaders), and configurable ap- plications for performing event-building, data- logging, and data-dispatch to downstream online data quality monitoring processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Customized BoardReaders acquire data fragments from the TPC, PMT, and CRT read- out electronics, and from the trigger and White Rabbit timing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' They then assign ap- propriate event counters and timestamps to each fragment and then queue that data for transfer to a configurable number of EventBuilder appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For each triggered event, the ICARUS trigger BoardReader sends its data fragment to an EventBuilder, triggering a request for data from all other configured BoardReaders in the DAQ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Events are written using the art event-processing framework [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Data are writ- ten on separate file streams using simple filters on trigger type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each event in ICARUS, after lossless data compression, is approximately 160 MB, with the majority of data corresponding to the TPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The DAQ system is capable of stably supporting trigger rates in excess of 5 Hz, though typical operational trigger rates are of roughly 1 Hz or below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The BoardReader for the trigger system sends a single fragment containing the trigger and beam-gate timing, the type of beam gate, a global trigger counter, and a counter for the number of beam gates of each type in that DAQ – 20 – BnB NuM 4 Background .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='50 Ms 2140 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 425 Beam gate:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 μs 400 Data 375 lashes lashes 100 329 Background DL Beamgate:9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5μs 275 Data Pmt flash start tinmne [us] PMT flash start tirme Lus]run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The global trigger counter and time are used for collection of data from other subsys- tems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the latter derives from the common White Rabbit timing system, and is checked for validity against the network protocol time of the trigger BoardReader server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The number of beam gates of each type in the run is used offline for proper accounting of the total number of POT and de- tector exposure within a run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In order to handle large data volumes stored on tape, the Fermilab based SAM (Serial Ac- cess to Metadata) system is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For this purpose, a set of metadata is associated to each data file using Python scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The metadata al- low users to create large data sets for the analysis by requiring matching with data’s relevant infor- mation such as run number, data type (raw or reconstructed), run configuration, date, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 First operations with the BNB and NuMI The ICARUS-T600 detector was first fully op- erational in June 2021 before the summer shut- down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' It restarted data collection when beam returned November 5, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 23 shows the amounts of POT delivered by the accelerator and collected by the detector during its commis- sioning phase, concluded in June 2022, for a total of 296 · 1018 and 503 · 1018 POT collected for BNB and NuMI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Beam utiliza- tion - defined as the amount of POT collected divided by the delivered - of 89% for BNB and 88% for NuMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 23, daily variations of the beam utilization are also visible: periods with low utilization (less than 60%) correspond to days where the data acquisition was suspended in order to proceed with detector commission- ing activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Apart from this, the utilization is an average over 91% per day for both beams, which corresponds to a downtime of less than two hours per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The most frequent causes of operation downtime are data acquisition issues and less commonly hardware problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The detector and data collection status are continu- ously supervised with fully-remote shifts staffed by collaborators and with the support of on-call experts for each of the main detector subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7 Observation and reconstruction of neutrino events The data collected by the detector are processed by offline software to obtain information neces- sary for reconstruction and analysis of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The procedure to reconstruct the TPC wire and PMT signals is briefly described in the following Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The detector behavior was first investigated by a visual selection of neutrino interactions in the active liquid argon, as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These sample were an important component of the development and validation of an automated event selection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 Wire signal reconstruction The ICARUS wire signal processing chain fol- lows a logic similar to other LAr-TPC experi- ments, based on the deconvolution of the wire signal waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This procedure, explained in more detail in [34], has the goal to recover the original time structure of the current of drift electrons generating the signal on each wire, up- stream of the distortions produced by the electric field in the wire region and the shaping by the front-end electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Mathematically, this is obtained by inverting the response functions de- scribing both the electric field and the electronics effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' the resulting deconvolved signal shape is approximately Gaussian for all wire planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After the removal of the coherent noise (de- scribed in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1), the deconvolution is performed on each wire waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Segments of wave- forms corresponding to physical signals (hits) are searched for in the deconvolved waveform with a threshold-based hit finding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each hit – 21 – 06-01 2021 11-19 2021 12-29 2021 02-14 2022 03-27 2022 05-06 2022 20 40 60 80 100 Beam utilization [%] 06-01 2021 11-19 2021 12-29 2021 02-14 2022 03-27 2022 05-06 2022 20 40 60 80 100 Beam utilization [%] 0 50 100 150 200 250 300 POT (1018) BNB Delivered: 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 1018 POT Collected: 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 1018 POT 0 100 200 300 400 500 POT (1018) NuMI Delivered: 573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 1018 POT Collected: 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 1018 POT Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Cumulative sum of POT delivered by the accelerator and collected by the detector and daily beam utilization coefficient as a function of the operation time for BNB (NuMI) on the left (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The dotted black line marks the separation between the two operation periods of the detector: the full month of June 2021 and between November 5, 2021 and June 1, 2022 (the long break between the two periods is hidden in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' is then fit with a Gaussian, whose area is propor- tional to the number of drift electrons generating the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Globally, the efficiency for identifying a wire signal and associating it with the corre- sponding track that generated is exceeding 90% for all three wire planes when the 3D track seg- ment length contributing to each hit (pitch) is larger than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 mm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Hit efficiency as a function of wire "pitch": blue, red and green points correspond to Induction 1, Induction 2 and Collection wires respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Measurement made by means of a sample of cosmic muon tracks crossing the cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 PMT signal reconstruction The reconstruction of the scintillation light as- sociated with the event of interest is based on the recorded PMTs signals in the event, sam- pled at 500 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For any event triggered in coincidence with the beam spill, all 360 PMTs digitized signals are recorded in 30 µs long time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In addition, for cosmic rays crossing the detector in ±1 ms around the beam gate and identified by the trigger logic, all 180 PMTs be- longing to the ICARUS module containing the event are recorded in 10 µs long time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A threshold-based algorithm is applied to each recorded signal, to identify fired PMTs and to reconstruct the characteristics of the detected light to be used in the event analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Whenever a PMT signal exceeds the baseline by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 phe, a new OpHit object is created, characterized by a start time, a time interval for the signal to re- turn back to baseline, a maximal amplitude, and an integral of the signal over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' As a second stage all OpHits in coincidence within 100 ns are clustered together into an OpFlash object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Opflash is then expanded to include also OpHits within 1 µs after the first OpHit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Nominally, an OpFlash should correspond to the – 22 – Efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7 efficiency profile 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7 pitch [cm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8total detected light associated to each interac- tion, either due to cosmic rays or to a neutrino interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The distribution of the PMT signals in an OpFlash (time, amplitudes, integrals and geometrical positions) is clearly determined by the associated interaction in the TPC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The PMTs associated with a cosmic ray muon crossing the cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Initially, a very simple association between the event in the TPC and the corresponding de- tected light that is based on the comparison of the track and the light barycentre along the lon- gitudinal z axis (zTPC, zPMT) has been adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A correlation within few tens of centimeters was observed for the TPC and light barycen- tre (Δz = zTPC − zPMT) for both cosmic muons crossing the cathode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 26) and for a sample of BNB neutrino interactions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 27) selected by visual scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' By requiring |Δz| < 100 cm it is possible to restrict the analysis of the event to a detector slide that is approximately 5% of the total active LAr, with a corresponding reduction of randomly overlapping cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='3 CRT reconstruction The CRT hit reconstruction algorithm was vali- dated during the commissioning phase [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The first step in the reconstruction chain is to con- struct CRT hits defined as points in space and time corresponding to a muon track crossing the CRT volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' CRT data coming from Front End Board (FEB) read-outs in a given event are or- dered in time and grouped by CRT region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Due Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Distribution of Δz = zTPC − zPMT for a sample of cosmic ray muons crossing the cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Distribution of Δz = zTPC − zPMT for a sample BNB 𝜈 interactions identified by visual scan- ning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' to the differences in design of the side and top CRT systems, the Side and Top CRT Hits have to be handled differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The coincidence logic in the Side CRTs is performed offline in the reconstruction stage due to the inner and outer CRT modules being con- nected to FEBs in adjacent layers, whereas each top CRT module is a self-contained coincidence unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In order to identify a coincident grouping of CRT data objects, a software-based coincidence gate is performed (the hardware-based coinci- dence gate width is 150 ns and this value is the minimum for the software gate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The reason for not making the coincidence window too large is to avoid introducing fake coincidences from – 23 – PMTs (behindthe Fired wires) PMTs Central cathode PMTs(behind thewires 50 z axis24000 Cosmic Entries 22000 282361 Mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7743 20000 muons RMS 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='16 18000 16000 14000 12000 10000 8000 6000 4000 2000 0 400 200 0 200 400#events BNB vuCC 12 candidates 10 8 rms=41 cm 6 50 0 50 100 150 200 cmlow energy events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Studies are underway to es- tablish a gate width that optimizes the tagging efficiency while avoiding introducing fake coin- cidences with low energy events if the gate is too wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After the creation of coincident groupings of CRT data, the spatial information is extracted to reconstruct the position of the crossing track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The channel with the largest amplitude is the channel that generated the FEB trigger signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The channel position is identified and extracted from the geometry based on the global coordi- nates of the ICARUS building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The hit position is taken as the mean strip position where a track crosses multiple strips in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' When the charge amplitude exceeds the dis- criminator threshold, a CRT hit is acquired by the front-end electronics recording the values of two different time counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The first counter, T0, is reset every second by means of the PPS signal (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4) and it provides the global timing of the recorded hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The second counter, T1, is reset by the event trigger signal and is used to determine the hit relative timing with respect to the event trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Each CRT hit timestamp is corrected to account for cable delays and light propagation in the scintillator and in the WLS fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Top CRT hit is defined by the FEB inter- nal triggering logic (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 4) where a signal threshold of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 phe is applied to each chan- nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The position within a module is determined by selecting the four channels with the largest amplitude and projected in the global detector coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The CRT timing system has been cross- calibrated with the PMT signals, using the com- mon trigger pulse recorded by the CRT and PMT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A preliminary evaluation of the Time- Of-Flight (TOF) of cosmic muons has been per- formed by selecting particles entering the detec- tor from the Top CRT and generating a flash in the active argon volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The preliminary distri- Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Time difference between matched CRT hits and PMT flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The plot refers to Top CRT data in time with the BNB spill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' bution of the time differences between Top CRT hits and PMT signals is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 28: the measured average TOF of 24±9 ns is in agree- ment with the expected ∼ 26 ns evaluated from the distance between the Top CRT plane and the first PMT row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 10 − 8 − 6 − 4 − 2 − 0 2 4 6 8 10 s) µ CRT Hit T0 - gate start time ( 0 100 200 300 400 500 600 700 800 900 Number of CRT Hits BNB, Side, South s µ bin size = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2 Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' CRT hit time relative to the neutrino gate start time in the south wall (side CRT) for the BNB beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 29 shows the CRT hit time relative to the neutrino gate start time in the south side CRT wall for the BNB neutrino beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Using 11 days of commissioning data, a clear peak can be observed, showing activity in the 4 µs trigger coincidence window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Additional activity due to the beam appears inside the smaller BNB gate – 24 – Bins 1400 Events/100 I Fit parameters: 1200 mean= -24 ns sigma= 9 ns 1000 800 600 400 200 80 60 40 20 20 40 60 0 80 100 CRT Hit timestamp - PMT Flash [ns](1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 µs within the 4 µs window), the rest of the activity outside the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6 µs window is due to cos- mic ray triggering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='4 Event display study As a first check of the general behavior of the de- tector, a visual study campaign was performed to select and identify neutrino interactions in the active liquid argon using a graphical event dis- play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' As a first step, all the events recorded in the BNB and NuMI beam for some runs were studied selecting the tracks in the cryostat where the trig- ger signal has been produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' An interaction was classified as a neutrino candidate if a clear vertex with more than one track was visually identified: electron neutrino CC candidate events require the presence of a clear electromagnetic shower connected to the primary vertex, while the muon neutrino CC events are selected by requiring the presence of a long track (at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 m) from the primary vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In addition, only events with the primary vertex at least 5 cm from top/bottom TPC sides, 50 cm from the upstream/downstream TPC wall, and 5 cm from the anode position have been initially selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' An example of a 𝜈𝜇CC candidate is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 30, with an estimated total deposited energy of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The CC muon candidate is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8 m long, while the highly ionizing track from the pri- mary vertex is identified as a 20 cm long proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The full wire signal calibration is in the finaliza- tion stage, but by a very preliminary wire signal conversion to estimate the deposited energy, it is possible to reconstruct the dE/dx associated to the individual hits of the muon candidate in the same event, distributed as expected for a MIP particle particle, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Visual scanning also permitted identifica- tion of 𝜈𝑒CC candidates in the NuMI beam: a remarkable example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 32 for an event of ∼ 600 MeV deposited energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='5 Event reconstruction For a given cryostat, hits identified and passing a multi-plane matching algorithm are passed as input to Pandora [36]: a pattern reconstruction code that performs a 3D reconstruction of the full image recorded in the collected event, in- cluding the identification of interaction vertices and of tracks and showers inside the TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These are organized into a hierarchical structure (called a slice) of particles generated starting from a pri- mary interaction vertex or particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The analysis uses information reconstructed in Pandora to tag and reject “clear cosmic” slices by identifying straight tracks crossing the full ac- tive liquid argon volume or that are clearly out of time with respect to the beam gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In Monte Carlo studies, selection criteria require that the reconstructed vertex is in the fiducial volume and that PMT timing signals and the reconstructed angle of the muon track are inconsistent with that of a cosmic ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These requirements re- ject 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='7% of cosmic rays, while accepting more than 82% of true 𝜈𝜇CC events in the fiducial vol- ume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Requiring that a particle identified as a pro- ton be reconstructed in the event further reduces background from cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' After all criteria are applied, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='8% of a selected 𝜈𝜇CC contained sample is made up of background from cosmic rays, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='6% coming from intime cosmic rays and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='2% coming from out-of-time cosmic rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Further tagging and rejection of cosmic rays out of time with respect to the beam spill is possi- ble with the CRT detector, which can provide a few nanosecond absolute time measurement for the TPC tracks when they are unambiguously matched to signals on the CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This TPC track- CRT hit matching algorithm is still being tuned and validated with cosmic ray data collected off- beam, but is expected to facilitate improved ef- ficiency and allow further optimization of the cosmic rejection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pandora and a set of algorithms to iden- – 25 – Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A visually selected 𝜈𝜇CC candidate from the BNB beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Distribution of the measured dE/dx of the muon candidate in the event shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' dE/dx is reconstructed on each wire applying a preliminary calibration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' tify, measure and reconstruct tracks and show- ers can be exploited for the event reconstruction and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These reconstruction tools repre- sent a legacy from past efforts and made avail- able within the LArSoft framework [37], com- plemented by new efforts carried out within the joint SBN effort for a common near and far detec- tor analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This set of algorithms is applied to Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A visually selected 𝜈𝑒CC candidate from the NuMI beam .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' tracks and showers from any slice in the event to perform particle identification and estimate the momentum from range, calorimetry and multiple Coulomb Scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' A dedicated visual study of events was per- formed to select ∼ 600 𝜈𝜇CC interactions from BNB in the active liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' These events have been used for validation of the Pandora – 26 – Collection plane p Primary vertex Beam direction Cathode90 80 Mean 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='235 70 RMS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='211 60 50 40 30 20 10 0 9 1 2 3 4 5 6 8 dE/dx[MeV/cm]NuMI veCC candidate Track 1 Track 2 e-shower (~600MeV) COLL 1 m Wiresreconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In order to reduce the manual effort, events to be visually studied are first se- lected by requiring, offline, the absence of signals in the CRT in coincidence with the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' In ad- dition, full 3D reconstruction was performed for the events and only reconstructed tracks longer than 30 cm, fully contained in the detector, and whose barycenter was in agreement within 1 m with the barycenter of the light signal generating the trigger, have been visually studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For this sample, the neutrino interaction vertex was iden- tified and measured in 3D coordinates as well as the final point associated with the muon can- didate track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Out of the full selected sample, 476 neutrino events present in the analysis files showed a reasonable match with a reconstructed object based on vertex location and were adopted as a benchmark for the validation of the recon- struction tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' As an example, in ∼ 90% of these events the reconstruction reasonably iden- tifies the neutrino interaction vertex along the beam direction, meaning the difference between the two estimates is within 3 cm, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Comparison of the visual study to auto- mated reconstruction, along with studies of Monte Carlo simulation, will enable further un- derstanding of where to focus efforts and im- provements in the automatic reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' For example, in some cases inefficiencies in a wire plane for a given event reconstruction leading to loss of hits may impact some 3D steps and lead to a track broken into one or more smaller pieces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' or algorithms may lead to improper clustering or determination of particle types, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Further tuning of the reconstruction is progressing, as well as the complete calibration of the detec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' However the first results are quite promis- ing, demonstrating that the basic tools for the event reconstruction and the event selection are operational and allow an initial identification and measurement of neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 10 − 8 − 6 − 4 − 2 − 0 2 4 6 8 10 (cm) vertex Z ∆ 0 20 40 60 80 100 120 Slices (scan-reco) ∆ Figure 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Difference Δ𝑍 between the automatic and manual measured longitudinal (beam) coordinate of the neutrino interaction vertex for a sample of 476 𝜈𝜇CC candidates from the BNB beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Conclusions After the successful three-year physics run at the underground LNGS laboratories studying neutrino oscillations with the CERN Neutrino to Gran Sasso beam, the ICARUS T600 LAr- TPC detector underwent a significant overhaul at CERN and was then installed at Fermilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Detector activation began in 2020 with the cryo- genic commissioning and, despite serious chal- lenges and delays caused by prolonged restric- tions related to the COVID-19 pandemic, it started operations in 2021 and successfully com- pleted its commissioning phase in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' It col- lected neutrino events from both the Booster Neutrino Beam (BNB) and the Main Injector (NuMI) beam off-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Data taking started in June 2021 with the beam data acquisition, with the detector commissioning activities being con- ducted in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' An event sample correspond- ing to ∼ 3 · 1020 and 5 · 1020 POT of the Booster and NuMI beam respectively has been collected with an efficiency exceeding 91% during the normal operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This data set was used to study the single detector subsystems calibration and to test the ICARUS event selection and re- construction procedure and analysis algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' – 27 – ICARUS has already started the first year of reg- ular data taking devoted to a sensitive study of the claim by Neutrino-4 short baseline reactor exper- iment both in the 𝜈𝜇 channel with the BNB and in the 𝜈𝑒 channel with NuMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' ICARUS will also ad- dress other fundamental studies such as neutrino cross sections with the NuMI beam and a number of Beyond Standard Model searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The search for evidence of a sterile neutrino jointly with the Short-Baseline Near Detector, within the Short- Baseline Neutrino program, will follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Acknowledgements This document was prepared by the ICARUS Collaboration using the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Department of Energy, Office of Science, HEP User Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Fermilab is managed by Fermi Research Alliance, LLC (FRA), acting under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' DE-AC02-07CH11359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' This work was supported by the US Depart- ment of Energy, INFN, EU Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 734303, 822185, 858199, and 101003460 and Horizon Europe Program research and innova- tion programme under the Marie Sklodowska- Curie Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 101081478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Part of the work resulted from the implementation of the research Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' 2019/33/N/ST2/02874 funded by the National Science Centre, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The ICARUS Collaboration would like to 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Pandora multi-algorithm approach to automated pattern recognition of cosmic-ray muon and neutrino events in the MicroBooNE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content='03135v1, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' [37] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Pordes and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' Snider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' The Liquid Argon Software Toolkit (LArSoft): Goals, Status and Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' PoS, ICHEP2016:182, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} +page_content=' – 30 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFAT4oBgHgl3EQfoR3-/content/2301.08634v1.pdf'} diff --git a/9tAzT4oBgHgl3EQfFPrO/vector_store/index.pkl b/9tAzT4oBgHgl3EQfFPrO/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..38af84af261dde86afc4cf8af15c834c4c109a89 --- /dev/null +++ b/9tAzT4oBgHgl3EQfFPrO/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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sets of weakly convex functions for the HJB flow +7 +3 +Second order bounds for Schr¨odinger potentials +10 +3.1 +Weak semiconvexity of ψ implies weak semiconcavity of ϕ . . . . . . . . . . . . . . . +10 +3.2 +Weak semiconcavity of ϕ implies weak semiconvexity of ψ . . . . . . . . . . . . . . . +11 +3.3 +Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4 +Logarithmic Sobolev inequality for Schr¨odinger bridges +15 +5 +Appendix +18 +∗CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France E-mail address: +giovanni.conforti@polytechnique.edu. Research supported by the ANR project ANR-20-CE40-0014. +1 + +Abstract +We investigate the quadratic Schr¨odinger bridge problem, a.k.a. Entropic Optimal Trans- +port problem, and obtain weak semiconvexity and semiconcavity bounds on Schr¨odinger poten- +tials under mild assumptions on the marginals that are substantially weaker than log-concavity. +We deduce from these estimates that Schr¨odinger bridges satisfy a logarithmic Sobolev inequal- +ity on the product space. Our proof strategy is based on a second order analysis of coupling +by reflection on the characteristics of the Hamilton-Jacobi-Bellman equation that reveals the +existence of new classes of invariant functions for the corresponding flow. +Mathematics Subject Classification (2020) +49Q22,49L12,35G50,60J60,39B62 +1 +Introduction and statement of the main results +The Schr¨odinger problem [36] (SP) is a statistical mechanics problem that consists in finding the +most likely evolution of a cloud of independent Brownian particles conditionally to observations. +Also known as Entopic Optimal Transport (EOT) problem and formulated with the help of large +deviations theory as a constrained entropy minimization problem, it stands nowadays at the cross +of several research lines ranging from functional inequalities [13, 25], statistical machine learning +[15, 35], control engineering [9, 10], and numerics for PDEs [5, 4]. Given two probability distributions +µ, ν on Rd, the corresponding (quadratic) Schr¨odinger problem is +inf +π∈Π(µ,ν) H(π|R0T ), +(1) +where Π(µ, ν) represents the set of couplings of µ and ν and H(π|R0T ) is the relative entropy of a +coupling π computed against the joint law R0T at times 0 and T of a Brownian motion with initial +law µ. It is well known that under mild conditions on the marginals, the optimal coupling ˆπ, called +(static) Schr¨odinger bridge, is unique and admits the representation +ˆπ(dx dy) = exp(−ϕ(x) − ψ(y)) exp +� +− |x − y|2 +2T +� +dxdy +(2) +where ϕ, ψ are two functions, known as Schr¨odinger potentials [31] that can be regarded as prox- +ies for the Brenier potentials of optimal transport, that are recovered in the short-time (T → 0) +limit [34, 12]. In this article we seek for convexity and concavity estimates for Schr¨odinger po- +tentials. Such estimates have been recently established in [11] and [24] working under a set of +assumptions that implies in particular log-concavity of at least one of the two marginals. Such +assumption is crucial therein as it allows to profit from classical functional inequalities such as +Pr´ekopa-Leindler inequality and Brascamp-Lieb inequality. In particular, the estimates obtained +in the above-mentioned works yield alternative proofs of Caffarelli’s contraction Theorem [8] in the +short-time limit. The purpose of this work is twofold: in first place we show at Theorem 1.2 that, +for any fixed T > 0 it is possible to leverage the probabilistic interpretation of (1) to establish lower +and upper bounds on the functions +⟨∇ϕ(x) − ∇ϕ(y), x − y⟩ +and +⟨∇ψ(x) − ∇ψ(y), x − y⟩ +that are valid for all x, y ∈ Rd and do not require strict log concavity of the marginals to hold, but +still allow to recover the results of [11] as a special case. The second main contribution is to apply +2 + +these bounds to prove that static Schr¨odinger bridges satisfy the logarithmic Sobolev inequality +(LSI for short) at Theorem 1.3. In our main results we shall quantify the weak semiconvexity of a +potential U : Rd −→ R appealing to the function κU, defined as follows: +κU : (0, +∞) −→ R, +κU(r) = inf{|x − y|−2⟨∇U(x) − ∇U(y), x − y⟩ : |x − y| = r}. +(3) +κU(r) may be regarded as an averaged or integrated convexity lower bound for U for points that +are at distance r. This function is often encountered in applications of the coupling method to the +study of the long time behavior of Fokker-Planck equations [22, 32]. Obviously κU ≥ 0 is equivalent +to the convexity of U, but working with non-uniform lower bounds on κU allows to design efficient +generalizations of the classical notion of convexity. A commonly encountered sufficient condition +on κU ensuring the exponential trend to equilibrium of the Fokker-Planck equation +∂tµt − 1 +2∆µt − ∇ · +� +∇U µt +� += 0 +is the following +κU(r) ≥ +� +α, +if r > R, +α − L′, +if r ≤ R, +(4) +for some α, L′, R > 0. In this work, we refer to assumptions of the form (4) and variants thereof +as to weak convexity assumptions and our main result require an assumption of this kind, namely +(6) below, that is shown to be no more demanding than (4) (see Proposition 5.1), and is expressed +through a rescaled version of the hyperbolic tangent function. These functions play a special role in +this work since, as we show at Theorem 2.1, they define a weak convexity property that propagates +backward along the flow of the Hamilton-Jacobi-Bellman (HJB) equation +∂tϕt + 1 +2∆ϕt − 1 +2|∇ϕt|2 = 0. +Such invariance property represents the main innovation in our proof strategy: the propagation +of the classical notion of convexity along the HJB equation used in [24] as well as the Brascamp-Lieb +inequality employed in [11] are both consequences of the Pr´ekopa-Leindler inequality, see [7]. In +the framework considered here, such powerful tool becomes ineffective due to the possible lack of +log-concavity in both marginals. To overcome this obstacle we develop a probabilistic approach +based on a second order analysis of coupling by reflection on the solutions of the SDE +dXt = −∇ϕt(Xt)dt + dBt, +also known as characteristics of the HJB equation, that enables to establish the above mentioned +propagation of weak convexity (Theorem 2.1). This property is a key ingredient the proof of the +semiconvexity bounds of Theorem 1.2. Static Schr¨odinger bridges are not log-concave probability +measures in general, not even in the case when both marginals are strongly log-concave. For this +reason, one cannot infer LSI directly from Theorem 1.2 and the Bakry-´Emery criterion. However, +reintroducing a dynamical viewpoint and representing Schr¨odinger bridges as Doob h-transforms +of Brownian motion [21] reveals all the effectiveness of Theorem 1.2 that gives at once gradient +estimates and local (or conditional, or heat kernel) logarithmic Sobolev inequalities and gradient +estimates for the h-transform semigroup. By carefully mixing the local inequalities with the help +of the gradient estimates, we finally establish at Theorem 1.3 LSI for ˆπ, that is our second main +3 + +contribution. It is worth noticing that in the T → +∞ asymptotic regime, our approach to LSI +can be related to the techniques recently developed in [33] to construct Lipschitz transports be- +tween the Gaussian distribution and probability measures that are approximately log-concave in +a suitable sense. Because of the intrinsic probabilistic nature of our proof strategy, our ability to +compensate for the lack of log-concavity in the marginals depends on the size of the regularization +parameter T , and indeed vanishes as T → 0. Thus, our main results do not yield any sensible +convexity/concavity estimate on Brenier potentials that improves on Caffarelli’s Theorem. On the +other hand, the semiconvexity bounds of 1.2 find applications beyond LSI, that we shall address in +future works. For example, following classical arguments put forward in [20], they can be shown +to imply transport-entropy (a.k.a. Talagrand) inequalities on path space for dynamic Schr¨odiner +bridges. Moreover, building on the results of [13], they shall imply new semiconvexity estimates +for the Fisher information along entropic interpolations. It is also natural to conjecture that these +bounds will provide with new stability estimates for Schr¨odinger bridges under marginal perturba- +tions, thus addressing a question that has recently drawn quite some attention, see [19, 12, 23, 26, 3] +for example. Finally, we point out that Hessian bounds for potentials can play a relevant role in +providing theoretical guarantees for learning algorithms that make use of dynamic Schr¨odinger +bridges and conditional processes. In this framework, leveraging Doob’s h-transform theory and +time reversal arguments, they directly translate into various kinds of quantitative stability estimates +for the diffusion processes used for sampling, see e.g. [18, 17, 37]. +Organization +The document is organized as follows. In remainder of the first section we state +and comment our main hypothesis and results. In Section 2 we study invariant sets for the HJB +flow. Sections 3 and 4 are devoted to the proof our two main results, Theorem 1.2 and Theorem +1.3. Technical results and background material are collected in the Appendix section. +Assumption 1.1. We assume µ, ν admit a positive density against the Lebesgue measure which +can be written in the form exp(−U µ) and exp(−U ν) respectively. U µ, U ν are of class C2(Rd). +(H1) µ has finite second moment and finite relative entropy against the Lebsegue measure. More- +over, there exists βµ > 0 such that +⟨v, ∇2U µ(x)v⟩ ≤ βµ|v|2 +∀x, v ∈ Rd. +(5) +One of the following holds +(H2) There exist αν, L > 0 such that +κUν(r) ≥ αν − r−1fL(r) +∀r > 0, +(6) +where the function fL is defined for any L > 0 by: +fL : [0, +∞] −→ [0, +∞], +fL(r) = (2L)1/2 tanh +�1 +2(2L)1/2r +� +. +(H2′) There exist αν, L′ > 0 such that +κUν(r) ≥ +� +αν, +if r > R, +αν − L′, +if r ≤ R. +In this case, we set +L = inf{¯L : R−1f¯L(R) ≥ L′}. +(7) +4 + +Clearly, imposing (6) is less restrictive than asking that ν is strongly log-concave. +Remark 1.1. We show that (H2′) implies (H2) at Proposition 5.1. +Remark 1.2. The requirement that the density of ν is strictly positive everywhere could be dropped +at the price of additional technicalities. For µ, such requirement is a consequence of (5). +The Schr¨odinger system +Let (Pt)t≥0 the semigroup generated by a d-dimensional Brownian +motion. For given marginals, µ, ν and T > 0 the Schr¨odinger system, whose unknowns ϕ, ψ we +shall refer to as Schr¨odinger potentials, is given by +� +ϕ(x) = U µ(x) + log PT exp(−ψ)(x), +x ∈ Rd, +ψ(y) = U ν(y) + log PT exp(−ϕ)(y), +y ∈ Rd. +(8) +Under Assumption 1.1, it is known that the Schr¨odinger system admits a solution, and that if ( ¯ϕ, ¯ψ) +is another solution, then there exists c ∈ R such that (ϕ, ψ) = ( ¯ϕ + c, ¯ψ − c), see [34, sec. 2][31] and +references therein. +Weak semiconvexity and semiconcavity bounds for Schr¨odinger potentials +In the rest +of the article, given a scalar function U, any pointwise lower bound on κU implying in particular +that +lim inf +r→+∞ κU(r) > −∞ +shall be called a weak semiconvexity bound for U. Next, in analogy with (3) we introduce for a +differentiable U : Rd −→ R the function ℓU as follows: +ℓU : (0, +∞) −→ R, +ℓU(r) = sup{|x − y|−2⟨∇U(x) − ∇U(y), x − y⟩ : |x − y| = r}, +and call a weak semiconcavity bound for U any pointwise upper bound for ℓU implying in particular +that +lim sup +r→+∞ ℓU(r) < +∞. +Our first main result is about weak semiconvexity and weak semiconcavity bounds for Schr¨odinger +potentials. +Theorem 1.2. Let Assumption 1.1 hold and (ϕ, ψ) be solutions of the Schr¨odinger system. Then +ϕ, ψ are twice differentiable and for all r > 0 we have +κψ(r) ≥ αψ − r−1fL(r), +(9) +ℓϕ(r) ≤ βµ − +α +(1 + T α) + r−1fL(r) +(1 + T α)2 , +(10) +where αψ > αν − 1/T can be taken to be the smallest solution of the fixed point equation +α = αν − 1 +T + G(α, 2) +2T 2 +, +α ∈ (αν − 1/T, +∞) +(11) +5 + +with +G(α, u) = inf{s ≥ 0 : F(α, s) ≥ u}, +u > 0 +(12) +and +F(α, s) = βµs + +s +T (1 + T α) + s1/2fL(s1/2) +(1 + T α)2 , +s > 0. +Remark 1.3. It is proven at Lemma 3.2 that F(α, ·) is increasing on (0, +∞) for all α > −1/T . +G(α, ·) is therefore its inverse. +Remark 1.4. We conjecture that αψ can be taken to be the largest solution of the fixed point +equation (11). +To prove so, it would suffice to show that Sinkhorn’s iterates (see [34, Sec 6]) +converge to solutions of the Schr¨odinger system under Assumption 1.1 for a large set of initial +conditions. We could not find such result in the existing literature. +Remark 1.5. It is possible to check that if (H2) holds with L = 0, Theorem 1.2 recovers the +conclusion of [11, Theorem 4],after a change of variable. To be more precise, the potentials (ϕε, ψε) +considered there are related to the couple (ϕ, ψ) appearing in (8) by choosing ε = T and setting +ϕε = ε +� +ϕ − U µ + | · |2 +2ε +� +, +ψε = ε +� +ψ − U ν + | · |2 +2ε +� +. +Remark 1.6. The rescaled potential T ϕ converges to the Brenier potential in the small noise limit +[34]. As explained in the introduction, one cannot deduce from Theorem 1.2 an improvement over +Caffarelli’s Theorem [8] by letting T → 0 in Theorem 1.2. +Our second main result is that the static Schr¨odinger bridge ˆπ satisfies LSI with an explicit +constant. We recall here that a probability measure ρ on Rd satisfies LSI with constant C if and +only if for all positive differentiable function f +Entρ(f) ≤ C +2 +� |∇f|2 +f +dρ, +where +Entρ(f) = +� +f log fdρ − +� +fdρ log +� � +fdρ +� +. +Theorem 1.3. Let Assumption 1.1 hold and assume furthermore that µ satisfies LSI with constant +Cµ. Then the static Schr¨odinger bridge ˆπ satisfies LSI with constant +max +� +Cµ, CµC0,T + +� T +0 +Ct,T dt +� +, +where for all t ≤ T +Ct,T := exp +� +− +� T +t +αψ +s ds +� +, +αψ +t := +αψ +1 + (T − t)αψ +− +L +(1 + (T − t)αψ)2 , +and αψ is as in Theorem 1.2. +It is well known that LSI has a number of remarkable consequences including, but certainly not +limited to, spectral gaps and concentration of measure inequalities for Lipschitz observables. +Remark 1.7. By taking µ to be a Gaussian distribution, we obtain as a corollary of Theorem 1.3 +that any probability ν fulfilling (6) satisfies a logarithmic Sobolev inequality. To the best of our +knowledge, this is a new result. It is worth noticing (6) does not imply that ν is a bounded or +Lipschitz perturbation of a log-concave distribution: therefore the results of [28][1] do not apply in +this context. +6 + +2 +Invariant sets of weakly convex functions for the HJB flow +We introduce the notation +U T,g +t +(x) := − log PT −t exp(−g)(x) = − log +� +1 +(2π(T − t))d/2 +� +exp +� +− |y − x|2 +2(T − t) − g(y) +� +dy +� +. (13) +With this notation at hand, (8) rewrites as follows: +� +ϕ = U µ − U T,ψ +0 +, +ψ = U ν − U T,ϕ +0 +. +(14) +It is well known that under mild conditions on g, the map [0, T ] × Rd ∋ (t, x) �→ U T,g +t +(x) is a +classical solution of th HJB equation +� +∂tϕt(x) + 1 +2∆ϕt(x) − 1 +2|∇ϕt|2(x) = 0, +ϕT (x) = g(x). +(15) +In the next theorem, we construct for any L > 0 a set of weakly convex functions FL that is shown +to be invariant for the HJB flow. In the proof, and in the rest of the paper we shall denote by [·, ·] +the quadratic covariation of two Itˆo processes. +Theorem 2.1. Fix L > 0 and define +FL = {g ∈ C1(Rd) : κg(r) ≥ −r−1fL(r) +∀r > 0}. +Then for all 0 ≤ t ≤ T < +∞ we have +g ∈ FL ⇒ U T,g +t +∈ FL. +(16) +In the proof of the Theorem we profit from the fact that fL solves the ODE +ff ′(r) + 2f ′′(r) = 0 +∀r > 0, +f(0) = 0, f ′(0) = L. +(17) +To verify the above, it suffices to compute +f ′ +L(r) = +L +cosh2( 1 +2(2L)1/2r), +f ′′ +L(r) = 2−1/2L3/2 sinh( 1 +2(2L)1/2r) +cosh3( 1 +2(2L)1/2r) +Moreover, we recall here some useful properties of fL: +fL(r) > 0, f ′ +L(r) > 0, f ′′ +L(r) < 0, fL(r) ≥ rf ′ +L(r) +∀r > 0. +(18) +We are now in position to prove Theorem 2.1. As anticipated above, the proof relies on the analysis +of coupling by reflection along the characteristics of the HJB equation. In the recent article [14, Thm +1.3] Hessian bounds for HJB equations originating from stochastic control problems are obtained +by means of coupling techniques. These are two-sided bounds that require an a priori knowledge +of global Lipschitz bounds on solutions of the HJB equation to hold. +The one-sided estimates +of Theorem 2.1 do not require any Lipschitz property of solutions and their proof require finer +arguments than those used in [14]. +7 + +Proof. We first assume w.l.o.g. that t = 0 and work under the additional assumption that +g ∈ C3(Rd), +sup +x∈Rd |∇2g|(x) < +∞. +(19) +Combining the above with g ∈ FL, we can justify differentiation under the integral sign in (13) and +establish that +[0, T ] × Rd ∋ (t, x) �→ U T,g +t +(x) +is a classical solution of (15) such that +[0, T ] × Rd ∋ (t, x) �→ ∇U T,g +t +(x) +is continuously differentiable in t as well as twice continuously differentiable and uniformly Lipschitz +in x. Under these regularity assumptions, for given x, ˆx ∈ Rd, coupling by reflection of two diffusions +started at x and ˆx respectively and whose drift field is −∇U T,g +t +is well defined, see [22]. That is +to say, there exist a stochastic process (Xt, ˆXt)0≤t≤T with (X0, ˆX0) = (x, ˆx) and two Brownian +motions (Bt, ˆBt)0≤t≤T all defined on the same probability space and such that +� +dXt = −∇U T,g +t +(Xt)dt + dBt, +for 0 ≤ t ≤ T , +d ˆXt = −∇U T,g +t +( ˆXt)dt + d ˆBt, +for 0 ≤ t ≤ τ, Xt = ˆXt for t > τ, +where +et = r−1 +t +(Xt − ˆXt), +rt = |Xt − ˆXt|, +d ˆBt = dBt − 2et⟨et, dBt⟩ +and +τ = inf{t ∈ [0, T ] : Xt = ˆXt} ∧ T. +We now define +U : [0, T ] × Rd × Rd −→ R, +Ut(x, ˆx) = +� +|x − ˆx|−1⟨∇U T,g +t +(x) − ∇U T,g +t +(ˆx), x − ˆx⟩, +if x ̸= ˆx, +0 +if x = ˆx, +and proceed to prove that (U(Xt, ˆXt))0≤t≤T is a supermartingale. To this aim, we first deduce from +(15) and Itˆo’s formula that +d∇U T,g +t +(Xt) = dMt, +d∇U T,g +t +( ˆXt) = d ˆ +Mt +(20) +where M·, ˆ +M· are square integrable martingales. Indeed we find from Itˆo’s formula +d∇U T,g +t +(Xt) = +� +∂t∇U T,g +t +(Xt) − ∇2U T,g +t +∇U T,g +t +(Xt) + 1 +2∆∇U T,g +t +(Xt) +� +dt + ∇2U T,g +t +(Xt) · dBt +(15) += ∇2U T,g +t +(Xt) · dBt, +and a completely analogous argument shows that ∇U T,g +t +( ˆ +Xt) is a square integrable martingale. We +shall also prove separately at Lemma 2.1 that +det = −r−1 +t proje⊥ +t (∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))dt +∀t < τ, +(21) +8 + +where proje⊥ +t denotes the orthogonal projection on the orthogonal complement of the linear subspace +generated by et. Combining together (20) and(21) we find that dUt(Xt, ˆXt) = 0 for t ≥ τ, whereas +for t < τ +dUt(Xt, ˆXt) = ⟨∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt), det⟩ ++ ⟨et, d(∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))⟩ + d[(∇U T,g +· +(X·) − ∇U T,g +· +( ˆX·)), e·]t +(20)+(21) += +−r−1 +t +|proje⊥ +t (∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))|2dt + d ˜ +Mt. +proving that (U(Xt, ˆXt))0≤t≤T is a supermartingale. In the above, ˜ +M· denotes a square integrable +martingale and to obtain the last equality we used that the quadratic variation term vanishes +because of (21). Next, arguing exactly as in [22, Eq. 60] (see also (25) below for more details) on +the basis of Itˆo’s formula and invoking (17) we get +dfL(rt) = [−f ′ +L(rt)Ut(Xt, ˆXt) + 2f ′′ +L(rt)]dt + dNt +(17) += −f ′ +L(rt)[Ut(Xt, ˆXt) + fL(rt)]dt + dNt, +where N· is a square integrable martingale. It then follows that +d +� +Ut(Xt, ˆXt) + fL(rt) +� +≤ −f ′ +L(rt) +� +Ut(Xt, ˆXt) + fL(rt) +� +dt + dNt + d ˜ +Mt. +(22) +from which we deduce that the process +Γt = exp +� � t +0 +f ′ +L(rs)ds +�� +Ut(Xt, ˆXt) + fL(rt) +� +is a supermartingale and in particular is decreasing on average. This gives +|x − ˆx|−1⟨∇U T,g +0 +(x) − ∇U T,g +0 +(ˆx), x − ˆx⟩ + fL(|x − ˆx|) = E[Γ0] +≥ E[ΓT ] ≥ E +� +exp( +� T +0 +f ′ +L(rs)ds) +� +|XT − ˆXT |κg(|XT − ˆXT |) + fL(|XT − ˆXT |) +�� +≥ 0, +where the last inequality follows from g ∈ FL. +We have thus completed the proof under the +additional assumption (19). In order to remove it, consider any g ∈ FL. Then there exist (gn) ⊆ FL +such that (19) holds for any of the gn, gn → g pointwise and gn is uniformly bounded below. From +this, one can prove that ∇U gn,T +0 +→ ∇U g,T +0 +pointwise by differentiating (13) under the integral sign. +Using this result in combination with the fact that (16) holds for any gn allows to reach the desired +conclusion. +Lemma 2.1. Under the same assumptions and notations of Theorem 2.1 we have +det = −r−1 +t proje⊥ +t (∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))dt +∀t < τ. +Proof. Recall that if θ : Rd → R is the map z �→ |z|, then we have +∇θ(z) = z +|z|, +∇2θ(z) = I +|z| − zz⊤ +|z|3 , +z ̸= 0. +(23) +9 + +The proof consist of several applications of Itˆo’s formula. We first observe that for t < τ +d(Xt − ˆXt) = −(∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))dt + 2etdWt, +with +dWt = ⟨et, dBt⟩. +(24) +Note that by L´evy characterization, (Wt)0≤t≤T is a Brownian motion. Thus, invoking (23) (or +refferring directly to [22, Eq. 60] we obtain +drt = −⟨∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt), et⟩dt + 2dWt, +(25) +whence +dr−1 +t += −r−2 +t drt + r−3 +t +d[r]t += +� +r−2 +t +⟨∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt), et⟩ + 4r−3 +t +� +dt − 2r−2 +t dWt. +(26) +Combining (24) with (26) we find that for t < τ +det = d +� +r−1 +t +(Xt − ˆXt)) += r−1 +t +d(Xt − ˆXt) + (Xt − ˆXt)d(r−1 +t +) + d[X· − ˆX·, r−1 +· +]t += −r−1 +t +(∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))dt + 2r−1 +t +etdWt ++ +� +r−2 +t ⟨∇U T,g +t +(Xt) − ∇U T,g +t +( ˆ +Xt), et⟩ + 4r−3 +t +� +(Xt − ˆXt)dt +− 2r−2 +t (Xt − ˆXt)dWt − 4r−2 +t etdt += −r−1 +t +� +∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt) − ⟨∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt), et⟩et +� +dt += −(r−1 +t +)proje⊥ +t (∇U T,g +t +(Xt) − ∇U T,g +t +( ˆXt))dt. +3 +Second order bounds for Schr¨odinger potentials +From now on Assumption 1.1 is in force, even if we do not specify it. Moreover, since we show at +Proposition 5.1 in the appendix that (H2′) implies (H2), we shall always assume that (H2) holds +in the sequel. The next two subsections are devoted to establish the key estimates needed in the +proof of Theorem 1.2, that is carried out immediately afterwards. +3.1 +Weak semiconvexity of ψ implies weak semiconcavity of ϕ +Lemma 3.1. Assume that α > −1/T exists such that +κψ(r) ≥ α − r−1fL(r) +∀r > 0. +Then we have +ℓϕ(r) ≤ βµ − +α +1 + T α + r−1fL(r) +(1 + T α)2 = r−2F(α, r2) − 1 +T +∀r > 0. +10 + +Proof. We define +ˆψ(·) = ψ(·) − α +2 | · |2. +and note by assumption ˆψ ∈ FL. We claim that +U T,ψ +0 +(x) = +α +2(1 + T α)|x|2 + U T/(1+T α), ˆψ +0 +((1 + T α)−1x) + C, +(27) +where C is some constant independent of x. Indeed we have +U T,ψ +0 +(x) − d +2 log(2πT ) = − log +� +exp +� +− |y − x|2 +2T +− α +2 |y|2 − ˆψ(y) +� +dy += − log +� +exp +� +− +α|x|2 +2(1 + T α) − 1 + T α +2T +|y − (1 + T α)−1x|2 − ˆψ(y) +� +dy += +α|x|2 +2(1 + T α) + U T/(1+T α), ˆ +ψ +0 +((1 + T α)−1x) − d +2 log(2πT/(1 + T α)) +Since ˆψ ∈ FL, we can invoke Theorem 2.1 to obtain +κUT,ψ +0 +(r) ≥ +α +1 + T α − r−1fL(r) +(1 + T α)2 +∀r > 0. +(28) +The desired conclusion is then obtained from (14) and Assumption 1.1. +3.2 +Weak semiconcavity of ϕ implies weak semiconvexity of ψ +We begin by recording some useful properties of the functions F(·, ·) and G(·, ·). +Lemma 3.2. Let T, βµ > 0, L ≥ 0 be given. +(i) For any α > −1/T the function +s �→ F(α, s) +is concave and increasing [0, +∞). +(ii) α �→ G(α, 2) is positive and non decreasing over (− 1 +T , +∞). +(iii) The fixed point equation (11) admits at least one solution on (αν − 1/T, +∞) and αν − 1/T +is not an accumulation point for the set of solutions. +Proof. We begin with the proof of (i). To this aim, we observe that fL is increasing on [0, +∞) +and therefore so is s �→ s1/2fL(s1/2). Therefore +d +dsF(α, s) ≥ βµ + +1 +T (1 + T α) > 0, +where we used α > −1/T in the last inequality. To prove concavity, we observe that +d2 +du2 +� +u1/2fL(u1/2) +���� +u=s = s−1/2 +4 +f +′′ +L(s1/2) + s−3/2 +4 +(f +′ +L(s1/2)s1/2 − fL(s1/2)) +(18) +< 0. +11 + +Thus s �→ s1/2fL(s1/2) is concave and so is F(α, ·). We now move on to the proof of (ii) by first +showing that G(·, 2) is positive and then showing that it is increasing. If this was not the case then +G(α, 2) = 0 for some α > −1/T and therefore there exists a sequence (sn)n≥0 such that sn → 0 +and F(α, sn) ≥ 2. But this is impossible since lims↓0 F(α, sn) = 0. Next, we observe that F(α, s) +is increasing in s from item (i) and decreasing in α for α ∈ (−1/T, +∞). For this reason, for any +u and α′ ≥ α we have +{s : F(α′, s) ≥ u} ⊆ {s : F(α, s) ≥ u} +and therefore +G(α′, u) ≥ G(α, u). +To prove (iii), we introduce +h : [αν − 1 +T , +∞) −→ R, +h(α) := α − +� +αν − 1 +T + G(α, 2) +2T 2 +� +Note that that h is continuous on its domain since G(·, 2) is so. Therefore, to reach the conclusion +it suffices to show that +h(αν − 1 +T ) < 0, +lim +α→+∞ h(α) = +∞. +(29) +The first inequality is a direct consequence of G(αν − 1/T ) > 0, that we have already proven. The +second inequality is proven if we can show that +lim sup +α→+∞ G(α, 2) ≤ +1 +2βµ +. +(30) +To see that this relation holds, observe that, using fL(r) ≥ 0 we obtain that for any α > −1/T +F(α, s) ≥ βµs +∀s > 0. +But then we obtain directly from (12) that +G(α, 2) ≤ +1 +2βµ +, +thus proving (30). +We shall now introduce the modified potential ¯ψ as follows +¯ψ(y) = T +� +ψ(y) − U ν(y) + |y|2 +2T +� +, +(31) +It has been proven at [11, Lemma 1] that the Hessian of ¯ψ relates to the covariance matrix of the +conditional distributions of the static Schr¨odinger bridge ˆπ. That is to say, +∇2 ¯ψ(y) = 1 +T CovX∼ˆπy(X) +(32) +where ˆπy is (a version of) the conditional distribution of ˆπ that, in view of (8) has the following +form: +ˆπy(dx) = exp(−V ˆπy(x)))dx +� +exp(−V ˆπy(¯x))d¯x, +V ˆπy(x) := ϕ(x) + |x|2 +2T − xy +T . +(33) +12 + +We shall give an independent proof of (32) under additional regularity assumptions at Proposition +5.2 in the Appendix for the readers’ convenience. A consequence of (32) is that ¯ψ is convex and we +obtain from (31) that +κψ(r) ≥ αν − 1 +T − r−1fL(r) +∀r > 0. +(34) +This is a first crude weak semiconvexity bound on ψ upon which Theorem 1.2 improves by means +of a recursive argument. We show in the forthcoming Lemma how to deduce weak semiconvexity +of ψ from weak semiconcavity of ϕ. In the L = 0 setting, this step is carried out in [11] invoking +the Cramer-Rao inequality, whose application is not justified in the present more general setup. +Lemma 3.3. Assume that α > −1/T exists such that +ℓϕ(r) ≤ − 1 +T + r−2F(α, r2) +∀r > 0. +(35) +Then +κψ(r) ≥ αν − 1 +T + G(α, 2) +2T 2 +− r−1fL(r) +∀r > 0. +Proof. Recalling the definition of V ˆπy given at (33) we observe that the standing assumptions imply +ℓV ˆπy (r) ≤ r−2F(α, r2) +∀r > 0. +(36) +In view of (32), we now proceed to bound VarX∼ˆπy(X1) from below for a given y, where we adopted +the notational convention X = (X1, . . . , Xd) for the components of random vectors. We first observe +that +VarX∼ˆπy(X1) ≥ EX∼ˆπy[VarX∼ˆπy(X1|X2, . . . , Xd)]. +(37) +Moreover, upon setting for any z = (z2, . . . , zd) +V ˆπy,z(·) := V ˆπy(·, z), +ˆπy,z(dx) = +exp(−V ˆπy,z(x))dx +� +exp(−V ˆπy,z(¯x))d¯x +we have +VarX∼ˆπy(X1|X2 = z2, . . . , Xd = zd) = 1 +2 +� +|x − ˆx|2ˆπy,z(dx)ˆπy,z(dˆx) +With this notation at hand, we find that, uniformly in z ∈ Rd−1, +1 = 1 +2 +� +(∂xV ˆπy,z(x) − ∂xV ˆπy,z(ˆx))(x − ˆx)ˆπy,z(dx)ˆπy,z(dˆx) += 1 +2 +� +⟨∇V ˆπy(x, z) − ∇V ˆπy(ˆx, z), (x, z) − (ˆx, z)⟩ˆπy,z(dx)ˆπy,z(dˆx) +(36) +≤ 1 +2 +� +F(α, |x − ˆx|2)ˆπy,z(dx)ˆπy,z(dˆx) +≤ 1 +2F(α, 2VarX∼ˆπy(X1|X2 = z2, . . . , Xd = zd)) +where to establish the last inequality we used Lemma 3.2(i) and Jensen’s inequality. Since α > +−1/T , invoking again Lemma 3.2(i) we have that s �→ F(α, s) is non decreasing. But then, we get +from (37) and the last bound that +VarX∼ˆπy(X1) ≥ 1 +2G(α, 2), +∀y ∈ Rd. +13 + +Next, we observe that, because of the fact that if ϕ(·) satisfies (35) then so does ϕ(O·) for any +orthonormal matrix O, repeating the argument above yields +VarX∼ˆπy(⟨v, X⟩) ≥ 1 +2G(α, 2), +∀y, v ∈ Rd s.t. |v| = 1. +In light of (32), this implies +⟨v, ∇2 ¯ψ(y)v⟩ ≥ G(α, 2) +2T +|v|2 +∀v, y ∈ Rd. +But then, since +ψ(y) = U ν(y) − |y|2 +2T + +¯ψ(y) +T +we immediately obtain +κψ(r) ≥ αν − 1 +T + G(α, 2) +2T 2 +− r−1fL(r)∀r > 0. +3.3 +Proof of Theorem 1.2 +The proof is obtained combining the results of the former two sections through a fixed point argu- +ment. +Proof of Theorem 1.2. We define a sequence (αn)n≥0 via +α0 = αν − 1 +T , +αn = αν − 1 +T + G(αn−1, 2) +2T 2 +, +n ≥ 1. +Using Lemma 3.2(ii) and an induction argument, we obtain that α1 ≥ α0 and (αn)n≥0 is a non +decreasing sequence. If we denote by α∗ the limit, then by continuity of G(·, 2), we know that +α∗ > αν − 1/T and α∗ satisfies the fixed point equation (11). To conclude the proof, we show by +induction that +κψ(r) ≥ αn − r−1fL(r) +∀n ≥ 1. +(38) +The case n = 0 is (34). For the inductive step, suppose (38) holds for a given n. Then Lemma 3.1 +gives that +ℓϕ(r) ≤ r−2F(αn, r2) − 1 +T +∀r > 0. +But then, an application of Lemma 3.1 proves that for all r > 0 we have +κψ(r) ≥ αν − 1 +T + G(αn, 2) +2T 2 +− r−1fL(r) = αn+1 − r−1fL(r). +The proof of (9) is now finished. To conclude, we observe that (10) follows directly from (9) and +Lemma 3.3. +14 + +4 +Logarithmic Sobolev inequality for Schr¨odinger bridges +This section is devoted to the proof of Theorem 1.3 and is structured as follows: we first recall +known facts about logaithmic Sobolev inequalities and gradient estimates for diffusion semigroups +whose proofs can be found e.g. in [2] and eventually prove at Lemma 4.1 a sufficient condition +for the two-times distribution of a diffusion process to satisfy LSI. Though such a result may not +appear surprising, we could not find it in this form in the existing literature. We then proceed to +elucidate the connection between Schr¨odinger bridges and Doob h-transforms at Lemma 4.2, and +then finally prove Theorem 1.3. +Local LSIs and gradient estimates +Let [0, T ] × Rd ∋ (t, x) �→ Ut(x) be continuous in the time +variable and twice differentiable in the space variable with +⟨v, ∇2Ut(x)v⟩ ≥ αt|v|2 +∀x, v ∈ Rd, t ∈ [0, T ] +for some function αt uniformly bounded below. We consider the time-inhomogeneous semigroup +(Ps,t)0≤s≤t≤T generated by the diffusion process whose generator at time t acts on smooth functions +with bounded support as follows +f �→ 1 +2∆f − ⟨∇Ut, ∇f⟩. +We now recall some basic fact about gradient estimates and local LSIs for the semigroup (Ps,t)0≤s≤t≤T . +For time-homogeneous semigroups these facts are well known and can be found e.g. in [2]: the +adaptation to the time-inhomogeneous setting is straightforward. The first result we shall need +afterwards is the gradient estimate (see [2, Thm. 3.3.18]) +|∇Pt,T f|(x) ≤ Ct,T Pt,T (|∇f|)(x), +Ct,T = exp +� +− +� T +t +αsds +� +, +(39) +that holds for all (t, x) ∈ [0, T ] × Rd and any continuously differentiable f. Moreover, the local +logarithmic Sobolev inequalities (see [2, Thm. 5.5.2]) +(P0,T f log f)(x) − (P0,T f)(x) log(P0,T f)(x) ≤ +˜C0,T +2 +P0,T (|∇f|2/f)(x), +˜C0,T = +� T +0 +Ct,T dt +(40) +hold for all x ∈ Rd and all positive continuously differentiable f. In the next Lemma we show how +obtain LSI for the joint law at times 0 and T of a diffusion process with initial distribution µ and +drift −∇Ut, that is to say for the coupling π defined by +� +Rd×Rd f(x, y)π(dxdy) = +� +Rd P0,T f(x, ·)(x)µ(dx) +∀f > 0. +(41) +Lemma 4.1. Assume that µ satisfies LSI with constant Cµ and let π be as in (41). Then π satisfies +LSI with constant +max{Cµ, CµC0,T + ˜C0,T }. +To proof is carried out ”mixing” carefully with the help of the gradient estimate the local +(conditional) LSIs (40). Similar arguments and ideas can be found e.g. in [6, 27]. +15 + +Proof. We recall the decomposition of the entropy formula (see [30, Thm. 2.4]) +Entπ(f) = Entµ(f0) + +� +Rd Entπx(f x)f0(x)µ(dx), +where we adopted the following conventions +f0(x) = (P0,T f(x, ·))(x), +f x(y) = f(x, y)/f0(x), +� +g(y)πx(dy) = +� +P0,T g +� +(x) ∀g > 0. +We know from (40) that uniformly in x, f we have +Entπx(f x) = P0,T +� +f x log f x� +(x) − +� +P0,T f x log P0,T f x� +(x) ≤ +˜C0,T +2f0(x) +� +|∇yf(x, y)|2/f(x, y)πx(dy). +This gives +� +Entπx(f x)f0(x)µ(dx) ≤ +˜C0,T +2 +� |∇yf(x, y)|2 +f(x, y) +π(dxdy). +(42) +Next, we use LSI for µ to obtain +Entµ(f0) ≤ Cµ +2 +� +|∇xf0(x)|2/f0(x)µ(dx) += Cµ +2 +� +|P0,T (∇xf(x, ·))(x)|2(P0,T f(x, ·))−1(x) µ(dx) ++ Cµ +2 +� +|∇zP0,T (f(x, ·))(z)|2��� +z=x(P0,T f(x, ·))−1(x) µ(dx) +(43) +For the first summand on the rhs of (43), we can argue on the basis of Jensen’s inequality applied +to the convex function a, b �→ a2/b to obtain +Cµ +2 +� +|P0,T (∇xf(x, ·))(x)|2(P0,T f(x, ·))−1(x)µ(dx) +≤ Cµ +2 +� +P0,T +� +|∇xf(x, ·)|2/f(x, ·) +� +(x)µ(dx) += Cµ +2 +� +|∇xf(x, y)|2/f(x, y)π(dxdy). +(44) +For the second summand on the rhs of (43), we first invoke the gradient estimate (39) and eventually +apply again Jensen’s inequality to obtain +Cµ +2 +� +|∇zP0,T (f(x, ·))(z)|2��� +z=x(P0,T f(x, ·))−1(x)µ(dx) +≤ CµC0,T +2 +� +(P0,T (|∇yf(x, ·)|)(x))2(P0,T f(x, ·))−1(x)µ(dx) +≤ CµC0,T +2 +� +P0,T +� +|∇yf(x, ·)|2/f(x, ·) +� +(x)µ(dx) += CµC0,T +2 +� +|∇yf(x, y)|2/f(x, y) π(dxdy). +(45) +Gathering (42)-(44)-(45) we obtain the desired result. +16 + +In the next lemma, we represent the static Schr¨odinger bridge (2) through a diffusion process. +It is a classical result saying that Schr¨odinger bridges are indeed Doob’s h-transforms, see e.g. [31, +Sec. 4][16]. +Lemma 4.2. Let Assumption 1.1 hold and ˆπ be the static Schr¨odinger bridge (2). Then ˆπ has the +form (41), where (Ps,t)0≤s≤t≤T is the time-inhomogeneous semigroup associated with the generator +acting on smooth test functions as follows +f �→ 1 +2∆f − ⟨∇U T,ψ +t +, ∇f⟩, +t ∈ [0, T ], +(46) +where U T,ψ +t +has been defined at (13). +Proof. Let ψ be the Schr¨odinger potential issued from (8) and denote by Q the law on C([0, T ]; Rd) +of a solution (Xt)t∈[0,T ] to the stochastic differential equation +dXt = −∇U T,ψ +t +(Xt)dt + dBt, +X0 ∼ µ. +Note that, because of Theorem 1.2, existence of strong solutions and pathwise uniqueness hold for +the above equation. Next, we denote by P the Wiener measure with initial distribution µ. By +Girsanov’s Theorem, see [29] for a version that applies in the current setting, we know that +dQ +dP (ω) = exp +� +− +� T +0 +∇U T,ψ +t +(ωt)dωt − 1 +2 +� T +0 +|∇U T,ψ +t +(ωt)|2dt +� +P − a.s., +where we denote by ω the typical element of the canonical space C([0, T ]; Rd). Using Itˆo formula +we rewrite the above as +dQ +dP (ω) = exp +� +U T,ψ +0 +(ω0) − U T,ψ +T +(ωT ) + +� T +0 +� +∂tU T,ψ +t ++ 1 +2∆U T,ψ +t +− 1 +2|∇U T,ψ +t +|2� +(ωt)dt +� += exp(U µ(ω0) − ϕ(ω0) − ψ(ωT )) +where we used the Schr¨odinger system (8) and the HJB equation (15) to obtain the last expression. +Indeed because of Theorem 1.2 one can deduce that [0, T ] × Rd ∋ (t, x) �→ U T,ψ +t +(x), is a classical +solution of (15) by differentiating under the integral sign in (13). From this, we deduce that +dQ0T +dP0T +(x, y) = exp +� +U µ(x) − ϕ(x) − ψ(y) +� +P0T − a.s., +where Q0T (resp. P0T ) denotes the joint distribution of Q (resp. P) at times 0 and T . Since +dP0T (dxdy) = (2πT )−d/2 exp(−U µ(x)) exp +� +− |y − x|2 +2T +� +dxdy, +we conclude that +dQ0T (dxdy) = (2πT )−d/2 exp +� +− ϕ(x) − ψ(y) − |y − x|2 +2T +� +dxdy. +But then Q0T = ˆπ, where ˆπ is defined at (2). To conclude, we observe that Q0T has the desired +form (41) where (Ps,t)0≤s≤t is indeed the semigroup generated by (46) +17 + +Proof of Theorem 1.3. We know by Lemma 4.2 that ˆπ has the form (41) for the inhomogeneous +semigroup generated by (46). We now set +αψ +t = +inf +x,v∈Rd,|v|=1⟨v, ∇2U ψ,T +t +(x), v⟩ +and proceed to estimate αψ +t from below. To do so, we observe that U ψ,T +t += U ψ,T −t +0 +and argue +exactly as we did to establish (28) to obtain that +κUT,ψ +t +(r) ≥ +αψ +1 + (T − t)αψ − +r−1fL(r) +(1 + (T − t)αψ)2 +∀r > 0. +From here, using the concavity of fL and f ′ +L(0) = L we obtain +αψ +t ≥ +αψ +1 + (T − t)αψ − +L +(1 + (T − t)αψ)2 . +At this point, the conclusion follows from Lemma 4.1 +5 +Appendix +Proposition 5.1. Assume that U satisfies (4) for some α > 0, L′, R ≥ 0. Then +κU(r) ≥ α − r−1fL(r) +∀r > 0. +with L given by (7). +Proof. If r > R the claim is a simple consequence of fL(r) ≥ 0. If r ≤ R, using (18) to get that +r′ �→ r′−1fL(r′) is non increasing on (0, +∞), we obtain +r−1fL(r) ≥ R−1fL(R) = L′, +from which the conclusion follows. +Proposition 5.2. Let Assumption 1.1 hold and assume furthermore that there exist ε, γ′ > 0 such +that +� +exp(γ′|x|1+ε)µ(dx) < +∞. +(47) +Moreover, let ¯ψ be as in (31). Then ¯ψ is twice differentiable and we have +∇2 ¯ψ(y) = 1 +T CovX∼ˆπy(X) +∀y ∈ Rd, +where ˆπy is given by (33). +Proof. From (8) we obtain that +¯ψ(y) + d +2 log(π) = T log +� +Rd exp +� +− ϕ(x) − |x|2 +2T + ⟨x, y⟩ +T +� +dx. +(48) +18 + +From Assumption 1.1, (8) and (47) it follows that +� +Rd×Rd exp +� +γ′|x|1+ε − ϕ(x) − ψ(y) − |x − y|2 +2T +� +dx dy < +∞, +whence the existence of some y′ such that +� +Rd×Rd exp +� +γ′|x|1+ε − ϕ(x) − |x|2 +2T + ⟨x, y′⟩ +T +� +dx < +∞. +From this, we easily obtain that for all γ < γ′ +� +Rd exp +� +γ|x|1+ε − ϕ(x) − |x|2 +2T + ⟨x, y⟩ +T +� +dx < +∞ +∀y ∈ Rd. +(49) +Thanks to (49) we can apply the dominated convergence theorem and differentiate under the integral +sign in (15) to obtain that ¯ψ is differentiable and +∇ ¯ψ(y) = +� +x exp(−ϕ(x) − |x|2 +2T + ⟨x,y⟩ +T +)dx +� +exp(−ϕ(¯x) − |¯x|2 +2T + ⟨¯x,y⟩ +T +)d¯x +(33) += EX∼ˆπy[X] +Using once again (49) to differentiate under the integral sign in (48) we conclude that ¯ψ is twice +differentiable and that (32) holds. +References +[1] Shigeki Aida, Takao Masuda, and Ichiro Shigekawa. +Logarithmic sobolev inequalities and +exponential integrability. 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Math., 144:144–153, 1931. +[37] Yuyang Shi, Valentin De Bortoli, George Deligiannidis, and Arnaud Doucet. +Conditional +simulation using diffusion schr¨odinger bridges. arXiv preprint arXiv:2202.13460, 2022. +21 + diff --git a/DtAyT4oBgHgl3EQfSPej/content/tmp_files/load_file.txt b/DtAyT4oBgHgl3EQfSPej/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..407f806a764ac5f024312e60bb3aa0abaa5d5bf6 --- /dev/null +++ b/DtAyT4oBgHgl3EQfSPej/content/tmp_files/load_file.txt @@ -0,0 +1,645 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf,len=644 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='00083v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='PR] 31 Dec 2022 Weak semiconvexity estimates for Schr¨odinger potentials and logarithmic Sobolev inequality for Schr¨odinger bridges Giovanni Conforti ∗, January 3, 2023 Contents 1 Introduction and statement of the main results 2 2 Invariant sets of weakly convex functions for the HJB flow 7 3 Second order bounds for Schr¨odinger potentials 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 Weak semiconvexity of ψ implies weak semiconcavity of ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 14 4 Logarithmic Sobolev inequality for Schr¨odinger bridges 15 5 Appendix 18 ∗CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France E-mail address: giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='conforti@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Research supported by the ANR project ANR-20-CE40-0014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 1 Abstract We investigate the quadratic Schr¨odinger bridge problem, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Entropic Optimal Trans- port problem, and obtain weak semiconvexity and semiconcavity bounds on Schr¨odinger poten- tials under mild assumptions on the marginals that are substantially weaker than log-concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We deduce from these estimates that Schr¨odinger bridges satisfy a logarithmic Sobolev inequal- ity on the product space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Our proof strategy is based on a second order analysis of coupling by reflection on the characteristics of the Hamilton-Jacobi-Bellman equation that reveals the existence of new classes of invariant functions for the corresponding flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Mathematics Subject Classification (2020) 49Q22,49L12,35G50,60J60,39B62 1 Introduction and statement of the main results The Schr¨odinger problem [36] (SP) is a statistical mechanics problem that consists in finding the most likely evolution of a cloud of independent Brownian particles conditionally to observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Also known as Entopic Optimal Transport (EOT) problem and formulated with the help of large deviations theory as a constrained entropy minimization problem, it stands nowadays at the cross of several research lines ranging from functional inequalities [13, 25], statistical machine learning [15, 35], control engineering [9, 10], and numerics for PDEs [5, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Given two probability distributions µ, ν on Rd, the corresponding (quadratic) Schr¨odinger problem is inf π∈Π(µ,ν) H(π|R0T ), (1) where Π(µ, ν) represents the set of couplings of µ and ν and H(π|R0T ) is the relative entropy of a coupling π computed against the joint law R0T at times 0 and T of a Brownian motion with initial law µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is well known that under mild conditions on the marginals, the optimal coupling ˆπ, called (static) Schr¨odinger bridge, is unique and admits the representation ˆπ(dx dy) = exp(−ϕ(x) − ψ(y)) exp � − |x − y|2 2T � dxdy (2) where ϕ, ψ are two functions, known as Schr¨odinger potentials [31] that can be regarded as prox- ies for the Brenier potentials of optimal transport, that are recovered in the short-time (T → 0) limit [34, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In this article we seek for convexity and concavity estimates for Schr¨odinger po- tentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Such estimates have been recently established in [11] and [24] working under a set of assumptions that implies in particular log-concavity of at least one of the two marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Such assumption is crucial therein as it allows to profit from classical functional inequalities such as Pr´ekopa-Leindler inequality and Brascamp-Lieb inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In particular, the estimates obtained in the above-mentioned works yield alternative proofs of Caffarelli’s contraction Theorem [8] in the short-time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The purpose of this work is twofold: in first place we show at Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 that, for any fixed T > 0 it is possible to leverage the probabilistic interpretation of (1) to establish lower and upper bounds on the functions ⟨∇ϕ(x) − ∇ϕ(y), x − y⟩ and ⟨∇ψ(x) − ∇ψ(y), x − y⟩ that are valid for all x, y ∈ Rd and do not require strict log concavity of the marginals to hold, but still allow to recover the results of [11] as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The second main contribution is to apply 2 these bounds to prove that static Schr¨odinger bridges satisfy the logarithmic Sobolev inequality (LSI for short) at Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In our main results we shall quantify the weak semiconvexity of a potential U : Rd −→ R appealing to the function κU, defined as follows: κU : (0, +∞) −→ R, κU(r) = inf{|x − y|−2⟨∇U(x) − ∇U(y), x − y⟩ : |x − y| = r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (3) κU(r) may be regarded as an averaged or integrated convexity lower bound for U for points that are at distance r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' This function is often encountered in applications of the coupling method to the study of the long time behavior of Fokker-Planck equations [22, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Obviously κU ≥ 0 is equivalent to the convexity of U, but working with non-uniform lower bounds on κU allows to design efficient generalizations of the classical notion of convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' A commonly encountered sufficient condition on κU ensuring the exponential trend to equilibrium of the Fokker-Planck equation ∂tµt − 1 2∆µt − ∇ · � ∇U µt � = 0 is the following κU(r) ≥ � α, if r > R, α − L′, if r ≤ R, (4) for some α, L′, R > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In this work, we refer to assumptions of the form (4) and variants thereof as to weak convexity assumptions and our main result require an assumption of this kind, namely (6) below, that is shown to be no more demanding than (4) (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1), and is expressed through a rescaled version of the hyperbolic tangent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' These functions play a special role in this work since, as we show at Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1, they define a weak convexity property that propagates backward along the flow of the Hamilton-Jacobi-Bellman (HJB) equation ∂tϕt + 1 2∆ϕt − 1 2|∇ϕt|2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Such invariance property represents the main innovation in our proof strategy: the propagation of the classical notion of convexity along the HJB equation used in [24] as well as the Brascamp-Lieb inequality employed in [11] are both consequences of the Pr´ekopa-Leindler inequality, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In the framework considered here, such powerful tool becomes ineffective due to the possible lack of log-concavity in both marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To overcome this obstacle we develop a probabilistic approach based on a second order analysis of coupling by reflection on the solutions of the SDE dXt = −∇ϕt(Xt)dt + dBt, also known as characteristics of the HJB equation, that enables to establish the above mentioned propagation of weak convexity (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' This property is a key ingredient the proof of the semiconvexity bounds of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Static Schr¨odinger bridges are not log-concave probability measures in general, not even in the case when both marginals are strongly log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For this reason, one cannot infer LSI directly from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 and the Bakry-´Emery criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' However, reintroducing a dynamical viewpoint and representing Schr¨odinger bridges as Doob h-transforms of Brownian motion [21] reveals all the effectiveness of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 that gives at once gradient estimates and local (or conditional, or heat kernel) logarithmic Sobolev inequalities and gradient estimates for the h-transform semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' By carefully mixing the local inequalities with the help of the gradient estimates, we finally establish at Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3 LSI for ˆπ, that is our second main 3 contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is worth noticing that in the T → +∞ asymptotic regime, our approach to LSI can be related to the techniques recently developed in [33] to construct Lipschitz transports be- tween the Gaussian distribution and probability measures that are approximately log-concave in a suitable sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Because of the intrinsic probabilistic nature of our proof strategy, our ability to compensate for the lack of log-concavity in the marginals depends on the size of the regularization parameter T , and indeed vanishes as T → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Thus, our main results do not yield any sensible convexity/concavity estimate on Brenier potentials that improves on Caffarelli’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' On the other hand, the semiconvexity bounds of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 find applications beyond LSI, that we shall address in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For example, following classical arguments put forward in [20], they can be shown to imply transport-entropy (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Talagrand) inequalities on path space for dynamic Schr¨odiner bridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Moreover, building on the results of [13], they shall imply new semiconvexity estimates for the Fisher information along entropic interpolations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is also natural to conjecture that these bounds will provide with new stability estimates for Schr¨odinger bridges under marginal perturba- tions, thus addressing a question that has recently drawn quite some attention, see [19, 12, 23, 26, 3] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Finally, we point out that Hessian bounds for potentials can play a relevant role in providing theoretical guarantees for learning algorithms that make use of dynamic Schr¨odinger bridges and conditional processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In this framework, leveraging Doob’s h-transform theory and time reversal arguments, they directly translate into various kinds of quantitative stability estimates for the diffusion processes used for sampling, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' [18, 17, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Organization The document is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In remainder of the first section we state and comment our main hypothesis and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In Section 2 we study invariant sets for the HJB flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Sections 3 and 4 are devoted to the proof our two main results, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Technical results and background material are collected in the Appendix section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We assume µ, ν admit a positive density against the Lebesgue measure which can be written in the form exp(−U µ) and exp(−U ν) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' U µ, U ν are of class C2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (H1) µ has finite second moment and finite relative entropy against the Lebsegue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' More- over, there exists βµ > 0 such that ⟨v, ∇2U µ(x)v⟩ ≤ βµ|v|2 ∀x, v ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (5) One of the following holds (H2) There exist αν, L > 0 such that κUν(r) ≥ αν − r−1fL(r) ∀r > 0, (6) where the function fL is defined for any L > 0 by: fL : [0, +∞] −→ [0, +∞], fL(r) = (2L)1/2 tanh �1 2(2L)1/2r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (H2′) There exist αν, L′ > 0 such that κUν(r) ≥ � αν, if r > R, αν − L′, if r ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In this case, we set L = inf{¯L : R−1f¯L(R) ≥ L′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (7) 4 Clearly, imposing (6) is less restrictive than asking that ν is strongly log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We show that (H2′) implies (H2) at Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The requirement that the density of ν is strictly positive everywhere could be dropped at the price of additional technicalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For µ, such requirement is a consequence of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The Schr¨odinger system Let (Pt)t≥0 the semigroup generated by a d-dimensional Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For given marginals, µ, ν and T > 0 the Schr¨odinger system, whose unknowns ϕ, ψ we shall refer to as Schr¨odinger potentials, is given by � ϕ(x) = U µ(x) + log PT exp(−ψ)(x), x ∈ Rd, ψ(y) = U ν(y) + log PT exp(−ϕ)(y), y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (8) Under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1, it is known that the Schr¨odinger system admits a solution, and that if ( ¯ϕ, ¯ψ) is another solution, then there exists c ∈ R such that (ϕ, ψ) = ( ¯ϕ + c, ¯ψ − c), see [34, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 2][31] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Weak semiconvexity and semiconcavity bounds for Schr¨odinger potentials In the rest of the article, given a scalar function U, any pointwise lower bound on κU implying in particular that lim inf r→+∞ κU(r) > −∞ shall be called a weak semiconvexity bound for U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Next, in analogy with (3) we introduce for a differentiable U : Rd −→ R the function ℓU as follows: ℓU : (0, +∞) −→ R, ℓU(r) = sup{|x − y|−2⟨∇U(x) − ∇U(y), x − y⟩ : |x − y| = r}, and call a weak semiconcavity bound for U any pointwise upper bound for ℓU implying in particular that lim sup r→+∞ ℓU(r) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Our first main result is about weak semiconvexity and weak semiconcavity bounds for Schr¨odinger potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Let Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 hold and (ϕ, ψ) be solutions of the Schr¨odinger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then ϕ, ψ are twice differentiable and for all r > 0 we have κψ(r) ≥ αψ − r−1fL(r), (9) ℓϕ(r) ≤ βµ − α (1 + T α) + r−1fL(r) (1 + T α)2 , (10) where αψ > αν − 1/T can be taken to be the smallest solution of the fixed point equation α = αν − 1 T + G(α, 2) 2T 2 , α ∈ (αν − 1/T, +∞) (11) 5 with G(α, u) = inf{s ≥ 0 : F(α, s) ≥ u}, u > 0 (12) and F(α, s) = βµs + s T (1 + T α) + s1/2fL(s1/2) (1 + T α)2 , s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is proven at Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 that F(α, ·) is increasing on (0, +∞) for all α > −1/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' G(α, ·) is therefore its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We conjecture that αψ can be taken to be the largest solution of the fixed point equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To prove so, it would suffice to show that Sinkhorn’s iterates (see [34, Sec 6]) converge to solutions of the Schr¨odinger system under Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 for a large set of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We could not find such result in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is possible to check that if (H2) holds with L = 0, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 recovers the conclusion of [11, Theorem 4],after a change of variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To be more precise, the potentials (ϕε, ψε) considered there are related to the couple (ϕ, ψ) appearing in (8) by choosing ε = T and setting ϕε = ε � ϕ − U µ + | · |2 2ε � , ψε = ε � ψ − U ν + | · |2 2ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The rescaled potential T ϕ converges to the Brenier potential in the small noise limit [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' As explained in the introduction, one cannot deduce from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 an improvement over Caffarelli’s Theorem [8] by letting T → 0 in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Our second main result is that the static Schr¨odinger bridge ˆπ satisfies LSI with an explicit constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We recall here that a probability measure ρ on Rd satisfies LSI with constant C if and only if for all positive differentiable function f Entρ(f) ≤ C 2 � |∇f|2 f dρ, where Entρ(f) = � f log fdρ − � fdρ log � � fdρ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Let Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 hold and assume furthermore that µ satisfies LSI with constant Cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then the static Schr¨odinger bridge ˆπ satisfies LSI with constant max � Cµ, CµC0,T + � T 0 Ct,T dt � , where for all t ≤ T Ct,T := exp � − � T t αψ s ds � , αψ t := αψ 1 + (T − t)αψ − L (1 + (T − t)αψ)2 , and αψ is as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is well known that LSI has a number of remarkable consequences including, but certainly not limited to, spectral gaps and concentration of measure inequalities for Lipschitz observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' By taking µ to be a Gaussian distribution, we obtain as a corollary of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3 that any probability ν fulfilling (6) satisfies a logarithmic Sobolev inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To the best of our knowledge, this is a new result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is worth noticing (6) does not imply that ν is a bounded or Lipschitz perturbation of a log-concave distribution: therefore the results of [28][1] do not apply in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 6 2 Invariant sets of weakly convex functions for the HJB flow We introduce the notation U T,g t (x) := − log PT −t exp(−g)(x) = − log � 1 (2π(T − t))d/2 � exp � − |y − x|2 2(T − t) − g(y) � dy � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (13) With this notation at hand, (8) rewrites as follows: � ϕ = U µ − U T,ψ 0 , ψ = U ν − U T,ϕ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (14) It is well known that under mild conditions on g, the map [0, T ] × Rd ∋ (t, x) �→ U T,g t (x) is a classical solution of th HJB equation � ∂tϕt(x) + 1 2∆ϕt(x) − 1 2|∇ϕt|2(x) = 0, ϕT (x) = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (15) In the next theorem, we construct for any L > 0 a set of weakly convex functions FL that is shown to be invariant for the HJB flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In the proof, and in the rest of the paper we shall denote by [·, ·] the quadratic covariation of two Itˆo processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Fix L > 0 and define FL = {g ∈ C1(Rd) : κg(r) ≥ −r−1fL(r) ∀r > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then for all 0 ≤ t ≤ T < +∞ we have g ∈ FL ⇒ U T,g t ∈ FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (16) In the proof of the Theorem we profit from the fact that fL solves the ODE ff ′(r) + 2f ′′(r) = 0 ∀r > 0, f(0) = 0, f ′(0) = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (17) To verify the above, it suffices to compute f ′ L(r) = L cosh2( 1 2(2L)1/2r), f ′′ L(r) = 2−1/2L3/2 sinh( 1 2(2L)1/2r) cosh3( 1 2(2L)1/2r) Moreover, we recall here some useful properties of fL: fL(r) > 0, f ′ L(r) > 0, f ′′ L(r) < 0, fL(r) ≥ rf ′ L(r) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (18) We are now in position to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' As anticipated above, the proof relies on the analysis of coupling by reflection along the characteristics of the HJB equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In the recent article [14, Thm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3] Hessian bounds for HJB equations originating from stochastic control problems are obtained by means of coupling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' These are two-sided bounds that require an a priori knowledge of global Lipschitz bounds on solutions of the HJB equation to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The one-sided estimates of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 do not require any Lipschitz property of solutions and their proof require finer arguments than those used in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We first assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' that t = 0 and work under the additional assumption that g ∈ C3(Rd), sup x∈Rd |∇2g|(x) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (19) Combining the above with g ∈ FL, we can justify differentiation under the integral sign in (13) and establish that [0, T ] × Rd ∋ (t, x) �→ U T,g t (x) is a classical solution of (15) such that [0, T ] × Rd ∋ (t, x) �→ ∇U T,g t (x) is continuously differentiable in t as well as twice continuously differentiable and uniformly Lipschitz in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Under these regularity assumptions, for given x, ˆx ∈ Rd, coupling by reflection of two diffusions started at x and ˆx respectively and whose drift field is −∇U T,g t is well defined, see [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' That is to say, there exist a stochastic process (Xt, ˆXt)0≤t≤T with (X0, ˆX0) = (x, ˆx) and two Brownian motions (Bt, ˆBt)0≤t≤T all defined on the same probability space and such that � dXt = −∇U T,g t (Xt)dt + dBt, for 0 ≤ t ≤ T , d ˆXt = −∇U T,g t ( ˆXt)dt + d ˆBt, for 0 ≤ t ≤ τ, Xt = ˆXt for t > τ, where et = r−1 t (Xt − ˆXt), rt = |Xt − ˆXt|, d ˆBt = dBt − 2et⟨et, dBt⟩ and τ = inf{t ∈ [0, T ] : Xt = ˆXt} ∧ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We now define U : [0, T ] × Rd × Rd −→ R, Ut(x, ˆx) = � |x − ˆx|−1⟨∇U T,g t (x) − ∇U T,g t (ˆx), x − ˆx⟩, if x ̸= ˆx, 0 if x = ˆx, and proceed to prove that (U(Xt, ˆXt))0≤t≤T is a supermartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To this aim, we first deduce from (15) and Itˆo’s formula that d∇U T,g t (Xt) = dMt, d∇U T,g t ( ˆXt) = d ˆ Mt (20) where M·, ˆ M· are square integrable martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Indeed we find from Itˆo’s formula d∇U T,g t (Xt) = � ∂t∇U T,g t (Xt) − ∇2U T,g t ∇U T,g t (Xt) + 1 2∆∇U T,g t (Xt) � dt + ∇2U T,g t (Xt) · dBt (15) = ∇2U T,g t (Xt) · dBt, and a completely analogous argument shows that ∇U T,g t ( ˆ Xt) is a square integrable martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We shall also prove separately at Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 that det = −r−1 t proje⊥ t (∇U T,g t (Xt) − ∇U T,g t ( ˆXt))dt ∀t < τ, (21) 8 where proje⊥ t denotes the orthogonal projection on the orthogonal complement of the linear subspace generated by et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Combining together (20) and(21) we find that dUt(Xt, ˆXt) = 0 for t ≥ τ, whereas for t < τ dUt(Xt, ˆXt) = ⟨∇U T,g t (Xt) − ∇U T,g t ( ˆXt), det⟩ + ⟨et, d(∇U T,g t (Xt) − ∇U T,g t ( ˆXt))⟩ + d[(∇U T,g (X·) − ∇U T,g ( ˆX·)), e·]t (20)+(21) = −r−1 t |proje⊥ t (∇U T,g t (Xt) − ∇U T,g t ( ˆXt))|2dt + d ˜ Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' proving that (U(Xt, ˆXt))0≤t≤T is a supermartingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In the above, ˜ M· denotes a square integrable martingale and to obtain the last equality we used that the quadratic variation term vanishes because of (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Next, arguing exactly as in [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 60] (see also (25) below for more details) on the basis of Itˆo’s formula and invoking (17) we get dfL(rt) = [−f ′ L(rt)Ut(Xt, ˆXt) + 2f ′′ L(rt)]dt + dNt (17) = −f ′ L(rt)[Ut(Xt, ˆXt) + fL(rt)]dt + dNt, where N· is a square integrable martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It then follows that d � Ut(Xt, ˆXt) + fL(rt) � ≤ −f ′ L(rt) � Ut(Xt, ˆXt) + fL(rt) � dt + dNt + d ˜ Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (22) from which we deduce that the process Γt = exp � � t 0 f ′ L(rs)ds �� Ut(Xt, ˆXt) + fL(rt) � is a supermartingale and in particular is decreasing on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' This gives |x − ˆx|−1⟨∇U T,g 0 (x) − ∇U T,g 0 (ˆx), x − ˆx⟩ + fL(|x − ˆx|) = E[Γ0] ≥ E[ΓT ] ≥ E � exp( � T 0 f ′ L(rs)ds) � |XT − ˆXT |κg(|XT − ˆXT |) + fL(|XT − ˆXT |) �� ≥ 0, where the last inequality follows from g ∈ FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We have thus completed the proof under the additional assumption (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In order to remove it, consider any g ∈ FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then there exist (gn) ⊆ FL such that (19) holds for any of the gn, gn → g pointwise and gn is uniformly bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' From this, one can prove that ∇U gn,T 0 → ∇U g,T 0 pointwise by differentiating (13) under the integral sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Using this result in combination with the fact that (16) holds for any gn allows to reach the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Under the same assumptions and notations of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 we have det = −r−1 t proje⊥ t (∇U T,g t (Xt) − ∇U T,g t ( ˆXt))dt ∀t < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Recall that if θ : Rd → R is the map z �→ |z|, then we have ∇θ(z) = z |z|, ∇2θ(z) = I |z| − zz⊤ |z|3 , z ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (23) 9 The proof consist of several applications of Itˆo’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We first observe that for t < τ d(Xt − ˆXt) = −(∇U T,g t (Xt) − ∇U T,g t ( ˆXt))dt + 2etdWt, with dWt = ⟨et, dBt⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (24) Note that by L´evy characterization, (Wt)0≤t≤T is a Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Thus, invoking (23) (or refferring directly to [22, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 60] we obtain drt = −⟨∇U T,g t (Xt) − ∇U T,g t ( ˆXt), et⟩dt + 2dWt, (25) whence dr−1 t = −r−2 t drt + r−3 t d[r]t = � r−2 t ⟨∇U T,g t (Xt) − ∇U T,g t ( ˆXt), et⟩ + 4r−3 t � dt − 2r−2 t dWt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (26) Combining (24) with (26) we find that for t < τ det = d � r−1 t (Xt − ˆXt)) = r−1 t d(Xt − ˆXt) + (Xt − ˆXt)d(r−1 t ) + d[X· − ˆX·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' r−1 ]t = −r−1 t (∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t (Xt) − ∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t ( ˆXt))dt + 2r−1 t etdWt + � r−2 t ⟨∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t (Xt) − ∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t ( ˆ Xt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' et⟩ + 4r−3 t � (Xt − ˆXt)dt − 2r−2 t (Xt − ˆXt)dWt − 4r−2 t etdt = −r−1 t � ∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t (Xt) − ∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t ( ˆXt) − ⟨∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t (Xt) − ∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t ( ˆXt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' et⟩et � dt = −(r−1 t )proje⊥ t (∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t (Xt) − ∇U T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g t ( ˆXt))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 3 Second order bounds for Schr¨odinger potentials From now on Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 is in force, even if we do not specify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Moreover, since we show at Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 in the appendix that (H2′) implies (H2), we shall always assume that (H2) holds in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The next two subsections are devoted to establish the key estimates needed in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2, that is carried out immediately afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 Weak semiconvexity of ψ implies weak semiconcavity of ϕ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Assume that α > −1/T exists such that κψ(r) ≥ α − r−1fL(r) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then we have ℓϕ(r) ≤ βµ − α 1 + T α + r−1fL(r) (1 + T α)2 = r−2F(α, r2) − 1 T ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We define ˆψ(·) = ψ(·) − α 2 | · |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' and note by assumption ˆψ ∈ FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We claim that U T,ψ 0 (x) = α 2(1 + T α)|x|2 + U T/(1+T α), ˆψ 0 ((1 + T α)−1x) + C, (27) where C is some constant independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Indeed we have U T,ψ 0 (x) − d 2 log(2πT ) = − log � exp � − |y − x|2 2T − α 2 |y|2 − ˆψ(y) � dy = − log � exp � − α|x|2 2(1 + T α) − 1 + T α 2T |y − (1 + T α)−1x|2 − ˆψ(y) � dy = α|x|2 2(1 + T α) + U T/(1+T α), ˆ ψ 0 ((1 + T α)−1x) − d 2 log(2πT/(1 + T α)) Since ˆψ ∈ FL, we can invoke Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 to obtain κUT,ψ 0 (r) ≥ α 1 + T α − r−1fL(r) (1 + T α)2 ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (28) The desired conclusion is then obtained from (14) and Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 Weak semiconcavity of ϕ implies weak semiconvexity of ψ We begin by recording some useful properties of the functions F(·, ·) and G(·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Let T, βµ > 0, L ≥ 0 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (i) For any α > −1/T the function s �→ F(α, s) is concave and increasing [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (ii) α �→ G(α, 2) is positive and non decreasing over (− 1 T , +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (iii) The fixed point equation (11) admits at least one solution on (αν − 1/T, +∞) and αν − 1/T is not an accumulation point for the set of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We begin with the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To this aim, we observe that fL is increasing on [0, +∞) and therefore so is s �→ s1/2fL(s1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Therefore d dsF(α, s) ≥ βµ + 1 T (1 + T α) > 0, where we used α > −1/T in the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To prove concavity, we observe that d2 du2 � u1/2fL(u1/2) ���� u=s = s−1/2 4 f ′′ L(s1/2) + s−3/2 4 (f ′ L(s1/2)s1/2 − fL(s1/2)) (18) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 11 Thus s �→ s1/2fL(s1/2) is concave and so is F(α, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We now move on to the proof of (ii) by first showing that G(·, 2) is positive and then showing that it is increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' If this was not the case then G(α, 2) = 0 for some α > −1/T and therefore there exists a sequence (sn)n≥0 such that sn → 0 and F(α, sn) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' But this is impossible since lims↓0 F(α, sn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Next, we observe that F(α, s) is increasing in s from item (i) and decreasing in α for α ∈ (−1/T, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For this reason, for any u and α′ ≥ α we have {s : F(α′, s) ≥ u} ⊆ {s : F(α, s) ≥ u} and therefore G(α′, u) ≥ G(α, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To prove (iii), we introduce h : [αν − 1 T , +∞) −→ R, h(α) := α − � αν − 1 T + G(α, 2) 2T 2 � Note that that h is continuous on its domain since G(·, 2) is so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Therefore, to reach the conclusion it suffices to show that h(αν − 1 T ) < 0, lim α→+∞ h(α) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (29) The first inequality is a direct consequence of G(αν − 1/T ) > 0, that we have already proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The second inequality is proven if we can show that lim sup α→+∞ G(α, 2) ≤ 1 2βµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (30) To see that this relation holds, observe that, using fL(r) ≥ 0 we obtain that for any α > −1/T F(α, s) ≥ βµs ∀s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' But then we obtain directly from (12) that G(α, 2) ≤ 1 2βµ , thus proving (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We shall now introduce the modified potential ¯ψ as follows ¯ψ(y) = T � ψ(y) − U ν(y) + |y|2 2T � , (31) It has been proven at [11, Lemma 1] that the Hessian of ¯ψ relates to the covariance matrix of the conditional distributions of the static Schr¨odinger bridge ˆπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' That is to say, ∇2 ¯ψ(y) = 1 T CovX∼ˆπy(X) (32) where ˆπy is (a version of) the conditional distribution of ˆπ that, in view of (8) has the following form: ˆπy(dx) = exp(−V ˆπy(x)))dx � exp(−V ˆπy(¯x))d¯x, V ˆπy(x) := ϕ(x) + |x|2 2T − xy T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (33) 12 We shall give an independent proof of (32) under additional regularity assumptions at Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 in the Appendix for the readers’ convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' A consequence of (32) is that ¯ψ is convex and we obtain from (31) that κψ(r) ≥ αν − 1 T − r−1fL(r) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (34) This is a first crude weak semiconvexity bound on ψ upon which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 improves by means of a recursive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We show in the forthcoming Lemma how to deduce weak semiconvexity of ψ from weak semiconcavity of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In the L = 0 setting, this step is carried out in [11] invoking the Cramer-Rao inequality, whose application is not justified in the present more general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Assume that α > −1/T exists such that ℓϕ(r) ≤ − 1 T + r−2F(α, r2) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (35) Then κψ(r) ≥ αν − 1 T + G(α, 2) 2T 2 − r−1fL(r) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Recalling the definition of V ˆπy given at (33) we observe that the standing assumptions imply ℓV ˆπy (r) ≤ r−2F(α, r2) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (36) In view of (32), we now proceed to bound VarX∼ˆπy(X1) from below for a given y, where we adopted the notational convention X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' , Xd) for the components of random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We first observe that VarX∼ˆπy(X1) ≥ EX∼ˆπy[VarX∼ˆπy(X1|X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' , Xd)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (37) Moreover, upon setting for any z = (z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' , zd) V ˆπy,z(·) := V ˆπy(·, z), ˆπy,z(dx) = exp(−V ˆπy,z(x))dx � exp(−V ˆπy,z(¯x))d¯x we have VarX∼ˆπy(X1|X2 = z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' , Xd = zd) = 1 2 � |x − ˆx|2ˆπy,z(dx)ˆπy,z(dˆx) With this notation at hand, we find that, uniformly in z ∈ Rd−1, 1 = 1 2 � (∂xV ˆπy,z(x) − ∂xV ˆπy,z(ˆx))(x − ˆx)ˆπy,z(dx)ˆπy,z(dˆx) = 1 2 � ⟨∇V ˆπy(x, z) − ∇V ˆπy(ˆx, z), (x, z) − (ˆx, z)⟩ˆπy,z(dx)ˆπy,z(dˆx) (36) ≤ 1 2 � F(α, |x − ˆx|2)ˆπy,z(dx)ˆπy,z(dˆx) ≤ 1 2F(α, 2VarX∼ˆπy(X1|X2 = z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' , Xd = zd)) where to establish the last inequality we used Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2(i) and Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Since α > −1/T , invoking again Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2(i) we have that s �→ F(α, s) is non decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' But then, we get from (37) and the last bound that VarX∼ˆπy(X1) ≥ 1 2G(α, 2), ∀y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 13 Next, we observe that, because of the fact that if ϕ(·) satisfies (35) then so does ϕ(O·) for any orthonormal matrix O, repeating the argument above yields VarX∼ˆπy(⟨v, X⟩) ≥ 1 2G(α, 2), ∀y, v ∈ Rd s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' |v| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In light of (32), this implies ⟨v, ∇2 ¯ψ(y)v⟩ ≥ G(α, 2) 2T |v|2 ∀v, y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' But then, since ψ(y) = U ν(y) − |y|2 2T + ¯ψ(y) T we immediately obtain κψ(r) ≥ αν − 1 T + G(α, 2) 2T 2 − r−1fL(r)∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 The proof is obtained combining the results of the former two sections through a fixed point argu- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We define a sequence (αn)n≥0 via α0 = αν − 1 T , αn = αν − 1 T + G(αn−1, 2) 2T 2 , n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2(ii) and an induction argument, we obtain that α1 ≥ α0 and (αn)n≥0 is a non decreasing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' If we denote by α∗ the limit, then by continuity of G(·, 2), we know that α∗ > αν − 1/T and α∗ satisfies the fixed point equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To conclude the proof, we show by induction that κψ(r) ≥ αn − r−1fL(r) ∀n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (38) The case n = 0 is (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For the inductive step, suppose (38) holds for a given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 gives that ℓϕ(r) ≤ r−2F(αn, r2) − 1 T ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' But then, an application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 proves that for all r > 0 we have κψ(r) ≥ αν − 1 T + G(αn, 2) 2T 2 − r−1fL(r) = αn+1 − r−1fL(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The proof of (9) is now finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To conclude, we observe that (10) follows directly from (9) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 14 4 Logarithmic Sobolev inequality for Schr¨odinger bridges This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3 and is structured as follows: we first recall known facts about logaithmic Sobolev inequalities and gradient estimates for diffusion semigroups whose proofs can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' in [2] and eventually prove at Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 a sufficient condition for the two-times distribution of a diffusion process to satisfy LSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Though such a result may not appear surprising, we could not find it in this form in the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We then proceed to elucidate the connection between Schr¨odinger bridges and Doob h-transforms at Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2, and then finally prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Local LSIs and gradient estimates Let [0, T ] × Rd ∋ (t, x) �→ Ut(x) be continuous in the time variable and twice differentiable in the space variable with ⟨v, ∇2Ut(x)v⟩ ≥ αt|v|2 ∀x, v ∈ Rd, t ∈ [0, T ] for some function αt uniformly bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We consider the time-inhomogeneous semigroup (Ps,t)0≤s≤t≤T generated by the diffusion process whose generator at time t acts on smooth functions with bounded support as follows f �→ 1 2∆f − ⟨∇Ut, ∇f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We now recall some basic fact about gradient estimates and local LSIs for the semigroup (Ps,t)0≤s≤t≤T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' For time-homogeneous semigroups these facts are well known and can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' in [2]: the adaptation to the time-inhomogeneous setting is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' The first result we shall need afterwards is the gradient estimate (see [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='18]) |∇Pt,T f|(x) ≤ Ct,T Pt,T (|∇f|)(x), Ct,T = exp � − � T t αsds � , (39) that holds for all (t, x) ∈ [0, T ] × Rd and any continuously differentiable f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Moreover, the local logarithmic Sobolev inequalities (see [2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2]) (P0,T f log f)(x) − (P0,T f)(x) log(P0,T f)(x) ≤ ˜C0,T 2 P0,T (|∇f|2/f)(x), ˜C0,T = � T 0 Ct,T dt (40) hold for all x ∈ Rd and all positive continuously differentiable f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' In the next Lemma we show how obtain LSI for the joint law at times 0 and T of a diffusion process with initial distribution µ and drift −∇Ut, that is to say for the coupling π defined by � Rd×Rd f(x, y)π(dxdy) = � Rd P0,T f(x, ·)(x)µ(dx) ∀f > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (41) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Assume that µ satisfies LSI with constant Cµ and let π be as in (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then π satisfies LSI with constant max{Cµ, CµC0,T + ˜C0,T }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To proof is carried out ”mixing” carefully with the help of the gradient estimate the local (conditional) LSIs (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Similar arguments and ideas can be found e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' in [6, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We recall the decomposition of the entropy formula (see [30, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='4]) Entπ(f) = Entµ(f0) + � Rd Entπx(f x)f0(x)µ(dx), where we adopted the following conventions f0(x) = (P0,T f(x, ·))(x), f x(y) = f(x, y)/f0(x), � g(y)πx(dy) = � P0,T g � (x) ∀g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We know from (40) that uniformly in x, f we have Entπx(f x) = P0,T � f x log f x� (x) − � P0,T f x log P0,T f x� (x) ≤ ˜C0,T 2f0(x) � |∇yf(x, y)|2/f(x, y)πx(dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' This gives � Entπx(f x)f0(x)µ(dx) ≤ ˜C0,T 2 � |∇yf(x, y)|2 f(x, y) π(dxdy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (42) Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' we use LSI for µ to obtain Entµ(f0) ≤ Cµ 2 � |∇xf0(x)|2/f0(x)µ(dx) = Cµ 2 � |P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T (∇xf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·))(x)|2(P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·))−1(x) µ(dx) + Cµ 2 � |∇zP0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T (f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·))(z)|2��� z=x(P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·))−1(x) µ(dx) (43) For the first summand on the rhs of (43),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' we can argue on the basis of Jensen’s inequality applied to the convex function a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' b �→ a2/b to obtain Cµ 2 � |P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T (∇xf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·))(x)|2(P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·))−1(x)µ(dx) ≤ Cµ 2 � P0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='T � |∇xf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·)|2/f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' ·) � (x)µ(dx) = Cµ 2 � |∇xf(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' y)|2/f(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' y)π(dxdy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (44) For the second summand on the rhs of (43), we first invoke the gradient estimate (39) and eventually apply again Jensen’s inequality to obtain Cµ 2 � |∇zP0,T (f(x, ·))(z)|2��� z=x(P0,T f(x, ·))−1(x)µ(dx) ≤ CµC0,T 2 � (P0,T (|∇yf(x, ·)|)(x))2(P0,T f(x, ·))−1(x)µ(dx) ≤ CµC0,T 2 � P0,T � |∇yf(x, ·)|2/f(x, ·) � (x)µ(dx) = CµC0,T 2 � |∇yf(x, y)|2/f(x, y) π(dxdy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (45) Gathering (42)-(44)-(45) we obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 16 In the next lemma, we represent the static Schr¨odinger bridge (2) through a diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' It is a classical result saying that Schr¨odinger bridges are indeed Doob’s h-transforms, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' [31, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' 4][16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Let Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 hold and ˆπ be the static Schr¨odinger bridge (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then ˆπ has the form (41), where (Ps,t)0≤s≤t≤T is the time-inhomogeneous semigroup associated with the generator acting on smooth test functions as follows f �→ 1 2∆f − ⟨∇U T,ψ t , ∇f⟩, t ∈ [0, T ], (46) where U T,ψ t has been defined at (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Let ψ be the Schr¨odinger potential issued from (8) and denote by Q the law on C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Rd) of a solution (Xt)t∈[0,T ] to the stochastic differential equation dXt = −∇U T,ψ t (Xt)dt + dBt, X0 ∼ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Note that, because of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2, existence of strong solutions and pathwise uniqueness hold for the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Next, we denote by P the Wiener measure with initial distribution µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' By Girsanov’s Theorem, see [29] for a version that applies in the current setting, we know that dQ dP (ω) = exp � − � T 0 ∇U T,ψ t (ωt)dωt − 1 2 � T 0 |∇U T,ψ t (ωt)|2dt � P − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=', where we denote by ω the typical element of the canonical space C([0, T ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Using Itˆo formula we rewrite the above as dQ dP (ω) = exp � U T,ψ 0 (ω0) − U T,ψ T (ωT ) + � T 0 � ∂tU T,ψ t + 1 2∆U T,ψ t − 1 2|∇U T,ψ t |2� (ωt)dt � = exp(U µ(ω0) − ϕ(ω0) − ψ(ωT )) where we used the Schr¨odinger system (8) and the HJB equation (15) to obtain the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Indeed because of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 one can deduce that [0, T ] × Rd ∋ (t, x) �→ U T,ψ t (x), is a classical solution of (15) by differentiating under the integral sign in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' From this, we deduce that dQ0T dP0T (x, y) = exp � U µ(x) − ϕ(x) − ψ(y) � P0T − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=', where Q0T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' P0T ) denotes the joint distribution of Q (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' P) at times 0 and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Since dP0T (dxdy) = (2πT )−d/2 exp(−U µ(x)) exp � − |y − x|2 2T � dxdy, we conclude that dQ0T (dxdy) = (2πT )−d/2 exp � − ϕ(x) − ψ(y) − |y − x|2 2T � dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' But then Q0T = ˆπ, where ˆπ is defined at (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To conclude, we observe that Q0T has the desired form (41) where (Ps,t)0≤s≤t is indeed the semigroup generated by (46) 17 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We know by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2 that ˆπ has the form (41) for the inhomogeneous semigroup generated by (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' We now set αψ t = inf x,v∈Rd,|v|=1⟨v, ∇2U ψ,T t (x), v⟩ and proceed to estimate αψ t from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' To do so, we observe that U ψ,T t = U ψ,T −t 0 and argue exactly as we did to establish (28) to obtain that κUT,ψ t (r) ≥ αψ 1 + (T − t)αψ − r−1fL(r) (1 + (T − t)αψ)2 ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' From here, using the concavity of fL and f ′ L(0) = L we obtain αψ t ≥ αψ 1 + (T − t)αψ − L (1 + (T − t)αψ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' At this point, the conclusion follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 5 Appendix Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Assume that U satisfies (4) for some α > 0, L′, R ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then κU(r) ≥ α − r−1fL(r) ∀r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' with L given by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' If r > R the claim is a simple consequence of fL(r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' If r ≤ R, using (18) to get that r′ �→ r′−1fL(r′) is non increasing on (0, +∞), we obtain r−1fL(r) ≥ R−1fL(R) = L′, from which the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Let Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1 hold and assume furthermore that there exist ε, γ′ > 0 such that � exp(γ′|x|1+ε)µ(dx) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (47) Moreover, let ¯ψ be as in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Then ¯ψ is twice differentiable and we have ∇2 ¯ψ(y) = 1 T CovX∼ˆπy(X) ∀y ∈ Rd, where ˆπy is given by (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' From (8) we obtain that ¯ψ(y) + d 2 log(π) = T log � Rd exp � − ϕ(x) − |x|2 2T + ⟨x, y⟩ T � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (48) 18 From Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content='1, (8) and (47) it follows that � Rd×Rd exp � γ′|x|1+ε − ϕ(x) − ψ(y) − |x − y|2 2T � dx dy < +∞, whence the existence of some y′ such that � Rd×Rd exp � γ′|x|1+ε − ϕ(x) − |x|2 2T + ⟨x, y′⟩ T � dx < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' From this, we easily obtain that for all γ < γ′ � Rd exp � γ|x|1+ε − ϕ(x) − |x|2 2T + ⟨x, y⟩ T � dx < +∞ ∀y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DtAyT4oBgHgl3EQfSPej/content/2301.00083v1.pdf'} +page_content=' (49) Thanks to (49) we can apply the dominated convergence theorem and differentiate under the integral sign in (15) to 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a/ENE1T4oBgHgl3EQfEQP0/vector_store/index.pkl b/ENE1T4oBgHgl3EQfEQP0/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..6c8fda62cf5aa141e6b5695c430075de2842a7a6 --- /dev/null +++ b/ENE1T4oBgHgl3EQfEQP0/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a0304c4e22e91c4882af1b1b0a979b109104530de11aa39e6ff23e791bc9a040 +size 238394 diff --git a/EdE3T4oBgHgl3EQfVQpt/content/tmp_files/2301.04458v1.pdf.txt b/EdE3T4oBgHgl3EQfVQpt/content/tmp_files/2301.04458v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..57094667db4888285f86b0f77ae729fa09045dad --- /dev/null +++ b/EdE3T4oBgHgl3EQfVQpt/content/tmp_files/2301.04458v1.pdf.txt @@ -0,0 +1,1536 @@ +Femtosecond laser-induced sub-wavelength plasma inside dielectrics: III. +Terahertz radiation emission +Kazem Ardaneh,1, 2, a) Ken-Ichi Nishikawa,3 Remo Giust,1 Benoit Morel,1 Pierre-Jean Charpin,1 Arnaud +Couairon,4 Guy Bonnaud,5 and Francois Courvoisier1, b) +1)FEMTO-ST Institute, Univ. Bourgogne Franche-Comt´e, CNRS, 15B avenue des Montboucons, 25030, +Besan¸con Cedex, France +2)Sorbonne University, Pierre and Marie Curie Campus, 4 place Jussieu, 75252, Paris Cedex 5, +France +3)Department of Physics, Chemistry and Mathematics, V. Murry Chambers Bld., Alabama A&M University, +Huntsville, AL 35810, USA +4)CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, F-91128 Palaiseau, +France +5)CEA, Centre de Paris-Saclay, DRF, Univ. +Paris-Saclay, 91191 Gif-sur-Yvette, +France +(Dated: 12 January 2023) +Electromagnetic radiation within the terahertz (THz) frequency range is of great interest for applications +in remote sensing and time-domain spectroscopy. +The laser-induced plasmas are promising mediums for +generating THz radiation. +It has been recently reported that focusing femtosecond Bessel pulses inside +dielectrics induces a high aspect ratio over-critical plasmas. Here we show that the intense resonantly driven +electrostatic fields at the so-called critical surface lead to THz radiation emission. Through three-dimensional +particle-in-cell simulation and analytical derivation, we have investigated the emission of THz radiation. We +show that the THz radiation is associated with a hot population of electrons trapped in ambipolar electric +fields of the double layers. +I. +INTRODUCTION +Terahertz (THz) radiation, typically referred to as the +frequency band 100 GHz − 10 THz, in the infrared and +microwaves ranges, has been attracting ongoing interest +because of its broad applications ranging from biomedical +imaging, security or packaged goods inspection, to time- +domain spectroscopy.1–4 Using femtosecond laser pulses, +the techniques for generating THz radiation are gen- +erally classified as either optical rectification,5–11 tran- +sient current sources,2,12–23 or a combination of these two +mechanisms.24 +Optical rectification can induce THz radiation in +non-centrosymmetric crystals, e.g., ZnTe and GaAs, in +which the fundamental frequency of an infrared fem- +tosecond laser pulse is down-converted to the THz fre- +quency via the second-order susceptibility where the po- +larization reads P (ωTHz) = χ(2)E(ω + ωTHz)E∗(ω).5,7 +The frequency of the rectified pulse envelope is in +the range of 3-10 THz.7 In the centrosymmetric me- +dia, e.g., gases, two-color illumination can be used +to mix the fundamental frequency with the second- +harmonic in a four-wave mixing process as P (ωTHz) = +χ(3)E(2ω −ωTHz) E∗(ω)E∗(ω).8,11 Correspondingly, the +THz component might match with the fundamental one +in an inverse process and leads to second-harmonic gen- +eration, which is a method for THz detection. A THz +emission with an estimated field strength ∼ 400 kV/cm +a)Electronic mail: kazem.arrdaneh@gmail.com +b)Electronic mail: francois.courvoisier@femto-st.fr +has been reported in which the presence of plasma was +essential for the high-efficiency process.10 +Laser-induced plasmas are attractive for THz radia- +tion generation because of their ability to sustain ex- +tremely intense electromagnetic fields.13,14,25–27 In this +context, femtosecond laser-induced breakdown of gases +is investigated widely,28,29 mostly using the two-color +approach.8 The peak of the THz field however satu- +rates for laser intensities higher than 1015 W/cm2 be- +cause of the strong THz absorption in the long (∼7 mm) +air plasma.17 Plasma generation in laser-solid interac- +tions offer an alternative: experiments of ultrashort laser +pulses-solid interaction have shown a monotonous in- +crease in THz radiation with the incident laser intensity +up to 1019 W/cm2.18,19,21 +Illumination of solid targets by intense ultrashort laser +beams results in the generation of hot electron currents +that are the source of the secondary electromagnetic ra- +diation ranging from x-rays30–33 to THz radiation.19 Un- +der p−polarized laser illumination of a short-scale inho- +mogeneous plasma, for moderate laser intensities (bellow +1014 W/cm2), resonance absorption is the main mecha- +nism for hot electron generation.34–36 +In previous papers, we have reported that over-critical +plasmas, i.e. with density above the reflection density for +the incident laser wavelength, are generated by focusing +Bessel beams with moderate intensities on the order of +1014 W/cm2 inside sapphire.37–39 A Bessel beam is a so- +lution of the wave equation in which the wave amplitude +is defined by the Bessel function of the first kind.40 Im- +portantly, the axial intensity profile of the Bessel beam is +propagation invariant. Therefore, all segments of the di- +electric along the Bessel zone will receive simultaneously +arXiv:2301.04458v1 [physics.plasm-ph] 11 Jan 2023 + +2 +FIG. 1. The energy spectrum of the radiation emitted by a population of hot electrons: (a) the spatially averaged, (b) angular +distribution, and (c) spatial distribution averaged in the frequency range of 1-30 rad/ps. +The hot electrons are randomly +selected. +the same amount of energy which results in a high aspect +ratio plasma rod. +In the first article of this series (Ardaneh et al.,38 Pa- +per I hereafter), we confirmed that the resonance of the +plasma waves can explain the experimental diagnostics +of total absorption, and far-field intensity pattern. We +reported electron acceleration up to several keV while +surfing the plasma waves. In the outward propagation of +hot electrons, electrostatic ambipolar fields form at the +plasma surface due to the different inertia of the electron +and ion. Moreover, in the second article of this series +(Ardaneh et al.,39 Paper II hereafter), we reported the +second-harmonic generation by a second-order current +of hot electrons near the critical surface. The electron +currents form by the resonance absorption and radiation +force of the incident laser wave. +In the current work as the third in this series, we estab- +lish a link between resonance absorption-driven currents +and THz radiation. This is based on calculating the co- +herent radiation spectrum of the hot electrons for the +performed Particle-In-Cell (PIC) simulation. The sim- +ulation consists of electron-ion plasma initially induced +by multi-photon and collisional ionizations. The dipole +moments are induced due to the radiation force of the +resonance fields. For a laser field with frequency ω0, this +force induces a second-harmonic component at 2ω0 and +a low-frequency component by separating the light elec- +trons from the heavy ions. We have developed an ana- +lytical model for THz generation in laser-plasma inter- +actions to explain the underlying physics, in particular, +how the dipole moment is created in the plasma and the +characteristics of the radiation spectrum. +We organized the paper as follows. In Sec. II, we re- +call the setup of the PIC simulation as discussed in Paper +I38, we detail the radiation diagnostic, and the results of +the simulation. Then, in Sec. III, we derive an analyti- +cal solution for the current source of THz radiation, and +the radiated electromagnetic fields with their frequency +spectrums. +TABLE I. Simulation setup. +Parameter +Value +Simulation volume +15 × 15 × 30 µm3 +Grid resolution +∆x:yk0 +r = 0.04a, ∆zk0 +z = 0.1b +FDTD order +Second-order +BC for fieldsc +Perfectly matched layers +BC for particles +Outflow +Pulse energy (Ep) +1.2 µJ +Pulse frequency (ω0) +2.35 rad/fs +Pulse cone angle (θ) +25◦ +Pulse temporal profile +exp[−(t − tc)2/T 2] +Central time (tc) +130 fs +Pulse FWHM = +√ +2 ln 2T +100 fs +Pulse spatial profile +exp(−r2/w2 +0) +Pulse spatial waist (w0) +10 µm +Maximum density (nmax) +5 nc +Density profile (axial) +tanh(zµm) +Mass ratio (mi/me) +102 × 1836d +Plasma distribution [f(ve:i)] Maxwellian +Plasma temperature (Te:i) +1 eV +Particles per cell per species 32 +Particle shape function +triangle +Time step +∆tω0 = 0.07 +Simulation time +320 fs +a k0 +r = k0 sin θ. +b k0 +z = k0 cos θ. +c BC: Boundary condition. +d 102 is the sapphire molar mass. +II. +PIC SIMULATION +We performed self-consistent PIC simulation using +the three-dimensional massively parallel electromag- +netic code EPOCH41. +In our simulation, we used +the plasma parameters that could reproduce our ex- +perimental measurements (far-field, near-field, absorp- + +dW(WTHz)/dQ[a.u. ] +dW(WTHz)/dΩ[a. u. ] +0.25 +0.50 +0.75 +1.00 +0.25 +0.50 +0.75 +1.00 +180 +1 +100↓ +(a) +(b) +(c) +u. +135 +esin Φ +10-1. +deg] +dW/dw [ +90 +0 +sin( +10-2 +45 - +10-3 +0 +-1 +-2 +0 +90 +180 +270 +360 +-1 +0 +2 +0 +[0m]3 +Φ[deg] +sin Ocos Φ3 +tion) as reported in Paper I,38 and II.39 The simula- +tion setup is summarized in Table I. The plasma is +fully ionized and composed of electrons and ions with +equal densities (to preserve electric neutrality) given +by n = nmax exp(−x2/w2 +x) exp(−y2/w2 +y) tanh(zµm) with +FWHMx = +√ +2 ln 2wx = 250 nm, and FWHMy += +√ +2 ln 2wy = 600 nm. +There are initially 32 particles +per cell per species leading to the total number of parti- +cles in the simulation ∼ 109. The collisions are modeled +through a binary model as presented in Refs.41,42. +We +injected +from +the +zmin +boundary +a +linearly +x−polarized Gaussian pulse propagating along the pos- +itive z−direction. We applied a phase to the Gaussian +beam to create a Bessel-Gauss beam.43 The peak inten- +sity in the Bessel zone is 6 × 1014 W/cm2 in absence of +plasma. The time step is limited by the Courant con- +dition. +The minimum frequency in the simulation is +1.5 rad/ps which is well below the peak frequency of the +THz spectrum at 30 rad/ps. +One of the primary advantages of PIC codes is the pos- +sibility to access full information about the particles. We +have developed a radiation diagnostic that utilizes the +position and momentum of particles over time and cal- +culates the radiated fields and energy. For this purpose, +let us consider a particle at position r (t) at time t. At the +same time, we observe the radiated electromagnetic fields +from the particle at position x. Due to the finite velocity +of light, we observe the particle at an earlier position r (t′) +where it was at the retarded time t′ = t−R (t′) /c, where +R (t′) = |x − r (t′)| is the distance from the charged par- +ticle (at the retarded time t′) to the observer. The mag- +netic and electric fields produced from a moving point +charge can be calculated directly from their scalar and +vector potentials known as the Li´enard–Wiechert poten- +tials. The electric field reads:44 +E(x, t) = +Velocity field +� +�� +� +e +� +n − β +γ2(1 − β · n)3R2 +� +ret ++ +Acceleration field +� +�� +� +e +c +� +n × {(n − β) × ˙β} +(1 − β · n)3R +� +ret +(1) +where n = R (t′) / |R (t′)| is a unit vector pointing +from the particle retarded position to the observer, β = +v/c the particle instantaneous velocity, +˙β = dβ/dt is +the acceleration divided by c, γ is the Lorentz fac- +tor. The spatial spectra are obtained by the choice of +n +� +n2 +x + n2 +y + n2 +z = 1 +� +. The field in Eq. (1) divides itself +into ”velocity fields,” which are independent of acceler- +ation, and ”acceleration fields,” which depend linearly +on ˙β. The velocity field is a static field decreasing as +R−2 while the acceleration field is a radiation field, be- +ing transverse to the radius vector and falling off as R−1. +The total energy W radiated per unit solid angle dΩ per +unit frequency dω from the accelerated charged particle +reads:44 +d2W +dωdΩ = e2ω2 +4π2c +���� +� ∞ +−∞ +dt′ˆn × (ˆn × β)ejω(t′+R(t′)/c) +���� +2 +(2) +In our simulations, we collected the THz radiation +emissions from the hot electrons as follows. +We have +tracked 105 electrons in the simulations and recorded the +information of these electrons. We calculated the energy +spectrum of the radiation emitted by 100 randomly se- +lected electrons according to Eq. (2). The result is shown +in Fig. +1(a). +We see two peaks around ω = 0, with +a width of typically 50 rad/ps. We note that, because +of computing memory limitations, the time resolution of +the particle positions is insufficient to capture the second +harmonic emission. +The angular and spatial distribu- +tions of the THz emission are obtained by averaging the +energy spectrum in the frequency range of 1-30 rad/ps +[Figs. 1(b) and 1(c)]. +As one expects, the energy spectrum has sharp max- +ima at the laser frequency ω0 due to strong electron ac- +celeration in resonantly driven plasma waves at the crit- +ical surfaces. +The spectrum also has maxima at ω ≈ +30 rad/ps. The angular distribution of this THz radia- +tion in Fig. 1(b) shows maxima around (θ, φ) ≈ (0, π/2) +and (0, 3π/2), perpendicular to the electron acceleration +which is mainly in the x−direction. +The small tilt in +Fig. 1(c) is due to the asymmetric distribution of the +randomly selected electrons in xy−space over the inte- +grated time (a similar deviation occurred for another set +of 100 electrons). +We select some representative electrons to calculate +the radiated fields in a spatial window |x| ⩽ 45 µm and +|y| ⩽ 45 µm at z = 0. Using a time window, we also ex- +amined in which part of the trajectory, the electron emits +electromagnetic radiation in the THz frequency range +[Figs. 2(a)]. For each time window, we calculated the +intensity distribution I(ωTHz, x, y) by performing a dis- +crete Fourier transform on each component of the electric +field, Ex:y:z(t, x, y) and averaging in the frequency range +of 1-30 rad/ps [Figs. 2(b)-2(f)]. +Figure 2(a) shows the time evolution of the Ex com- +ponent, parallel to the incident laser polarization over- +plotted with the trajectory of a representative electron. +One can see the resonance plasma waves induced at the +critical surfaces (x = ±0.2 µm), in the time between 70- +200 fs (See Fig. 4 in Paper I38 for more details). Near +the peak of the laser field, the intense ambipolar fields +propagating with the sound speed are generated at the +surface of the plasma (dashed lines). The ambipolar field +sign is positive for x > 0 and negative for x < 0. The ra- +diation force (See Appendix A) due to the intense, local- +ized resonance field ejects electrons from the resonance +region. The electrons are ejected from the critical sur- +faces in the positive x−direction where x > 0 and nega- +tive x−direction where x < 0 as shown in Fig. 6 Paper I. +Therefore, the electrons ejected with energy less than the +potential barrier of the ambipolar field will be reflected by + +4 +FIG. 2. THz radiation from an electron trapped in the ambipolar electric fields of double layers. Shown are: (a) x−component +of the electric field over-plotted by the trajectory of a representative electron, electron emission for a time window between: +(b) 20-150 fs [shown by blue � symbols in panel (a), and blue line in panel (b)], and 182-312 fs [shown by red � symbols in +panel (a) and red line in panel (b)], (c-f) cE2 +x/8π, cE2 +y/8π, cE2 +z /8π and the total intensity of the THz radiation emitted by +the electron for the time window between 182-224 fs. The dashed lines in panel (a) show the expansion of the plasma at the +sound velocity. The color in the electron trajectory reflects its energy based on the color bar of the panel (b). +−eE force. Consequently, these electrons will be trapped +between the ambipolar electric fields on either side of the +plasma. An ejected electron oscillates between the am- +bipolar fields with a period that increases with time due +to the energy exchange between the electrons and ions. +We monitored the electron radiation using a time win- +dow of 130 fs. The electron emission between 20-150 fs +[shown by blue � symbols in panel (a)], is sharply +peaked at the laser frequency ω0 [blue line in panel (b)]. +This emission is due to the electron acceleration while +surfing the resonantly driven plasma waves (See Paper +I38 for more details). During the time interval 182-312 fs +[shown by red � symbols in panel (a)], the electron is +trapped between two ambipolar fields and emits the THz +radiation with a peak frequency at ω = 24 rad/ps [red +line in panel (b)]. Figures 2(c-f) show respectively the +intensity of the electric field components computed using +Eq. (1), c(E2 +x, E2 +y, E2 +z )/8π, and the total intensity radi- +ated by the electron during the time between 182-312 fs. +The THz radiation is mainly polarized in the x−direction +because Ex component is dominant in the radiated field. +This polarization is the same as the incident pulse and +second-harmonic detailed in Paper II39. In Sec. III, we +will show that the emission pattern corresponds with an +electron current in the x−direction. +III. +ELECTRON THZ EMISSION IN AMBIPOLAR +ELECTRIC FIELDS +The starting point in understanding the mechanism +responsible for THz radiation is the identification of its +current sources. We have seen earlier, in Figs. 2, that +the electrons emit THz radiation while they are trapped +in the ambipolar electric fields of +plasma double lay- +ers. Taking the strength of the resonance electric field +of about 50 GV/m integrated over its width of 70 nm +(See Paper I38), one arrives at a potential of about a few +keV which corresponds to the temperature of the hottest +electrons in the simulation. The hot electrons propagate +outside of the plasma and consequently, the separation of +charges forms an electric double layer where an ambipo- + +Ex[GV/m] +Energy[keV] +-50 +0 +50 0 +2 +4 +(a) +(b) +-1.0 +0.8' +[μm] +0.6 +0 +xy +X +0.4 +0.2 +-1 +0.0 +50 +100 +150 +200 +250 +300 +-1 +0 +0 +1 +2 +t[fs] +[m]m +0.0 +2.0 +4.00.0 +0.2 +0.50.0 +0.2 +0.50.0 +2.0 +4.0 +(c) +(d) +(e) +(f) +30 +[un] +0o +0 +y +-30 +cE2/8r [W/m?] +cE/8π [W/m²] +cE3/8π [W/m2] +I [W/m?] +0 +-30 +-30 +0 +-30 +0 +0 +30 +30 +30 +-30 +30 +x[μm] +x[μm] +x[μm] +x[μm]5 +lar electric field is present. An analytical solution for this +field is possible by using the two-fluid plasma equations +for continuity and momentum (See Appendix B). +Here for simplicity, +we considered a s−polarized +monochromatic laser wave as E = Es(r) cos(ω0t) where +Es(r) includes the spatial dependence. +The ambipo- +lar electric field Ea is described by an inhomogeneous +second-order differential equation for a classical, damped, +driven harmonic oscillator given by [See Eq. (B7) in Ap- +pendix B]: +∂2 +t Ea + 2Γ∂tEa + Ω2 +pEa = Ω2 +p [E0 + E2 cos(2ω0t)] +(3) +where Γ = νei/2 (1 + Zme/mi), νei is the electron-ion +collision frequency, Ω2 +p += ω2 +pe (1 + Zme/mi), ωpe += +(4πnee2/me)1/2 is the electron plasma frequency (in cgs +units), and +E0 =4πe +Ω2p +� +∂x +� +Z Pi +mi +− Pe +me ++ Zniv2 +i − nev2 +e +�� +− +4πe +meΩ2p +ω2 +pe +ω2 +0 +∂x +�E2 +8π +� +(4a) +E2 = − +4πe +meΩ2p +ω2 +pe +ω2 +0 +∂x +�E2 +8π +� +(4b) +with the standard notation (t, x, v, P, ms, ns, Z) for the +time, space, velocity, pressure, mass and density of a +particle of species s, and ion charge respectively. +The +⟨⟩ denotes an average over a laser cycle. The coupling +to the laser was included in the momentum equation via +the radiation force density fRF = (ϵ − 1)/8π∇E2 where +ϵ is the plasma permittivity (See Appendix A). One can +find an equation similar to Eq. (3) in Refs.2,16,20,24 but +with a different right-hand side (different current sources +of the THz radiation). +The solution of Eq. (3) under the initial conditions of +(Ea, ∂tEa) = (0, 0) reads (See for example Ref.45): +Ea(t) = +Terahertz oscillation +� +�� +� +E0 +� +1 − exp (−Γt) +� +cos (ϖt) + Γ +ϖ sin (ϖt) +�� ++ +Second-harmonic oscillation +� +�� +� +Ω2 +pE2 +� +Ω2 +p − 4ω2 +0 +� +cos(2ω0t) + 4ω0Γ sin(2ω0t) +� +Ω2p − 4ω2 +0 +�2 + 16Γ2ω2 +0 +(5) +where ϖ2 = Ω2 +p − Γ2. This solution includes two com- +ponents. The first component oscillates with a frequency +close to the plasma frequency ωpe when ωpe ≫ νei. This +oscillation, however, decays exponentially at a rate close +to the collision frequency. This component is established +by the spatial gradients of the pressure difference between +the light electrons and the heavy ions as represented in +Eq. (4a). This part induces the dipole moment in the +plasma by separating the electrons from the ions. After +a time t ≫ 1/νei, neglecting the electron and ion veloc- +ities and assuming Te ≫ Ti (See Paper I38), a nearly +constant electric field remains eEa ≈ −1/nedPe/dx = +−γTed ln ne/dx, considering an adiabatic equation of +state with the adiabatic index γ. Therefore, the ambipo- +lar field oscillations are driven by the electron density +gradient. The work function of the electrons that moved +from the plasma interior (density n1) to the exterior (den- +sity n2) then reads −e∆φ = γTe ln(n1/n2) ≈ 4 keV. +The second part in Eq. +5 arises where gradients of +the laser intensity induce a second harmonic longitu- +dinal field oscillation. +This term has a resonance at +2ω = Ωp ≈ ωpe (four times the critical density) for +the evanescent part of the wave causing a very steep in- +crease of the oscillation amplitude. This resonance for +s−polarized lasers is different from the Denisov reso- +nance absorption occurring under oblique incidence of +p−polarized lasers.46,47 +An example of the electric field given by Eq. (5) is +shown in Fig. 3(a) for ϖ = 30 rad/ps (corresponds to an +edge plasma density of n2/nc = 10−4), and Γ = 3 rad/ps +where we have supposed that the electron collision fre- +quency is small compared to the plasma frequency. The +wave shows 2-3 oscillations and damps out within a time +scale of ∼ 2 ps. +The Fourier spectrum of the elec- +tric current associated with the quasi-static electric field, +4πJa(ω) = jωEa(ω)/(1 + jνei/ω), is shown in Fig. 3(b). +As one can see, it has a maximum at ω ≈ ϖ = 30 rad/ps. +To compare with the numerical results of the radi- +ated emission in Fig. 1(b), we derive the angular dis- +tribution of the radiated energy from a current source. +For a number of accelerated charges, the integrand +in Eq. +(2) involves the replacement eβejωR(t′)/c → +�N +m=1 emβmejωRm(t′)/c. +In the limit of a continu- +ous distribution of charge, the summation becomes an +integral over the current density as eβejωR(t′)/c +→ +1/c +� +d3r′ J(r′, t′)ejωR(t′)/c. +Hence, the radiation en- +ergy per solid angle per frequency of the current source +reads:44 +d2W +dωdΩ = +ω2 +4π2c3 +���� +� +dt′ +� +d3r′ ˆn × [ˆn × J(r′, t′)] ejω[t′+R(t′)/c] +���� +2 +(6) +We consider an emission length of L for the plasma +rod oriented parallel to the z−direction, and a cur- +rent of electron in the x−direction as J(r′, t′) += +ˆxJa(t′)δ(x′)δ(y′) exp (jkz′). +For a coordinate system +with the spherical angle θ = cos−1 (z/r) and the azimuth +angle φ = tan−1 (y/x) defining the direction of observa- +tion n, Eq. (6) reduces to: + +6 +FIG. 3. Temporal profile of the THz component of the ambipolar electric field from Eq. 5, panel (a), the frequency spectrum +of the electromagnetic radiation, panel (b), angular distributions of the radiated energy from Eq. 7(b), panel (c), and electric +field components, panels (d)-(f). +d2W +dωdΩ =|Ja(ω)|2 +π2c +f(φ, θ) +(7a) +f(φ, θ) =sin2 θ sin2 φ + cos2 θ +(1 − cos θ)2 +sin2 +� +Lk sin2 θ +2 +� +(7b) +where k is the emission wave-vector. +The angular distribution given in Eq. (7b) is shown in +Fig. 3(c) for an emission length of L = 10 µm. It fits well +with the distribution obtained from the PIC simulation +in Fig. 1(b) where the emission is beamed in the posi- +tive z−direction and has two maxima in the y−direction, +perpendicular to the current source. The vector poten- +tial for this current source is in the x−direction and at +far-field, it reads:44 +A(x, ω) =1 +c +ejkR +R +� +d3r′J (r′, ω) e−jkn·r′ +=ˆx2Ja(ω) +c +ejkR +kR +sin +� +Lk sin2 θ +2 +� +1 − cos θ +(8) +One can derive the scalar potential Φ using the Lorenz +gauge and then the components of the electric field as +follows: +∇ · A = − 1 +c +∂Φ +∂t +(9a) +E = − 1 +c +∂A +∂t − ∇Φ +(9b) +The components of the radiated electric field calculated +using Eq. (9b) are shown in Fig. 3(d)-(f). In agreement +with the results of the PIC simulation [Figs. 2(c-d)], the +radiated emission is polarized in the x−direction. More- +over, the angular distributions of the electric field compo- +nents agree with the results of the PIC simulation [Figs. +2(c)-(e)]. +The quadrupole pattern in Fig. +2(d) is not +symmetric like Fig. 3(e). The asymmetry is because the +trajectory of the electron is not symmetric as the electron +spends more time in the x > 0 region relative to x < 0 +[Fig. 2(a)]. +This analytical model allows us to explain the THz ra- +diation emission in PIC simulations. (i) We show that +the current source of the THz emission originates from +the electrons which are trapped between the double lay- +ers. (ii) The current source is parallel to the incident laser +polarization, and consequently, the THz radiation is po- +larized like the incident laser polarization. (iii) The THz +radiation shows a much higher signal along the angles +corresponding to the forward direction (0◦ < θ ≤ 90◦) +than for the backward direction (90◦ < θ ≤ 180◦). This + +180 +1.0 +(a) +(b) +(c) +1.0 +1.0 +135 +0.8 +0.8 +9 +a +0.6 +0.6 +[de +90- +-0.5 +f(Φ, +Ea(t) [ +0.4 +45 - +0.2 - +0.2 +0.0 - +0.0 +0 +0.0 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +25 +50 +75 +100 +0 +90 +180 +270 +360 +t[ps] +w [rad/ps] +Φ[deg] +5 +-1 +5 +5 +-1 +(d) +(e) +(f) +-2 +F-2 +-2 +8 +0 +0 +0 +y +-3 +-3 +-3 +log]Ey]?[a. u. ] +loglEx]²[a. u. ] +loglEz]2[a. u. ] +-5 +-5. +-4 +-5 +.4 +-5 +0 +5 +-5 +0 +5 +0 +5 +x[入] +x[入] +x[入]7 +FIG. 4. +The frequency spectrum of the THz wave [Eq. (5)] +is shown for different plasma frequencies, panel (a), different +electron-ion collision frequencies, panel (b), different pulse du- +rations, panel (c), and for different pulse central wavelengths, +panel (d). +is due to the coherence of the phases of the dipole mo- +ments induced along the plasma rod [Eq. (7)]. +It would be of interest to see what parameters affect the +THz radiation. The THz radiation forms due to the os- +cillating dipole moments in the plasma. Hence, the peak +frequency of the THz spectrum is determined by plasma +frequency (ϖ2 = Ω2 +p−Γ2) as shown in Fig. 4(a). A higher +plasma frequency will cause a greater radiation force and +increases the energy of THz radiation. The electron-ion +collision frequency Γ is one of the important factors af- +fecting the THz spectrum. Increasing the collision time +slows down the thermal equilibrium between the elec- +trons and ions and leads to a longer-lasting ambipolar +electric field and a broader THz spectrum as shown in +Fig. 4(b). +The THz wave amplitude is proportional to the radia- +tion force driven E0 field as expressed in Eq. (5). To have +an estimate of this amplitude, let us suppose a pulse in- +tensity given by I = I0/√π exp +� +−r2/w2 +0 +� +exp +� +−t2/T 2� +, +where T is the duration of the pulse, and w0 the waist +of the pulse. +Under the equilibrium between radia- +tion and space charge forces, the THz wave amplitude +reads E0 = 4E2 +p/(ecω2 +0T 3), where the pulse energy is +Ep = I0πw2 +0T. Hence, reducing the pulse duration while +keeping the pulse energy constant strongly increases the +radiated THz energy [Fig. 4(c)]. This is due to the higher +peak intensity of the incident field. Moreover, increasing +the laser wavelength enforces the exerted radiation force +on electrons during a laser cycle [Eq. (A2)]. It leads to +a stronger net electron current and amplification of the +THz radiation [Fig. 4(d)]. +IV. +DISCUSSION +In this work, we have extended our study of femtosec- +ond Bessel beam-induced plasmas inside the dielectrics. +A single-shot Bessel beam can generate a high aspect ra- +tio over-critical plasma inside the dielectric.37 The gen- +erated plasma offers a promising medium for the THz +radiation due to the current hot electrons driven by the +resonance absorption. +Based on an analytical approach, we derived the cur- +rent source, the electric field components, and the angu- +lar distribution of THz radiation. The analytical deriva- +tion reproduces the main characteristics of the THz ra- +diation calculated using the radiation diagnostic of PIC +simulation. Under the linear mode conversion, the radia- +tion force of the resonantly driven plasma waves kicks the +electrons from the critical surfaces. Due to the different +mobility of the plasma species, charge separations known +as double layers, and consequently, ambipolar electric +fields form at the plasma surfaces. Most of the ejected +electrons from the critical surfaces trap in the potentials +of the ambipolar electric fields at plasma edges. +The +trapped electrons oscillate with a period of around 130 fs +and radiate in the THz frequency domain. +Although in this work we have examined the over- +critical plasma, the presented study is valid for the sub- +critical plasma, because the radiation force of an intense +laser (≳ 1018 W/cm2) can also induce the quasi-static +fields and the associated THz radiation. +The second- +harmonic part of the ambipolar field offers an experi- +mental diagnostic for the detection of THz radiation. Its +pattern at the far-field (a central spot) differs from the +one generated at the critical surfaces (two lobes parallel +with the incident laser polarization discussed in Paper +II39). +To estimate power radiated within the THz range, +equating the radiation force with the force due to the +space charge field generated from electron-ion separa- +tion gives an acceleration a = eE0/m = 4E2 +p/(mcω2 +0T 3), +and the power using the Larmor formula44 +P += +2e2a2/3c3, is P += 32e2E2 +p/(3m2c5ω4 +0T 6), or PW +≈ +108 � +EµJ +p (λµm +0 )2/(T fs)3�2. +Assuming a microjoule laser pulse with a duration of +100 fs at 800 nm wavelength, the laser to THz efficiency +is predicted to be about ∼ 10−8. +This value appears +to be very small compared with the THz efficiency of +∼ 10−3 − 10−6 for femtosecond pulses with the energy of +∼ 10−3 − 1 Joules.19,23,48 Unlike our works, the absorp- +tion process for interactions of 10−3−1 Joules class lasers +with solids relies on the Brunel mechanism49,50 and THz +radiation generation is due to the surface currents18,48 +or highly relativistic particles passing through the differ- +ent dielectrics, the so-called transition radiation.23 The +THz energy might be improved using the Bessel beams. + +1.2 +1.2 +(a) +Ωp = 10rad/ps +(b) +『= 3rad/ps +1.0- +Qp = 20 rad/ps +1.0 + -- 「= 6rad/ps + -- Ωp = 30rad/ps + -- 「= 12rad/ps +0.8 +0.8 +"n'el(m) +0.6- +0.6 +0.4- +0.4 - +0.2 +0.2 +0.0 +0.0 +0 +25 +50 +75 +100 +25 +50 +75 +0 +100 +w[rad/ps] +w[rad/ps] +1.2 +2.5 +(c) +(d) + T=50fs +入o = 0.8 μm +卜 +1.0 +-- T= 75 fs +--- 入o = 1.0 μm +2.0 +--- T= 100fs +- - - 入o = 1.2 μm +0.8- +1.5 +0.6- +1.0 +0.4 - +0.5 +0.2 - +0.0 +0.0 +25 +50 +75 +100 +0 +25 +50 +75 +0 +100 +w[rad/ps] +w[rad/ps]8 +The long plasma (recently, we reached cm-scale over- +critical plasmas inside dielectrics)51 created by Bessel +beams yields a longer double layer at the plasma surface. +A longer double layer traps the ejected hot electrons from +the critical surface on a longer distance, for a longer time +which results in a longer THz pulse. +Assuming that individual electrons radiate incoher- +ently, we might estimate the THz intensity and conver- +sion efficiency in the presented PIC simulation. As re- +ported in Paper I38, the high-energy electrons represent +around 4% of the electrons in the simulation (∼ 109). +Considering the THz intensity for a single electron of +0.04 W/cm2 [Fig. +2(f)], the radiated THz intensity +amounts to about 106 W/cm2, corresponding to a con- +version efficiency of 10−8, in agreement with the above +predicted efficiency. +We require a picosecond timescale to observe the com- +plete process of the THz wave generation (for example, +a THz wave at 0.5 THz corresponds to a timescale of 2 +ps). However, the numerical heating appearing for high- +density plasmas in several picoseconds imposes a limita- +tion on the maximum duration of the PIC simulations.41 +For this reason, we did not run our simulations beyond +320 fs. The THz radiation corresponds to a frequency +around 30 rad/s. +Another challenge is calculating the +radiation integral using the whole set of electrons in the +PIC simulation. This requires the implementation of the +radiation integral in the MPI-based parallel PIC codes, +as done in Refs.52,53 +ACKNOWLEDGMENTS +We thank the EPOCH support team for help https: +//cfsa-pmw.warwick.ac.uk. +The authors acknowl- +edge the financial supports of: +European Research +Council (ERC) 682032-PULSAR, Region Bourgogne- +Franche-Comte and Agence Nationale de la Recherche +(EQUIPEX+ SMARTLIGHT platform ANR-21-ESRE- +0040), Labex ACTION ANR-11-LABX-0001-01, I-SITE +BFC project (contract ANR-15-IDEX-0003), and the +EIPHI Graduate School ANR-17-EURE-0002. +This +work was granted access to the PRACE HPC resources +MARCONI-KNL, MARCONI-M100, and GALILEO at +CINECA, Casalecchio di Reno, Italy, under the Project +”PULSARPIC” (PRA19 4980), PRACE HPC resource +Joliot-Curie Rome at TGCC, CEA, France under the +Project ”PULSARPIC” (RA5614), HPC resource Joliot- +Curie Rome/SKL/KNL at TGCC, CEA, France un- +der the projects A0070511001 and A0090511001, and +M´esocentre de Calcul de Franche-Comt´e. +Appendix A: Radiation force density +The radiation force per unit volume, force density fRF, +on the free electrons can be written:54 +fRF = ϵ − 1 +8π ∇E2 + ϵ − 1 +4πc ∂t(E × B) +(A1) +Usually, the average values of the force density dur- +ing one period of the laser wave are considered. +This +is because the time envelope of the laser wave is much +slower in comparison with the frequency of the laser +wave. +Hence, one can neglect the time average of the +Poynting term, the last term in Eq. (A1). Let us con- +sider the monochromatic solutions of the wave equation +E = Es(r) cos(ω0t) where Es(r) includes the field’s spa- +tial dependence. The radiation force density then reads +fRF = − ω2 +pe +8πω2 +0 +cos2(ω0t)∇E2 +s += − +ω2 +pe +16πω2 +0 +∇E2 +s − +ω2 +pe +16πω2 +0 +cos(2ω0t)∇E2 +s += − ω2 +pe +ω2 +0 +∇ +�E2 +8π +� +− ω2 +pe +ω2 +0 +cos(2ω0t)∇ +�E2 +8π +� +(A2) +We have used +� +E2� += E2 +s /2 in Eq. (A2). +Appendix B: Ambipolar electric field of double layer +Following Refs.,46,47,55 we use the two-fluid plasma +equations for continuity and momentum to derive an an- +alytical solution for the ambipolar electric field of the +double layer. The continuity equations read +∂t (neme) + ∂x (nemeve) =0 +(B1a) +∂t (nimi) + ∂x (nimivi) =0 +(B1b) +where indexes e and i refer to electrons and ions, re- +spectively. The equations for conservation of momentum +read: +∂t (nemeve) = − ∂x +� +nemev2 +e +� +− ∂xPe − eneEa +− nemeνei (ve − vi) + fRF +(B2a) +∂t (nimivi) = − ∂x +� +nimiv2 +i +� +− ∂xPi + eniZEa ++ nemeνei (ve − vi) +(B2b) +In Eq. (B2a), the radiation force density is given by Eq. +(A2). We have neglected the radiation force on the ions +Zme/mifRF in Eq. (B2b). +The Gauss law for the electric field Ea reads: +∂xEa = −4πe (ne − Zni) +(B3) +Taking the time derivative of the Gauss law, using the +equations of continuity in Eqs. (B1), and spatial integra- +tion gives: +∂tEa = 4πe (neve − Znivi) +(B4) + +9 +The second derivative in time results in: +∂2 +t Ea = 4πe [∂t (neve) − Z∂t (nivi)] +(B5) +Substituting from the equations of momentum in Eqs. +(B2) results in +1 +4πe∂2 +t Ea = −∂x +� +nev2 +e +� +− 1 +me +∂xPx − eneEa +me ++νeine (vi − ve) + fRF +me ++Z∂x +� +niv2 +i +� ++ Z +mi +∂xPi − Z2eniEa +mi ++Zνeine (vi − ve) me +mi +(B6) +The rearrangements of the terms result in the following +differential equation that described a damped oscillator +subjected to an external force (inhomogeneous second- +order differential equation). +∂2 +t Ea + 2Γ∂tEa + Ω2 +pEa = Ω2 +p [E0 + E2 cos(2ω0t)] +(B7) +where +Γ =νei +2 +� +1 + Zme +mi +� +(B8a) +Ω2 +p =ω2 +pe +� +1 + Zme +mi +� +(B8b) +E0 =4πe +Ω2p +� +∂x +� +Z Pi +mi +− Pe +me ++ Zniv2 +i − nev2 +e +�� +− +4πe +meΩ2p +ω2 +pe +ω2 +0 +∂x +�E2 +8π +� +(B8c) +E2 = − +4πe +meΩ2p +ω2 +pe +ω2 +0 +∂x +�E2 +8π +� +(B8d) +REFERENCES +1D. Grischkowsky, S. 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Hora, “First direct electron and ion fluid com- +putation of high electrostatic fields in dense inhomogeneous plas- +mas with subsequent nonlinear laser interaction,” Laser and Par- +ticle Beams 1, 283–304 (1983). + diff --git a/EdE3T4oBgHgl3EQfVQpt/content/tmp_files/load_file.txt b/EdE3T4oBgHgl3EQfVQpt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d2d7e3325df50aa8987fc868a7a5e5ef480e844 --- /dev/null +++ b/EdE3T4oBgHgl3EQfVQpt/content/tmp_files/load_file.txt @@ -0,0 +1,913 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf,len=912 +page_content='Femtosecond laser-induced sub-wavelength plasma inside dielectrics: III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Terahertz radiation emission Kazem Ardaneh,1, 2, a) Ken-Ichi Nishikawa,3 Remo Giust,1 Benoit Morel,1 Pierre-Jean Charpin,1 Arnaud Couairon,4 Guy Bonnaud,5 and Francois Courvoisier1, b) 1)FEMTO-ST Institute, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Bourgogne Franche-Comt´e, CNRS, 15B avenue des Montboucons, 25030, Besan¸con Cedex, France 2)Sorbonne University, Pierre and Marie Curie Campus, 4 place Jussieu, 75252, Paris Cedex 5, France 3)Department of Physics, Chemistry and Mathematics, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Murry Chambers Bld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=', Alabama A&M University, Huntsville, AL 35810, USA 4)CPHT, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Route de Saclay, F-91128 Palaiseau, France 5)CEA, Centre de Paris-Saclay, DRF, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Paris-Saclay, 91191 Gif-sur-Yvette, France (Dated: 12 January 2023) Electromagnetic radiation within the terahertz (THz) frequency range is of great interest for applications in remote sensing and time-domain spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The laser-induced plasmas are promising mediums for generating THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' It has been recently reported that focusing femtosecond Bessel pulses inside dielectrics induces a high aspect ratio over-critical plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Here we show that the intense resonantly driven electrostatic fields at the so-called critical surface lead to THz radiation emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Through three-dimensional particle-in-cell simulation and analytical derivation, we have investigated the emission of THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We show that the THz radiation is associated with a hot population of electrons trapped in ambipolar electric fields of the double layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' INTRODUCTION Terahertz (THz) radiation, typically referred to as the frequency band 100 GHz − 10 THz, in the infrared and microwaves ranges, has been attracting ongoing interest because of its broad applications ranging from biomedical imaging, security or packaged goods inspection, to time- domain spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='1–4 Using femtosecond laser pulses, the techniques for generating THz radiation are gen- erally classified as either optical rectification,5–11 tran- sient current sources,2,12–23 or a combination of these two mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='24 Optical rectification can induce THz radiation in non-centrosymmetric crystals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=', ZnTe and GaAs, in which the fundamental frequency of an infrared fem- tosecond laser pulse is down-converted to the THz fre- quency via the second-order susceptibility where the po- larization reads P (ωTHz) = χ(2)E(ω + ωTHz)E∗(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5,7 The frequency of the rectified pulse envelope is in the range of 3-10 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='7 In the centrosymmetric me- dia, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=', gases, two-color illumination can be used to mix the fundamental frequency with the second- harmonic in a four-wave mixing process as P (ωTHz) = χ(3)E(2ω −ωTHz) E∗(ω)E∗(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8,11 Correspondingly, the THz component might match with the fundamental one in an inverse process and leads to second-harmonic gen- eration, which is a method for THz detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' A THz emission with an estimated field strength ∼ 400 kV/cm a)Electronic mail: kazem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='arrdaneh@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='com b)Electronic mail: francois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='courvoisier@femto-st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='fr has been reported in which the presence of plasma was essential for the high-efficiency process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='10 Laser-induced plasmas are attractive for THz radia- tion generation because of their ability to sustain ex- tremely intense electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='13,14,25–27 In this context, femtosecond laser-induced breakdown of gases is investigated widely,28,29 mostly using the two-color approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8 The peak of the THz field however satu- rates for laser intensities higher than 1015 W/cm2 be- cause of the strong THz absorption in the long (∼7 mm) air plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='17 Plasma generation in laser-solid interac- tions offer an alternative: experiments of ultrashort laser pulses-solid interaction have shown a monotonous in- crease in THz radiation with the incident laser intensity up to 1019 W/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='18,19,21 Illumination of solid targets by intense ultrashort laser beams results in the generation of hot electron currents that are the source of the secondary electromagnetic ra- diation ranging from x-rays30–33 to THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='19 Un- der p−polarized laser illumination of a short-scale inho- mogeneous plasma, for moderate laser intensities (bellow 1014 W/cm2), resonance absorption is the main mecha- nism for hot electron generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='34–36 In previous papers, we have reported that over-critical plasmas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' with density above the reflection density for the incident laser wavelength, are generated by focusing Bessel beams with moderate intensities on the order of 1014 W/cm2 inside sapphire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='37–39 A Bessel beam is a so- lution of the wave equation in which the wave amplitude is defined by the Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='40 Im- portantly, the axial intensity profile of the Bessel beam is propagation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Therefore, all segments of the di- electric along the Bessel zone will receive simultaneously arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='04458v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='plasm-ph] 11 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The energy spectrum of the radiation emitted by a population of hot electrons: (a) the spatially averaged, (b) angular distribution, and (c) spatial distribution averaged in the frequency range of 1-30 rad/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The hot electrons are randomly selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' the same amount of energy which results in a high aspect ratio plasma rod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In the first article of this series (Ardaneh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=',38 Pa- per I hereafter), we confirmed that the resonance of the plasma waves can explain the experimental diagnostics of total absorption, and far-field intensity pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We reported electron acceleration up to several keV while surfing the plasma waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In the outward propagation of hot electrons, electrostatic ambipolar fields form at the plasma surface due to the different inertia of the electron and ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Moreover, in the second article of this series (Ardaneh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=',39 Paper II hereafter), we reported the second-harmonic generation by a second-order current of hot electrons near the critical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The electron currents form by the resonance absorption and radiation force of the incident laser wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In the current work as the third in this series, we estab- lish a link between resonance absorption-driven currents and THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This is based on calculating the co- herent radiation spectrum of the hot electrons for the performed Particle-In-Cell (PIC) simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The sim- ulation consists of electron-ion plasma initially induced by multi-photon and collisional ionizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The dipole moments are induced due to the radiation force of the resonance fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' For a laser field with frequency ω0, this force induces a second-harmonic component at 2ω0 and a low-frequency component by separating the light elec- trons from the heavy ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We have developed an ana- lytical model for THz generation in laser-plasma inter- actions to explain the underlying physics, in particular, how the dipole moment is created in the plasma and the characteristics of the radiation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We organized the paper as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' II, we re- call the setup of the PIC simulation as discussed in Paper I38, we detail the radiation diagnostic, and the results of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Then, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' III, we derive an analyti- cal solution for the current source of THz radiation, and the radiated electromagnetic fields with their frequency spectrums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Simulation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Parameter Value Simulation volume 15 × 15 × 30 µm3 Grid resolution ∆x:yk0 r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='04a, ∆zk0 z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='1b FDTD order Second-order BC for fieldsc Perfectly matched layers BC for particles Outflow Pulse energy (Ep) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 µJ Pulse frequency (ω0) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='35 rad/fs Pulse cone angle (θ) 25◦ Pulse temporal profile exp[−(t − tc)2/T 2] Central time (tc) 130 fs Pulse FWHM = √ 2 ln 2T 100 fs Pulse spatial profile exp(−r2/w2 0) Pulse spatial waist (w0) 10 µm Maximum density (nmax) 5 nc Density profile (axial) tanh(zµm) Mass ratio (mi/me) 102 × 1836d Plasma distribution [f(ve:i)] Maxwellian Plasma temperature (Te:i) 1 eV Particles per cell per species 32 Particle shape function triangle Time step ∆tω0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='07 Simulation time 320 fs a k0 r = k0 sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' b k0 z = k0 cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' c BC: Boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' d 102 is the sapphire molar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' PIC SIMULATION We performed self-consistent PIC simulation using the three-dimensional massively parallel electromag- netic code EPOCH41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In our simulation, we used the plasma parameters that could reproduce our ex- perimental measurements (far-field, near-field, absorp- dW(WTHz)/dQ[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ] dW(WTHz)/dΩ[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='00 180 1 100↓ (a) (b) (c) u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 135 esin Φ 10-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' deg] dW/dw [ 90 0 sin( 10-2 45 - 10-3 0 1 2 0 90 180 270 360 1 0 2 0 [0m]3 Φ[deg] sin Ocos Φ3 tion) as reported in Paper I,38 and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='39 The simula- tion setup is summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The plasma is fully ionized and composed of electrons and ions with equal densities (to preserve electric neutrality) given by n = nmax exp(−x2/w2 x) exp(−y2/w2 y) tanh(zµm) with FWHMx = √ 2 ln 2wx = 250 nm, and FWHMy = √ 2 ln 2wy = 600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' There are initially 32 particles per cell per species leading to the total number of parti- cles in the simulation ∼ 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The collisions are modeled through a binary model as presented in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='41,42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We injected from the zmin boundary a linearly x−polarized Gaussian pulse propagating along the pos- itive z−direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We applied a phase to the Gaussian beam to create a Bessel-Gauss beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='43 The peak inten- sity in the Bessel zone is 6 × 1014 W/cm2 in absence of plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The time step is limited by the Courant con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The minimum frequency in the simulation is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 rad/ps which is well below the peak frequency of the THz spectrum at 30 rad/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' One of the primary advantages of PIC codes is the pos- sibility to access full information about the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We have developed a radiation diagnostic that utilizes the position and momentum of particles over time and cal- culates the radiated fields and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' For this purpose, let us consider a particle at position r (t) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' At the same time, we observe the radiated electromagnetic fields from the particle at position x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Due to the finite velocity of light, we observe the particle at an earlier position r (t′) where it was at the retarded time t′ = t−R (t′) /c, where R (t′) = |x − r (t′)| is the distance from the charged par- ticle (at the retarded time t′) to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The mag- netic and electric fields produced from a moving point charge can be calculated directly from their scalar and vector potentials known as the Li´enard–Wiechert poten- tials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The electric field reads:44 E(x, t) = Velocity field � �� � e � n − β γ2(1 − β · n)3R2 � ret + Acceleration field � �� � e c � n × {(n − β) × ˙β} (1 − β · n)3R � ret (1) where n = R (t′) / |R (t′)| is a unit vector pointing from the particle retarded position to the observer, β = v/c the particle instantaneous velocity, ˙β = dβ/dt is the acceleration divided by c, γ is the Lorentz fac- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The spatial spectra are obtained by the choice of n � n2 x + n2 y + n2 z = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (1) divides itself into ”velocity fields,” which are independent of acceler- ation, and ”acceleration fields,” which depend linearly on ˙β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The velocity field is a static field decreasing as R−2 while the acceleration field is a radiation field, be- ing transverse to the radius vector and falling off as R−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The total energy W radiated per unit solid angle dΩ per unit frequency dω from the accelerated charged particle reads:44 d2W dωdΩ = e2ω2 4π2c ���� � ∞ −∞ dt′ˆn × (ˆn × β)ejω(t′+R(t′)/c) ���� 2 (2) In our simulations, we collected the THz radiation emissions from the hot electrons as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We have tracked 105 electrons in the simulations and recorded the information of these electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We calculated the energy spectrum of the radiation emitted by 100 randomly se- lected electrons according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The result is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We see two peaks around ω = 0, with a width of typically 50 rad/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We note that, because of computing memory limitations, the time resolution of the particle positions is insufficient to capture the second harmonic emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The angular and spatial distribu- tions of the THz emission are obtained by averaging the energy spectrum in the frequency range of 1-30 rad/ps [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1(b) and 1(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' As one expects, the energy spectrum has sharp max- ima at the laser frequency ω0 due to strong electron ac- celeration in resonantly driven plasma waves at the crit- ical surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The spectrum also has maxima at ω ≈ 30 rad/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The angular distribution of this THz radia- tion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1(b) shows maxima around (θ, φ) ≈ (0, π/2) and (0, 3π/2), perpendicular to the electron acceleration which is mainly in the x−direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The small tilt in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1(c) is due to the asymmetric distribution of the randomly selected electrons in xy−space over the inte- grated time (a similar deviation occurred for another set of 100 electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We select some representative electrons to calculate the radiated fields in a spatial window |x| ⩽ 45 µm and |y| ⩽ 45 µm at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Using a time window, we also ex- amined in which part of the trajectory, the electron emits electromagnetic radiation in the THz frequency range [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' For each time window, we calculated the intensity distribution I(ωTHz, x, y) by performing a dis- crete Fourier transform on each component of the electric field, Ex:y:z(t, x, y) and averaging in the frequency range of 1-30 rad/ps [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(b)-2(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Figure 2(a) shows the time evolution of the Ex com- ponent, parallel to the incident laser polarization over- plotted with the trajectory of a representative electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' One can see the resonance plasma waves induced at the critical surfaces (x = ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 µm), in the time between 70- 200 fs (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4 in Paper I38 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Near the peak of the laser field, the intense ambipolar fields propagating with the sound speed are generated at the surface of the plasma (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The ambipolar field sign is positive for x > 0 and negative for x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The ra- diation force (See Appendix A) due to the intense, local- ized resonance field ejects electrons from the resonance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The electrons are ejected from the critical sur- faces in the positive x−direction where x > 0 and nega- tive x−direction where x < 0 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 6 Paper I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Therefore, the electrons ejected with energy less than the potential barrier of the ambipolar field will be reflected by 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' THz radiation from an electron trapped in the ambipolar electric fields of double layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Shown are: (a) x−component of the electric field over-plotted by the trajectory of a representative electron, electron emission for a time window between: (b) 20-150 fs [shown by blue � symbols in panel (a), and blue line in panel (b)], and 182-312 fs [shown by red � symbols in panel (a) and red line in panel (b)], (c-f) cE2 x/8π, cE2 y/8π, cE2 z /8π and the total intensity of the THz radiation emitted by the electron for the time window between 182-224 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The dashed lines in panel (a) show the expansion of the plasma at the sound velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The color in the electron trajectory reflects its energy based on the color bar of the panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' −eE force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Consequently, these electrons will be trapped between the ambipolar electric fields on either side of the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' An ejected electron oscillates between the am- bipolar fields with a period that increases with time due to the energy exchange between the electrons and ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We monitored the electron radiation using a time win- dow of 130 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The electron emission between 20-150 fs [shown by blue � symbols in panel (a)], is sharply peaked at the laser frequency ω0 [blue line in panel (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This emission is due to the electron acceleration while surfing the resonantly driven plasma waves (See Paper I38 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' During the time interval 182-312 fs [shown by red � symbols in panel (a)], the electron is trapped between two ambipolar fields and emits the THz radiation with a peak frequency at ω = 24 rad/ps [red line in panel (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Figures 2(c-f) show respectively the intensity of the electric field components computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (1), c(E2 x, E2 y, E2 z )/8π, and the total intensity radi- ated by the electron during the time between 182-312 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The THz radiation is mainly polarized in the x−direction because Ex component is dominant in the radiated field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This polarization is the same as the incident pulse and second-harmonic detailed in Paper II39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' III, we will show that the emission pattern corresponds with an electron current in the x−direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ELECTRON THZ EMISSION IN AMBIPOLAR ELECTRIC FIELDS The starting point in understanding the mechanism responsible for THz radiation is the identification of its current sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We have seen earlier, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2, that the electrons emit THz radiation while they are trapped in the ambipolar electric fields of plasma double lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Taking the strength of the resonance electric field of about 50 GV/m integrated over its width of 70 nm (See Paper I38), one arrives at a potential of about a few keV which corresponds to the temperature of the hottest electrons in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The hot electrons propagate outside of the plasma and consequently, the separation of charges forms an electric double layer where an ambipo- Ex[GV/m] Energy[keV] 50 0 50 0 2 4 (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content="8' [μm] 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='6 0 xy X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 50 100 150 200 250 300 1 0 0 1 2 t[fs] [m]m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 (c) (d) (e) (f) 30 [un] 0o 0 y 30 cE2/8r [W/m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='] cE/8π [W/m²] cE3/8π [W/m2] I [W/m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='] 0 30 30 0 30 0 0 30 30 30 30 30 x[μm] x[μm] x[μm] x[μm]5 lar electric field is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' An analytical solution for this field is possible by using the two-fluid plasma equations for continuity and momentum (See Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Here for simplicity, we considered a s−polarized monochromatic laser wave as E = Es(r) cos(ω0t) where Es(r) includes the spatial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The ambipo- lar electric field Ea is described by an inhomogeneous second-order differential equation for a classical, damped, driven harmonic oscillator given by [See Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (B7) in Ap- pendix B]: ∂2 t Ea + 2Γ∂tEa + Ω2 pEa = Ω2 p [E0 + E2 cos(2ω0t)] (3) where Γ = νei/2 (1 + Zme/mi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' νei is the electron-ion collision frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Ω2 p = ω2 pe (1 + Zme/mi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ωpe = (4πnee2/me)1/2 is the electron plasma frequency (in cgs units),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' and E0 =4πe Ω2p � ∂x � Z Pi mi − Pe me + Zniv2 i − nev2 e �� − 4πe meΩ2p ω2 pe ω2 0 ∂x �E2 8π � (4a) E2 = − 4πe meΩ2p ω2 pe ω2 0 ∂x �E2 8π � (4b) with the standard notation (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ns,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Z) for the time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' pressure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' mass and density of a particle of species s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' and ion charge respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The ⟨⟩ denotes an average over a laser cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The coupling to the laser was included in the momentum equation via the radiation force density fRF = (ϵ − 1)/8π∇E2 where ϵ is the plasma permittivity (See Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' One can find an equation similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (3) in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2,16,20,24 but with a different right-hand side (different current sources of the THz radiation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (3) under the initial conditions of (Ea, ∂tEa) = (0, 0) reads (See for example Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='45): Ea(t) = Terahertz oscillation � �� � E0 � 1 − exp (−Γt) � cos (ϖt) + Γ ϖ sin (ϖt) �� + Second-harmonic oscillation � �� � Ω2 pE2 � Ω2 p − 4ω2 0 � cos(2ω0t) + 4ω0Γ sin(2ω0t) � Ω2p − 4ω2 0 �2 + 16Γ2ω2 0 (5) where ϖ2 = Ω2 p − Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This solution includes two com- ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The first component oscillates with a frequency close to the plasma frequency ωpe when ωpe ≫ νei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This oscillation, however, decays exponentially at a rate close to the collision frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This component is established by the spatial gradients of the pressure difference between the light electrons and the heavy ions as represented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This part induces the dipole moment in the plasma by separating the electrons from the ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' After a time t ≫ 1/νei, neglecting the electron and ion veloc- ities and assuming Te ≫ Ti (See Paper I38), a nearly constant electric field remains eEa ≈ −1/nedPe/dx = −γTed ln ne/dx, considering an adiabatic equation of state with the adiabatic index γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Therefore, the ambipo- lar field oscillations are driven by the electron density gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The work function of the electrons that moved from the plasma interior (density n1) to the exterior (den- sity n2) then reads −e∆φ = γTe ln(n1/n2) ≈ 4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The second part in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 5 arises where gradients of the laser intensity induce a second harmonic longitu- dinal field oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This term has a resonance at 2ω = Ωp ≈ ωpe (four times the critical density) for the evanescent part of the wave causing a very steep in- crease of the oscillation amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This resonance for s−polarized lasers is different from the Denisov reso- nance absorption occurring under oblique incidence of p−polarized lasers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='46,47 An example of the electric field given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (5) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 3(a) for ϖ = 30 rad/ps (corresponds to an edge plasma density of n2/nc = 10−4), and Γ = 3 rad/ps where we have supposed that the electron collision fre- quency is small compared to the plasma frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The wave shows 2-3 oscillations and damps out within a time scale of ∼ 2 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The Fourier spectrum of the elec- tric current associated with the quasi-static electric field, 4πJa(ω) = jωEa(ω)/(1 + jνei/ω), is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' As one can see, it has a maximum at ω ≈ ϖ = 30 rad/ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' To compare with the numerical results of the radi- ated emission in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1(b), we derive the angular dis- tribution of the radiated energy from a current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' For a number of accelerated charges, the integrand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (2) involves the replacement eβejωR(t′)/c → �N m=1 emβmejωRm(t′)/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In the limit of a continu- ous distribution of charge, the summation becomes an integral over the current density as eβejωR(t′)/c → 1/c � d3r′ J(r′, t′)ejωR(t′)/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Hence, the radiation en- ergy per solid angle per frequency of the current source reads:44 d2W dωdΩ = ω2 4π2c3 ���� � dt′ � d3r′ ˆn × [ˆn × J(r′, t′)] ejω[t′+R(t′)/c] ���� 2 (6) We consider an emission length of L for the plasma rod oriented parallel to the z−direction, and a cur- rent of electron in the x−direction as J(r′, t′) = ˆxJa(t′)δ(x′)δ(y′) exp (jkz′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' For a coordinate system with the spherical angle θ = cos−1 (z/r) and the azimuth angle φ = tan−1 (y/x) defining the direction of observa- tion n, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (6) reduces to: 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Temporal profile of the THz component of the ambipolar electric field from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 5, panel (a), the frequency spectrum of the electromagnetic radiation, panel (b), angular distributions of the radiated energy from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 7(b), panel (c), and electric field components, panels (d)-(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' d2W dωdΩ =|Ja(ω)|2 π2c f(φ, θ) (7a) f(φ, θ) =sin2 θ sin2 φ + cos2 θ (1 − cos θ)2 sin2 � Lk sin2 θ 2 � (7b) where k is the emission wave-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The angular distribution given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (7b) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 3(c) for an emission length of L = 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' It fits well with the distribution obtained from the PIC simulation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1(b) where the emission is beamed in the posi- tive z−direction and has two maxima in the y−direction, perpendicular to the current source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The vector poten- tial for this current source is in the x−direction and at far-field, it reads:44 A(x, ω) =1 c ejkR R � d3r′J (r′, ω) e−jkn·r′ =ˆx2Ja(ω) c ejkR kR sin � Lk sin2 θ 2 � 1 − cos θ (8) One can derive the scalar potential Φ using the Lorenz gauge and then the components of the electric field as follows: ∇ · A = − 1 c ∂Φ ∂t (9a) E = − 1 c ∂A ∂t − ∇Φ (9b) The components of the radiated electric field calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (9b) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 3(d)-(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' In agreement with the results of the PIC simulation [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(c-d)], the radiated emission is polarized in the x−direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' More- over, the angular distributions of the electric field compo- nents agree with the results of the PIC simulation [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(c)-(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The quadrupole pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(d) is not symmetric like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 3(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The asymmetry is because the trajectory of the electron is not symmetric as the electron spends more time in the x > 0 region relative to x < 0 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This analytical model allows us to explain the THz ra- diation emission in PIC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (i) We show that the current source of the THz emission originates from the electrons which are trapped between the double lay- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (ii) The current source is parallel to the incident laser polarization, and consequently, the THz radiation is po- larized like the incident laser polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (iii) The THz radiation shows a much higher signal along the angles corresponding to the forward direction (0◦ < θ ≤ 90◦) than for the backward direction (90◦ < θ ≤ 180◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This 180 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 (a) (b) (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8 9 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='6 [de 90- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 f(Φ, Ea(t) [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='4 45 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0 25 50 75 100 0 90 180 270 360 t[ps] w [rad/ps] Φ[deg] 5 1 5 5 1 (d) (e) (f) 2 F-2 2 8 0 0 0 y 3 3 3 log]Ey]?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ] loglEx]²[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ] loglEz]2[a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ] 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='4 5 0 5 5 0 5 0 5 x[入] x[入] x[入]7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The frequency spectrum of the THz wave [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (5)] is shown for different plasma frequencies, panel (a), different electron-ion collision frequencies, panel (b), different pulse du- rations, panel (c), and for different pulse central wavelengths, panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' is due to the coherence of the phases of the dipole mo- ments induced along the plasma rod [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (7)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' It would be of interest to see what parameters affect the THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The THz radiation forms due to the os- cillating dipole moments in the plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Hence, the peak frequency of the THz spectrum is determined by plasma frequency (ϖ2 = Ω2 p−Γ2) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' A higher plasma frequency will cause a greater radiation force and increases the energy of THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The electron-ion collision frequency Γ is one of the important factors af- fecting the THz spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Increasing the collision time slows down the thermal equilibrium between the elec- trons and ions and leads to a longer-lasting ambipolar electric field and a broader THz spectrum as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The THz wave amplitude is proportional to the radia- tion force driven E0 field as expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' To have an estimate of this amplitude, let us suppose a pulse in- tensity given by I = I0/√π exp � −r2/w2 0 � exp � −t2/T 2� , where T is the duration of the pulse, and w0 the waist of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Under the equilibrium between radia- tion and space charge forces, the THz wave amplitude reads E0 = 4E2 p/(ecω2 0T 3), where the pulse energy is Ep = I0πw2 0T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Hence, reducing the pulse duration while keeping the pulse energy constant strongly increases the radiated THz energy [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This is due to the higher peak intensity of the incident field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Moreover, increasing the laser wavelength enforces the exerted radiation force on electrons during a laser cycle [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (A2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' It leads to a stronger net electron current and amplification of the THz radiation [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' DISCUSSION In this work, we have extended our study of femtosec- ond Bessel beam-induced plasmas inside the dielectrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' A single-shot Bessel beam can generate a high aspect ra- tio over-critical plasma inside the dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='37 The gen- erated plasma offers a promising medium for the THz radiation due to the current hot electrons driven by the resonance absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Based on an analytical approach, we derived the cur- rent source, the electric field components, and the angu- lar distribution of THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The analytical deriva- tion reproduces the main characteristics of the THz ra- diation calculated using the radiation diagnostic of PIC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Under the linear mode conversion, the radia- tion force of the resonantly driven plasma waves kicks the electrons from the critical surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Due to the different mobility of the plasma species, charge separations known as double layers, and consequently, ambipolar electric fields form at the plasma surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Most of the ejected electrons from the critical surfaces trap in the potentials of the ambipolar electric fields at plasma edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The trapped electrons oscillate with a period of around 130 fs and radiate in the THz frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Although in this work we have examined the over- critical plasma, the presented study is valid for the sub- critical plasma, because the radiation force of an intense laser (≳ 1018 W/cm2) can also induce the quasi-static fields and the associated THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The second- harmonic part of the ambipolar field offers an experi- mental diagnostic for the detection of THz radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Its pattern at the far-field (a central spot) differs from the one generated at the critical surfaces (two lobes parallel with the incident laser polarization discussed in Paper II39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' To estimate power radiated within the THz range, equating the radiation force with the force due to the space charge field generated from electron-ion separa- tion gives an acceleration a = eE0/m = 4E2 p/(mcω2 0T 3), and the power using the Larmor formula44 P = 2e2a2/3c3, is P = 32e2E2 p/(3m2c5ω4 0T 6), or PW ≈ 108 � EµJ p (λµm 0 )2/(T fs)3�2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Assuming a microjoule laser pulse with a duration of 100 fs at 800 nm wavelength, the laser to THz efficiency is predicted to be about ∼ 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This value appears to be very small compared with the THz efficiency of ∼ 10−3 − 10−6 for femtosecond pulses with the energy of ∼ 10−3 − 1 Joules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='19,23,48 Unlike our works, the absorp- tion process for interactions of 10−3−1 Joules class lasers with solids relies on the Brunel mechanism49,50 and THz radiation generation is due to the surface currents18,48 or highly relativistic particles passing through the differ- ent dielectrics, the so-called transition radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='23 The THz energy might be improved using the Bessel beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 (a) Ωp = 10rad/ps (b) 『= 3rad/ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0- Qp = 20 rad/ps 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 -- 「= 6rad/ps -- Ωp = 30rad/ps -- 「= 12rad/ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8 "n\'el(m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0 25 50 75 100 25 50 75 0 100 w[rad/ps] w[rad/ps] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 (c) (d) T=50fs 入o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8 μm 卜 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 -- T= 75 fs --- 入o = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 --- T= 100fs - - 入o = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='8- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='6- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='0 25 50 75 100 0 25 50 75 0 100 w[rad/ps] w[rad/ps]8 The long plasma (recently, we reached cm-scale over- critical plasmas inside dielectrics)51 created by Bessel beams yields a longer double layer at the plasma surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' A longer double layer traps the ejected hot electrons from the critical surface on a longer distance, for a longer time which results in a longer THz pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Assuming that individual electrons radiate incoher- ently, we might estimate the THz intensity and conver- sion efficiency in the presented PIC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' As re- ported in Paper I38, the high-energy electrons represent around 4% of the electrons in the simulation (∼ 109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Considering the THz intensity for a single electron of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='04 W/cm2 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' 2(f)], the radiated THz intensity amounts to about 106 W/cm2, corresponding to a con- version efficiency of 10−8, in agreement with the above predicted efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We require a picosecond timescale to observe the com- plete process of the THz wave generation (for example, a THz wave at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='5 THz corresponds to a timescale of 2 ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' However, the numerical heating appearing for high- density plasmas in several picoseconds imposes a limita- tion on the maximum duration of the PIC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='41 For this reason, we did not run our simulations beyond 320 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The THz radiation corresponds to a frequency around 30 rad/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Another challenge is calculating the radiation integral using the whole set of electrons in the PIC simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This requires the implementation of the radiation integral in the MPI-based parallel PIC codes, as done in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='52,53 ACKNOWLEDGMENTS We thank the EPOCH support team for help https: //cfsa-pmw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The authors acknowl- edge the financial supports of: European Research Council (ERC) 682032-PULSAR, Region Bourgogne- Franche-Comte and Agence Nationale de la Recherche (EQUIPEX+ SMARTLIGHT platform ANR-21-ESRE- 0040), Labex ACTION ANR-11-LABX-0001-01, I-SITE BFC project (contract ANR-15-IDEX-0003), and the EIPHI Graduate School ANR-17-EURE-0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This work was granted access to the PRACE HPC resources MARCONI-KNL, MARCONI-M100, and GALILEO at CINECA, Casalecchio di Reno, Italy, under the Project ”PULSARPIC” (PRA19 4980), PRACE HPC resource Joliot-Curie Rome at TGCC, CEA, France under the Project ”PULSARPIC” (RA5614), HPC resource Joliot- Curie Rome/SKL/KNL at TGCC, CEA, France un- der the projects A0070511001 and A0090511001, and M´esocentre de Calcul de Franche-Comt´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Appendix A: Radiation force density The radiation force per unit volume, force density fRF, on the free electrons can be written:54 fRF = ϵ − 1 8π ∇E2 + ϵ − 1 4πc ∂t(E × B) (A1) Usually, the average values of the force density dur- ing one period of the laser wave are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' This is because the time envelope of the laser wave is much slower in comparison with the frequency of the laser wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Hence, one can neglect the time average of the Poynting term, the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Let us con- sider the monochromatic solutions of the wave equation E = Es(r) cos(ω0t) where Es(r) includes the field’s spa- tial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The radiation force density then reads fRF = − ω2 pe 8πω2 0 cos2(ω0t)∇E2 s = − ω2 pe 16πω2 0 ∇E2 s − ω2 pe 16πω2 0 cos(2ω0t)∇E2 s = − ω2 pe ω2 0 ∇ �E2 8π � − ω2 pe ω2 0 cos(2ω0t)∇ �E2 8π � (A2) We have used � E2� = E2 s /2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Appendix B: Ambipolar electric field of double layer Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=',46,47,55 we use the two-fluid plasma equations for continuity and momentum to derive an an- alytical solution for the ambipolar electric field of the double layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The continuity equations read ∂t (neme) + ∂x (nemeve) =0 (B1a) ∂t (nimi) + ∂x (nimivi) =0 (B1b) where indexes e and i refer to electrons and ions, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The equations for conservation of momentum read: ∂t (nemeve) = − ∂x � nemev2 e � − ∂xPe − eneEa − nemeνei (ve − vi) + fRF (B2a) ∂t (nimivi) = − ∂x � nimiv2 i � − ∂xPi + eniZEa + nemeνei (ve − vi) (B2b) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (B2a), the radiation force density is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' We have neglected the radiation force on the ions Zme/mifRF in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (B2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' The Gauss law for the electric field Ea reads: ∂xEa = −4πe (ne − Zni) (B3) Taking the time derivative of the Gauss law, using the equations of continuity in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (B1), and spatial integra- tion gives: ∂tEa = 4πe (neve − Znivi) (B4) 9 The second derivative in time results in: ∂2 t Ea = 4πe [∂t (neve) − Z∂t (nivi)] (B5) Substituting from the equations of momentum in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' (B2) results in 1 4πe∂2 t Ea = −∂x � nev2 e � − 1 me ∂xPx − eneEa me +νeine (vi − ve) + fRF me +Z∂x � niv2 i � + Z mi ∂xPi − Z2eniEa mi +Zνeine (vi − ve) me mi (B6) The rearrangements of the terms result in the following differential equation that described a damped oscillator subjected to an external force (inhomogeneous second- order differential equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' ∂2 t Ea + 2Γ∂tEa + Ω2 pEa = Ω2 p [E0 + E2 cos(2ω0t)] (B7) where Γ =νei 2 � 1 + Zme mi � (B8a) Ω2 p =ω2 pe � 1 + Zme mi � (B8b) E0 =4πe Ω2p � ∂x � Z Pi mi − Pe me + Zniv2 i − nev2 e �� − 4πe meΩ2p ω2 pe ω2 0 ∂x �E2 8π � (B8c) E2 = − 4πe meΩ2p ω2 pe ω2 0 ∂x �E2 8π � (B8d) REFERENCES 1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Grischkowsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Keiding, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' van Exter, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE3T4oBgHgl3EQfVQpt/content/2301.04458v1.pdf'} +page_content=' Fattinger, 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/dev/null +++ b/FNE2T4oBgHgl3EQf-Anw/content/tmp_files/2301.04235v1.pdf.txt @@ -0,0 +1,8490 @@ +Hyper-cores promote localization and efficient seeding in higher-order processes +Marco Mancastroppa,1 Iacopo Iacopini,2 Giovanni Petri,3 and Alain Barrat1 +1Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, +Turing Center for Living Systems, Marseille, France +2Department of Network and Data Science, Central European University, 1100 Vienna, Austria +3CENTAI, Corso Inghilterra 3, 10138 Turin, Italy +Going beyond networks, in order to include higher-order interactions involving groups of elements +of arbitrary sizes, has been recognized as a major step in reaching a better description of many +complex systems. In the resulting hypergraph representation, tools to identify particularly cohesive +structures and central nodes are still scarce. We propose here to decompose a hypergraph in hyper- +cores, defined as subsets of nodes connected by at least a certain number of hyperedges (groups +of nodes) of at least a certain size. We illustrate this procedure on empirical data sets described +by hypergraphs, showing how this suggests a novel notion of centrality for nodes in hypergraphs, +the hyper-coreness. We assess the role of the hyper-cores and of nodes with large hyper-coreness +values in several dynamical processes based on higher-order interactions. We show that such nodes +have large spreading power and that spreading processes are localized in hyper-cores with large +connectedness along groups of large sizes. In the emergence of social conventions moreover, very +few committed individuals with high hyper-coreness can rapidly overturn a majority convention. +Our work opens multiple research avenues, from fingerprinting and comparing empirical data sets +to model validation and study of temporally varying hypergraphs. +I. +INTRODUCTION +Network theory provides a powerful framework for de- +scribing a wide range of complex systems composed of +elements interacting in pairs [1–4]: over the years, this +theory has developed numerous concepts and techniques +to characterize the structure of complex networks at var- +ious scales, from the single element (node or link) to +groups of nodes to the whole system. +Moreover, net- +works are the support of dynamical processes of various +types, from spreading to synchronization phenomena [3], +thus understanding how network’s features impact such +processes, or which parts of a network play the most im- +portant role, is of primordial relevance. Several concepts +and results in this respect are now well established. For +instance, hubs, nodes with a very large number of con- +nections (degree), are known to influence processes such +as spreading or opinion dynamics, because of their ten- +dency to be reached easily and their ability to transmit to +many other nodes [1, 3]. The statistics of the individual +number of connections of nodes are however not a rich +enough characterization: the existence of well-connected +groups of nodes might indeed be even more relevant. In +this direction, the tendency of hubs –observed in real- +world networks– to be connected to each other far above +chance is quantified by the rich-club coefficient [5]. +A +more systematic way to decompose a network into a hi- +erarchy of subgraphs of increasing connectedness is given +instead by the k-core decomposition [6–9]: the k-core of +a network is by definition the maximal subgraph such +that all its nodes have degree (number of neighbours in +the subgraph) at least k. This decomposition provides a +fingerprint of the network’s structure [8, 10, 11] and grad- +ually focuses on more central and densely interconnected +parts of the network that were shown to play a crucial +role in spreading processes [12–14]. In fact, the coreness +of a node, defined as the largest value of k such that the +node belongs to the corresponding k-core, gives an alter- +native measure of centrality and largely determines the +impact of a spreading process initiated (seeded) in that +node [12]. Given the relevance of this decomposition, it +has also been extended to weighted networks [15], via the +s-core decomposition [16] (s representing the strength of +a node, i.e., the sum of the weights of its adjacent links), +to temporally evolving networks [17, 18], and to multi- +layer networks [19]. +Despite their convenience, network representations are +limited to systems composed of only binary interactions. +However, over the last few years, it has become clear that +many real systems include interactions between groups +of units [20, 21]. +Examples range from group conver- +sations among friends [22] to research teams and co- +authorship of scientific articles [23], from neural systems +[24] to interactions between multiple species in ecologi- +cal ones [25]. Analogously, considering a purely dyadic +network substrate for the unfolding of processes such +as consensus formation or (social) contagion could put +a limit on the ability to describe key mechanisms that +are at play. For instance, reinforcement mechanisms – +in which two or more people can convince others in a +group conversation– cannot be naturally accounted for +when considering only dyadic interactions [26–29]. +In +these cases, systems and processes can be effectively rep- +resented within the framework of hypergraphs, a “higher- +order” generalization of networks in which nodes can in- +teract in hyperedges, groups of arbitrary size [21, 30, 31]. +Higher-order interactions give rise to novel structures [32– +34] and phenomena [20, 35], highlighting the need for new +characterization tools able to detect hierarchies and rele- +vant subparts of all these systems that are better repre- +sented by hypergraphs. +Here, we contribute to this endeavour by proposing +the decomposition of a hypergraph in (k, m)-hyper-cores, +which we define as a series of subhypergraphs of increas- +ing connectivity k, ensured by hyperedges of increasing +sizes m. We apply this decomposition to a wide range of +data sets, representing systems of different nature, iden- +tifying non-trivial mesoscopic higher-order structures. In +so doing, we put forward the hyper-coreness, a new cen- +trality measure for nodes in hypergraphs based on their +inclusion in the hyper-cores. Finally, and crucially, we +investigate the role of the newly defined hyper-cores +arXiv:2301.04235v1 [physics.soc-ph] 10 Jan 2023 + +2 +and of the nodes with largest hyper-coreness in spread- +ing and consensus processes based on group interactions +[28, 36, 37]. We show that spreading processes tend to +be localized on hyper-cores associated to large k and +m. We then study the performance of hyper-coreness- +based strategies as opposed to both random and strength- +based ones [16] when it comes to identifying influential +nodes that sustain and drive higher-order processes. We +find that hyper-coreness can be effectively used to max- +imise the total outbreak size in non-linear spreading pro- +cesses [36] and help committed minorities reach the tip- +ping point leading to a systemic takeover in social con- +vention games [38]. +II. +RESULTS +A. +Hyper-core decomposition and hyper-coreness +We define the hyper-cores, i.e. the higher-order cores +of a hypergraph, through a systematic decomposition +of a hypergraph in a double hierarchy of nested sub- +hypergraphs of increasing connectedness and hyperedge +sizes. Let us consider a (static) hypergraph H = (V, E), +where V is the set of its N = |V| nodes and E is the +set of its hyperedges [21]. +We recall that a hyperedge +e = {i1, i2, ..., im} is a set of m nodes, which can thus +represent a group interaction between these nodes. We +denote by M = maxe∈E |e| the largest hyperedge size in +H. Each node i ∈ V can be characterized by a vector +of degrees d(i) = [d2(i), d3(i), ..., dm(i), ..., dM(i)] whose +component dm(i) denotes the m-hyper-degree of the node +i, i.e., the number of distinct hyperedges of size m to +which it belongs. +We denote by Dm(i) = � +p≥m dp(i) +the number of distinct hyperedges of size at least m to +which i belongs. +We define the (k, m)-hyper-core as the maximum sub- +hypergraph J induced by the set of nodes A ⊆ V and +with hyperedges of size at least m, such that ∀ i ∈ +A, DJ +m(i) ≥ k, where DJ +m(i) denotes the number of dis- +tinct hyperedges of size at least m in which i is involved +within the subhypergraph J . In other terms, all the nodes +in the (k, m)-hyper-core belong to at least k hyperedges +of size at least m, within the hyper-core itself. The set of +hyperedges of the subhypergraph J , induced by the set +A ⊆ V, is defined by S = {e ∩ A|e ∈ E ∧ |e ∩ A| ≥ m} +[39] (i.e., a hyperedge of S is a subset of a hyperedge of +E, of size at least m and containing only nodes of A). +To obtain the (k, m)-hyper-core of a hypergraph, one +can first remove from E all hyperedges of size smaller than +m. One then removes recursively from V all nodes i with +Dm(i) < k, until all the nodes in the remaining subhy- +pergraph are involved in at least k hyperedges of size at +least m. Note that this process does not correspond only +to the removal of nodes with Dm(i) < k in the original +hypergraph H: indeed, each time a node is removed, the +sizes of the hyperedges to which it belongs decrease by +one unit. Thus, the removal of a node can induce the +removal of some of the hyperedges to which it belongs, if +their size becomes less than m. In Fig. 1 we illustrate the +process on an example hypergraph and highlight some of +its (k, m)-hyper-cores. +As k and m increases, the (k, m)-hyper-cores progres- +FIG. 1. Sketch of the (k, m)-hyper-core decomposition. +We show a hypergraph and highlight some of its (k, m)-hyper- +cores. +Note the inclusions as k or m increase: the (1, 2)- +hyper-core contains the (1, 3)-hyper-core, which contains the +(2, 3)-hyper-core; similarly the (1, 2)-hyper-core contains the +(2, 2)-hyper-core which contains the (2, 3)-hyper-core. On the +other hand, the (1, 3)-hyper-core and the (2, 2)-hyper-core +share some nodes but neither is included in the other. The +green nodes belong to the (1, 2)-hyper-core but neither to the +(1, 3)- nor the (2, 2)- ones. +The blue nodes belong to the +(1, 3)-hyper-core but are excluded from the (2, 3) one. +Or- +ange nodes belong to the (2, 2)-hyper-core but are excluded +from the (2, 3) one because they belong only to hyperedges +of size 2. The (1, 4)-core and (1, 5)-core contain all the nodes +involved respectively in at least one interaction with m ≥ 4 +and m ≥ 5 (for simplicity these cores are not highlighted). +The (k, 2)-cores and (k, 3)-cores with k ≥ 3, and the (k, 4)- +cores and (k, 5)-cores with k ≥ 2 are all empty. Notice that +the node i does not belong to the (2, 3)-core even if D3(i) = 2 +because of the recursive and interaction downgrading mecha- +nisms of the decomposition; in the (1, 3)-core and (2, 3)-core +the pairwise interactions ei ∀i ∈ [1, 5] are excluded, thus the +(1, 3)-core is composed of two disjoint subhypergraphs. +sively identify groups of nodes increasingly connected +with each other through interactions of increasing or- +der (the (k, m)-hyper-core includes the (k, m + 1)- and +(k + 1, m)-hyper-cores). +We define the m-shell index +Cm(i) of a node i as the value of k such that i belongs +to the (k, m)-hyper-core but not to the (k + 1, m)-hyper- +core. The (k, m)-shell S(k,m) can then be defined as the +set of all nodes whose shell index Cm(i) at size m is k, +and we denote by km +max the maximum value of k such +that the shell S(k,m) is not empty. The ratio Cm(i)/km +max +thus quantifies how well-connected node i is in the hy- +pergraph. As it is a function of m, different nodes will +have different functions with potentially different func- +tional shapes, which makes it difficult to compare and +rank them (see the Supplemental Material, SM, for some +examples of Cm(i) functional shapes). We thus define for +each node i its hyper-coreness R(i) as: +R(i) = +M +� +m=2 +g(m)Cm(i)/km +max, +(1) +where g(m) is an arbitrary weight function, which can + +3 +weigh differently the various possible sizes of higher-order +interactions. Hereafter, for simplicity we will fix g(m) = +1: in this case, by definition R ∈ [0, M −1]. Other choices +could be considered, for example to emphasise hyperedges +of larger or smaller sizes. The R hyper-coreness gives thus +a summary of a node centrality with respect to the various +(k, m)-hyper-cores, taking into account how central the +node is for all orders of interactions and making it possible +to rank the nodes of the hypergraph. +B. +Hyper-core decomposition of empirical +hypergraphs +To illustrate the decomposition processes along (k, m)- +hyper-cores, we rely on a number of empirical hyper- +graphs, obtained from publicly available data sets, that +describe a variety of systems of agents interacting in dif- +ferent environments, both through online media and face- +to-face. In particular, we consider data sets of face-to-face +interactions provided by the SocioPatterns collaboration +[40–42] and by the Contacts among Utah’s School-age +Population (CUSP) project [43], collected in contexts +ranging from workplaces to schools. +We also use data +sets of email communication (email-EU, email-Enron [44– +46]) and of other types of online interactions, namely on- +line reviews of products (music-review [46, 47]) or online +opinion exchange on specific topics in scientific forums +[46, 48]. We moreover consider data describing commit- +tees membership (house-committees, senate-committees +[46, 49, 50]) and bills sponsorship (congress-bills, senate- +bills [46, 49, 51, 52]) in the US Congress. +These data +sets cover a wide range of system sizes and have also +very different interaction size distributions. We provide +a detailed description of each data set in the Methods +and in the SM. In the following, we give results on the +music-review, email-EU, house-committees and congress- +bills data sets while we refer to the SM for the other data +sets. +Figure 2 shows the results of the hyper-core decompo- +sition on two data sets. The relative size n(k,m) of the +(k, m)-hyper-cores exhibit distinct behaviors as a func- +tion of k and m, identifying structural differences between +data sets. In some cases, the decrease with k is rather +smooth (Fig. 2a and SM), showing that most shells are +populated. In other cases abrupt drops and plateaus can +be observed (Fig. 2c and SM), corresponding to alter- +natively empty and densely populated (k, m)-shells (see +also SM for figures showing the sizes of the (k, m)-shells +vs k and m). These differences indicate that the (k, m)- +hyper-cores could be used to provide a fingerprint of hy- +pergraphs, just as the k-core decomposition provides a +fingerprint of networks [8, 10, 11]. Also the distributions +of hyper-coreness values R differ across data sets, as illus- +trated in the rank-order plots of Fig. 2b,d and in the SM +for all data sets. While some data sets have an almost +uniform distribution of values, others feature few nodes +with high hyper-coreness and many nodes with medium +hyper-coreness, or vice-versa –many nodes having low or +high coreness and few with medium values (see SM). We +also show in the SM some typical examples of the nor- +malized m-shell index function Cm(i) as a function of m +for various nodes, to highlight the diversity of these func- +1 +16 31 46 61 76 91 106 +k +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +22 +24 +m +n(k, m) +a +email-EU +0 +250 +500 +750 +1000 +Rank(i) +0 +5 +10 +15 +20 +R(i) +b +1 +6 11 16 21 26 31 36 +k +2 +12 +22 +32 +42 +52 +62 +72 +82 +m +n(k, m) +c +music-review +0 +250 +500 +750 +1000 +Rank(i) +0 +20 +40 +60 +80 +R(i) +d +10 +1 +100 +0 +50 +100 +k +0.0 +0.5 +1.0 +n(k, m) +m = 2.0 +m = 4.0 +m = 6.0 +m = 14.0 +0 +500 +1000 +S(i) +0 +10 +20 +R(i) +10 +1 +100 +0 +20 +40 +k +0.0 +0.5 +1.0 +n(k, m) +m = 2.0 +m = 10.0 +m = 18.0 +m = 32.0 +0 +200 +S(i) +0 +40 +80 +R(i) +FIG. 2. Hyper-core decomposition of empirical hyper- +graphs. Panels a and c show colormaps giving the relative +size n(k,m) (number of nodes in the hyper-core, divided by the +total number of nodes N) of the (k, m)-hyper-core as a func- +tion of k and m (white regions correspond to n(k,m) = 0). In +the insets n(k,m) is shown as a function of k at fixed values of +m. In panels b and d the hyper-coreness R(i) is plotted as a +function of the corresponding node rank; the insets give scat- +terplots of the hyper-coreness R(i) vs. the s-coreness, S(i), +for all nodes. In panels a and b we consider the email-EU +data set: R(i) and S(i) have a Pearson correlation coefficient +of ρ = 0.90 (p-value p ≪ 0.001) and the corresponding rank- +ings have a Kendall’s τ coefficient of τ = 0.85 (p ≪ 0.001); in +panels c and d we consider the music-review data set: R(i) +and S(i) have a Pearson correlation coefficient of ρ = 0.74 +(p ≪ 0.001) and the corresponding rankings a Kendall’s τ +coefficient of τ = 0.58 (p ≪ 0.001). +tions and the need to define a summary index such as the +hyper-coreness. +We finally compare in the insets of Fig. 2b,d the hyper- +coreness with the centrality of nodes obtained by disre- +garding the higher-order character of the interactions and +projecting the hypergraph H onto a network. +To this +aim, we transform each hyperedge in a network clique, +and each edge (i, j) of the resulting network is weighted +by the number of distinct hyperedges in H involving both +i and j. We then perform the s-core decomposition of this +weighted network and assign its s-coreness S(i) to each +node i [16]. S(i) and R(i) are positively correlated, but +they do not provide exactly the same information. +In +particular the hyper-coreness enhances the information +given by the s-coreness by providing an internal hierar- +chy within the nodes of maximal s-coreness. This is evi- +dent from the scatter plots, as nodes presenting the same +s-coreness values correspond to values of hyper-coreness +that can span across a broad range (y axis). +Having illustrated the relevance of the newly defined +cores on empirical hypergraphs, we now move to study +the role of these substructures in dynamical processes +taking place on hypergraphs. In particular, we are go- + +4 +ing to investigate whether the (k, m)-hyper-cores and the +hyper-coreness centrality can be used to identify nodes +and structures relevant for spreading and consensus pro- +cesses whose mechanisms are explicitly defined on hyper- +edges. +C. +Higher-order contagion processes localize in +hyper-cores +Networks have been widely used to describe the sub- +strate on which contagion processes take place, such as +the spread of a pathogen or information. +In standard +diffusion modeling approaches, nodes represent individ- +uals that at any time can be in one of several possible +states, such as S (susceptible), I (infectious) or R (re- +covered); S nodes are typically thought to become I at +rate β when they share a link with an infectious (I) in- +dividual, while infected (I) nodes recover spontaneously +at rate µ, either becoming again susceptible (S), in what +is usually called the SIS model [53], or becoming recov- +ered (R) in the so-called SIR model. Recently, several +models have been proposed to take into account possible +higher-order mechanisms, that amount to reinforcement +mechanisms affecting the contagion probability due to the +simultaneous exposure to multiple sources of infections in +group interactions [26, 36, 54, 55]. For instance, in a so- +cial contagion process, the probability that an individual +is convinced upon separate exposures to two “infectious” +neighbours can be reinforced if these exposures occur dur- +ing a group discussion featuring the three individuals al- +together. +Here, we show that hyper-cores play a crucial role in +the dynamics of higher-order spreading processes. In or- +der to do this, we consider the recently proposed higher- +order non-linear contagion [36]. In this model, each sus- +ceptible node in a hyperedge of size m in which there +are i infected individuals becomes infectious with rate +λiν, where ν controls the non-linearity of the process (for +ν = 1 the usual linear contagion is recovered, while for +ν > 1 non-linearities are introduced) and λ ∈ [0, 1] (see +Methods for more details on the model). +Infected in- +dividuals (I) recover independently at constant rate µ, +becoming either susceptible S (SIS model) or R (SIR). +The higher-order nature of contagion produces novel +effects on the epidemic phenomenology, including abrupt +transitions with bistability in the SIS phase diagram, in- +termittent regimes [37, 54], and a mesoscopic localization +of the infection on large hyperedges [36]. +Against this +background, the connectivity properties of hyper-cores +we highlighted so far suggest that cores might play an +even stronger role in such localization. +To investigate this point, we perform numerical simula- +tions of the higher-order non-linear SIS model on empiri- +cal hypergraphs. The system is initialized with one single +seed of infection (a randomly chosen node in state I) in an +otherwise fully susceptible population. We let the process +evolve (see Methods for simulation details) until a steady +state is reached in which the number of infectious individ- +uals fluctuates (we consider parameter values such that +the system remains active and the epidemic does not die +out rapidly). We then consider a finite time-window T +and measure for each node j the time τ(j) that it spends +1 +6 11 16 21 26 31 36 +k +2 +12 +22 +32 +42 +52 +62 +72 +82 +m +/T +a +music-review +SIS +1 +6 11 16 21 26 31 36 +k +2 +12 +22 +32 +42 +52 +62 +72 +82 +m +R +b +SIR +1 3 5 7 9 1113151719 +k +2 +12 +22 +32 +42 +52 +62 +72 +82 +m +/T +c +house-committees +1 3 5 7 9 1113151719 +k +2 +12 +22 +32 +42 +52 +62 +72 +82 +m +R +d +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +0 +500 +1000 +n +0.5 +0.7 +0.9 +/T n +S-rank +R-rank +500 +600 +700 +800 +900 +0 +1000 +n +400 +600 +800 +R +n +S-rank +R-rank +0.675 +0.700 +0.725 +0.750 +0.775 +0.800 +0.825 +0.850 +0 +500 1000 +n +0.7 +0.8 +0.9 +/T n +S-rank +R-rank +100 +125 +150 +175 +200 +225 +0 +500 1000 +n +100 +200 +300 +R +n +S-rank +R-rank +FIG. 3. +Hyper-cores for seeding and localization in +higher-order non-linear contagion processes. For the +SIS model, panels a and c give the heatmap of the average +fraction of time ⟨τ/T⟩ of infected nodes in the steady state as +a function of k and m. Averages are computed over all the +nodes of each (k, m)-hyper-core. The insets represent ⟨τ/T⟩ +averaged over the first n nodes according to the coreness rank- +ings as a function of n. All results are obtained by averaging +the results of 103 numerical simulations, with an observation +window T = 103. For the SIR model, panels b and d show +the heatmap of the average final size of the epidemic ⟨R∞⟩ +as a function of k and m, where the process is seeded in a +single node belonging to the (k, m)-hyper-core (averaged over +all nodes of the hyper-core). The insets represent ⟨R∞⟩ as +a function of n averaged over the first n nodes according to +coreness rankings. All results are obtained by averaging the +results of 300 numerical simulations for each seed. Panels a +and b: music-review data set with ν = 1.25, λ = 5 × 10−4 +(a) and ν = 3, λ = 5 × 10−4 (b). Panels c and d: house- +committees data set with ν = 1.25, λ = 5 × 10−4 (c) and +ν = 4, λ = 5 × 10−5 (d). In all panels µ = 0.1. +in the I state during that window. This allows to iden- +tify the nodes on which the epidemic is mainly localized +in the steady state, i.e. the nodes that drive and sustain +the process. +Figure 3 reports results of simulations performed on +two data sets (the music-review and house-committees +data sets). Similar results are shown in the SM for the +other considered data sets. Panels 3a and 3c show that +nodes in (k, m)-hyper-cores with either increasing k or m +tend to be more often infectious, as the values of τ(j)/T +averaged over all nodes of each (k, m)-hyper-core increase +with k and m. This implies that the process is more local- +ized in the (k, m)-hyper-cores with large k (which favors +connectedness, hence mutual reachability) and m (i.e., +large hyperedges where large values of i can be obtained +yielding large infection rates). +The insets of the pan- +els moreover show the average of τ/T over the n nodes +with highest hyper-coreness R or highest s-coreness S. +The nodes with highest coreness tend to be more often in +the infectious state, and this tendency is stronger for the + +5 +hyper-coreness than for the s-coreness: among the nodes +with largest value of s-coreness, the hyper-coreness al- +lows to distinguish which ones are most involved in the +higher-order spreading processes. Moreover, in the SM +we show that a similar phenomenology is obtained with a +different model of contagion involving higher-order mech- +anisms [37, 54]. +D. +High hyper-coreness seeds increase total +outbreak size +Nodes belonging to large interaction groups have also +been shown to be optimal seeds of higher-order non-linear +contagions in terms of spreading speed at the beginning +of an SIS outbreak [36]. Which nodes have the largest +spreading power in the long run, i.e., in terms of final +size reached by a SIR process [12], remains however an +open question for higher-order spreading processes. We +thus consider the higher-order non-linear SIR model, in +which the dynamics, starting from a single seed, evolves +until no individual is in the state I anymore (only nodes +in states S or R remain). +To quantify the “spreading +power” of each node j considered separately as seed, we +average the final epidemic size R∞(j), i.e., the number +of nodes in state R at the end of the process, over 300 +stochastic runs for each seed. +Figure 3b and 3d show +that this average final epidemic size ⟨R∞(j)⟩, averaged +over all nodes of each (k, m)-hyper-core, increase with k +and m. The insets also show that the nodes with higher +hyper-coreness lead to larger epidemics, determining a +hierarchy even among the nodes with highest s-coreness. +In summary, nodes with higher connectedness along +groups of larger sizes can seed more efficiently, and the +hyper-coreness provides a good identification of the nodes +with highest spreading power in higher-order non-linear +contagion processes (similar results are shown in the SM +for another higher-order contagion model [37, 54]). +E. +Hypercore seeding facilitates systemic takeover +by minority norms +Group interactions can also play an important role in +the formation of a consensus and the emergence of shared +conventions in a population. According to critical mass +theory, regular individuals might then benefit –towards +addressing societal challenges– from the presence of a +committed minority that aims at overturning the sta- +tus quo [56]. Recently, it has been shown that groups +can modulate this takeover [28]. An important issue in +this respect concerns the best “seeding” strategy –where +should the committed minority start from in order to best +achieve the takeover? Here we show how hyper-coreness +can provide an answer. +We consider the well-known naming-game (NG) model +[38], which describes how a shared convention can emerge +in a population of locally interacting agents [57, 58], and +has been shown to account for the outcome of controlled +experiments of social coordination [59]. In the minimal +version of this model, recently modified to take group in- +teractions into account [28], individuals are represented +by the N nodes of a static hypergraph, and each node is +endowed with a dictionary that can contain at most two +names (representing conventions or norms), A and B. +At each time-step a hyperedge is chosen randomly and +a speaker is randomly selected within it. +The speaker +randomly chooses a name from its dictionary and com- +municates it to the other hyperedge members (the listen- +ers), who can agree or not on the proposed name. To +determine the possibility of an agreement within the hy- +peredge, we consider two distinct alternatives [28]: (i) +the union rule, for which an agreement can be reached if +at least one of the listeners has the proposed name in its +dictionary; (ii) the unanimity rule, for which the agree- +ment can be reached only if all the nodes in the group +have the proposed name in their dictionary. A parame- +ter β ∈ [0, 1] modulates the social influence by controlling +the propensity of the listeners to actually accept the local +consensus: the agreement in the group becomes effective +only with probability β. In this case, all nodes in the hy- +peredge add the accepted name to their dictionary, if it +was not already present, deleting all others. If instead no +agreement is reached, the listeners simply add the name +given by the speaker to their dictionaries. +Crucially, we include in the population a committed +minority of Np individuals who do not obey the aforemen- +tioned rules whenever they are listeners, but they instead +stick to their norm, a single name A, and their dictionary +is never updated. We initiate the process with the rest of +the population, i.e. the majority, having only the name +B in their dictionaries. The system can evolve towards +different regimes of co-existence of the two names or of +dominance of one name over the other, depending on β, +on the considered rule, and on the relative size of the mi- +nority p = Np/N. In particular, the committed minority +can overcome the majority, with the whole population +eventually converging on A, for a range of intermediate +values of β and for large enough p. +When committed +individuals are chosen at random in the population, this +range increases when the hypergraph contains hyperedges +of larger sizes [28]. This naturally raises the question of +whether the committed minority might also benefit from +belonging to specific substructures of a given hypergraph, +such as hyper-cores with large connectedness and group +sizes. +Here we investigate this issue through numerical simu- +lations of the higher-order NG process on empirical static +hypergraphs for varying values of the parameter β and of +the fraction p of committed individuals. In our numeri- +cal experiments committed individuals are selected with +different seeding strategies: (i) at random from the entire +population (random); (ii) as the Np ones with the high- +est hyper-coreness R (top hyper-coreness); (iii) as the Np +ones with the highest s-coreness (top s-coreness) in the +projected graph. In each experiment we measure the frac- +tion nA of nodes holding only A in their dictionary (both +committed or not), and focus on its large time limit n∗ +A. +This limit can be either 1, if the population reaches the +absorbing state in which all nodes agree on A, or, if the +absorbing state is not reached before tmax time-steps, we +average over 100 values of nA(t) sampled from the last T +time-steps. +Figure 4 reports the simulation results for two empir- +ical data sets, congress-bills (a-d) and the email-EU (e- +h). The results for the other data sets can be found in + +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +a +10 +2 +congress-bills +Random +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +b +10 +2 +s-coreness +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +c +10 +2 +n * +A +hyper-coreness +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +e +10 +2 +email-EU +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +f +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +g +10 +2 +n * +A +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +d +Random +R-rank +s-rank +103 +104 +105 +t +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +h +Random +R-rank +s-rank +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +FIG. 4. Comparison of seeding strategies for committed minorities in a higher-order naming-game process. In +the heatmaps a-c, e-g the stationary fraction n∗ +A of nodes supporting only the name A is shown as a function of the fraction +of committed nodes p and the agreement probability β (steps correspond to the fact that p varies by increments of 1/N). a-c: +congress-bills data set with unanimity rule. e-g: email-EU data set with union rule. Committed nodes are selected through +different strategies, in particular random seeding (a,e), top s-coreness (b,f), and top hyper-coreness (c,g). Panels d,h show +the temporal evolution of nA(t) according to the different seeding strategies and for fixed values of β and p (cross markers in +the heatmaps), (d): (β, p) = (0.48, 2.3 × 10−2); (h): (β, p) = (0.55, 9.2 × 10−3). The minority takeover, i.e. n∗ +A = 1, takes +place for 7.9% of the explored parameter space in panel a, 16.3% in b, 41.5% in c, 37.0% in e, 45.9% in f, and 56.4% in g. All +simulations are run until the absorbing state n∗ +A = 1 is reached or the dynamics has evolved for tmax = 5 × 105 time steps. The +stationary fraction n∗ +A is obtained by averaging over 100 values sampled in the last T = 5 × 104 time-steps. Results refer to the +median values obtained over 200 simulations for each pair of parameter values. +the SM. For the random strategy, we recover the results +of Ref. [28]: for low values of β, a co-existence state of +A and B is observed; at low fraction of committed and +large β values , the majority remains B. Contrarily, at +intermediate values of β, the minority takes over and the +whole population converges on A (in a way favored by the +union rule w.r.t. the unanimity rule). Non-random seed- +ing strategies yield the same phenomenology but enhance +the range of parameters in which the minority overturns +the majority. This is especially the case when we place +committed individuals in the most central nodes accord- +ing to their hyper-coreness. In this case, a tiny fraction +of committed is able to take over on a wide range of β +values, while for lower β values, a co-existence regime +is always observed –due to the small propensity to ac- +cept a local consensus [28]. For example, with the top +hyper-coreness strategy a fraction p = 1.51 × 10−2 in +the congress-bills data set with unanimity rule is enough +to allow the minority takeover over a range of β values +whose extension is ∆β ≳ 0.5. This cannot be achieved +with the other two seeding strategies, for which below +p = 2.8 × 10−2 only ∆β ≳ 0.2 can be reached (see +Fig. 4a-c). Analogously, in the email-EU data set with +the union rule a fraction p = 4.1 × 10−3 is enough to ob- +tain the minority dominance over ∆β ≳ 0.5 when seeded +according the top hyper-coreness strategy. In this case, +with the top s-coreness and the random strategies the +same result is obtained only for p = 1.33 × 10−2 and +p = 1.74 × 10−2 respectively (see Fig. 4e-g). To further +quantify the differences among these strategies we can +also calculate the value of critical mass pc necessary to +bring the system to the tipping point while keeping β con- +stant. In the congress-bills data set with unanimity rule +and β = 0.62 for instance, the critical mass for the top +hyper-coreness strategy is pc = 6.4 × 10−3 as compared +to pc = 2.68 × 10−2 and pc = 2.04 × 10−2 obtained with +the random and the top s-coreness strategies respectively +(see Fig. 4a-c); similarly, in the email-EU data set with +union rule and β = 0.83, these values are respectively +pc = 3.1 × 10−3, pc = 1.53 × 10−2, pc = 9.2 × 10−3 (see +Fig. 4e-g). +The hyper-coreness centrality is thus particularly effec- +tive in identifying nodes with a crucial role in higher-order +NG processes. Indeed, nodes belonging to (k, m)-hyper- +cores with large values of k and m, if committed, can +convince many others through their simultaneous pres- +ence in several large groups, and this can be efficiently +sustained by their large connectedness, favouring conver- +gence on their convention even outside the committed +minority. In addition, the temporal evolution diplayed in +Fig. 4d,h illustrate how, even when all seeding strategies +lead to the agreement on the convention initially sup- + +7 +ported by the minority, the convergence is much faster +for the hyper-coreness seeding strategy, followed by the +s-coreness and the random. +III. +DISCUSSION +We have considered here a systematic procedure to ex- +tract, from a given hypergraph, structures of increasing +connectedness along increasing group sizes: the (k, m)- +hyper-cores, in which each node is connected to the other +by at least k hyperedges (representing higher-order in- +teractions) of sizes at least m. Using the maximal con- +nectedness values of each node we define a new concept of +centrality in hypergraphs: a node hyper-coreness summa- +rizes its relative depth in the hierarchies of hyper-cores at +all orders (interaction sizes). Applying these concepts to +empirical data describing a variety of higher-order sys- +tems, we have shown how the (k, m)-hyper-cores pro- +vide a fingerprint of empirical hypergraphs. +Crucially, +we have also highlighted how hyper-cores with increas- +ing k and m play important roles in several dynamic +processes with higher-order mechanisms unfolding upon +hypergraphs, such as contagion processes and consensus +formation. +The hyper-coreness centrality in particular +identifies nodes with high spreading power and on which +stationary contagion processes tend to localize; moreover +nodes with high hyper-coreness, if belonging to a commit- +ted minority, can be particularly efficient at overturning +a majority convention. +Our work opens the door to several research directions +in the expanding field of hypergraphs structure and dy- +namics. It can provide an additional systematic charac- +terization of both empirical and model hypergraphs, and +thus potentially a model validation tool as well as a com- +parison method between hypergraphs (e.g. by computing +distances between the (k, m)-hypercore profiles of Fig. 2). +For specific systems where additional properties of the +nodes are known, the shell indices and hyper-coreness +values of nodes could be compared in more detail to pro- +vide insights into their relative positions and roles in the +system. +Moreover, while here we focused on static hypergraphs, +many such systems evolve in time [60, 61]. Hyper-cores +and hyper-coreness could be used to investigate the evo- +lution of the higher-order interactions at multiple scales, +from the global evolution of the structure described by +hyper-core sizes, to the shell indices and hyper-coreness +of individual nodes from one period to the next [8]. An +interesting case study in this direction could be for in- +stance the evolution of the hyper-core positions of scien- +tists in co-authorship “networks”, which are in fact by +construction evolving hypergraphs [60]. +IV. +MATERIALS AND METHODS +A. +Data description and preprocessing +Several data sets we considered are publicly available in +the form of static hypergraphs, thus they do not require +any preprocessing. These data sets describe: +• email communications: within a European institu- +tion (email-EU [44]), and within Enron, between +a core-set of workers (email-Enron [45, 46]). Each +node corresponds to an email address and a hy- +peredge includes the sender and all receivers of an +email. +• interactions in legislative bills in the U.S. Congress +(congress-bills) and in the U.S. Senate (senate-bills) +[46, 49, 51, 52]: each node corresponds to a member +of the U.S. Congress or Senate and a hyperedge +involves sponsors and co-sponsors of legislative bills +discussed in the Congress or Senate. +• interactions in committees in the U.S. House of +Representatives (house-committees) and in the U.S. +Senate (senate-committees) [46, 49, 50]: each node +corresponds to a member of the U.S. House of Rep- +resentatives or Senate and each hyperedge involves +nodes that share membership in a committee. +• online interactions (3 data sets): +exchanges be- +tween users of MathOverflow on algebra top- +ics +(algebra-questions) +or +on +geometry +topics +(geometry-questions), in which each node corre- +sponds to a user of MathOverflow and each hyper- +edge involves those users who have answered a spe- +cific question belonging to the topic of algebra or ge- +ometry [46, 48]; interactions between Amazon users +on music (music-review [46, 47]), in which each node +corresponds to an Amazon user and each hyperedge +involves users who have reviewed a specific product +belonging to the category of blues music. +Moreover, we built static hypergraphs from several +data sets of time-resolved face-to-face human interac- +tions, as in [26, 28]. The data sets are provided by the +SocioPatterns collaboration [40–42] and by the Contacts +among Utah’s School-age Population (CUSP) project +[43] and describe interactions between individuals in sev- +eral contexts : +a hospital (LH10 [62]), a workplace +(InVS15 [41, 63]), a conference (SFHH [41]), a high-school +(Thiers13 [64]), two primary-schools (LyonSchool [65], +Elem1 [43]) and a middle-school (Mid1 [43]). For these +data sets we carried out an aggregation procedure to ob- +tain static hypergraphs: (i) we aggregate the data over +time windows of 15 minutes; (ii) we identify the cliques +in each time window, i.e. groups of nodes forming a fully +connected cluster, (iii) we identify in each temporal win- +dow the maximum cliques, i.e. +cliques not completely +contained in a larger clique, and promote them to a hy- +peredge status. +Overall, the data sets considered describe interactions +in several different environments, mediated by different +mechanisms. They correspond to a wide variety of sta- +tistical properties (e.g. data set size, hyperedges size dis- +tributions), as shown in the SM where these statistical +properties of the data sets are reported in details. + +8 +B. +Models and stochastic simulations +1. +Higher-order non-linear contagion +We performed stochastic numerical simulations of the +higher-order non-linear contagion model on each empir- +ical static hypergraph. +The simulations are performed +with discrete time-steps. The S → I infection mecha- +nism is the same for the SIR and the SIS models: for +each time-step ∆t, given a hyperedge of size m contain- +ing i infected nodes, each of the susceptible nodes in it +can be infected with probability (1 − e−λiν). Therefore, +the probability that a node j is infected in a time-step +∆t is: +pj = 1 − +� +e∈E(j) +e−λiν +e , +(2) +where E(j) denotes the set of hyperedges in which the +node j is involved and ie is the number of infected nodes +in the hyperedge e. +Each infected node heals (return- +ing susceptible in SIS or gaining immunity in SIR) with +probability µ in each time-step. +In the SIS process the population is initialized with a +single infectious seed randomly selected in the population +and the process is iterated until the system reaches a +steady state with a fluctuating number of infectious. An +observation time window T is then considered and the +time τ spent in the infectious state is estimated for all +nodes over that time-window. The results are averaged +over 103 simulations. +In the SIR process the population is initialized with a +single infectious seed j and the dynamic process is iter- +ated until no more infectious nodes are present: the final +epidemic size R∞(j) obtained by seeding the infection in +j is defined as the final number of nodes in the R state. +The results are averaged over 300 simulations for each +infection seed j. +2. +Higher-order NG process +We also performed numerical simulations of the higher- +order NG process on the empirical hypergraphs. +The +system with N nodes is initialized by fixing Np nodes as +belonging to the committed minority (equivalently, with +a fraction p = Np/N of committed nodes), with only the +name A in their dictionary, and setting the dictionaries +of all the other nodes of the majority with only the name +B. The committed nodes are selected following one of +the three seeding strategies, i.e. randomly from the whole +population or as the Np nodes with highest s-coreness or +hyper-coreness. If several nodes have the same coreness +value, the committed nodes are randomly selected within +the coreness class. +The simulations are performed in discrete time-steps: +at each time-step a hyperedge is randomly selected (ac- +tivation of the group) and within it a node is randomly +chosen as the speaker, while the other nodes behave as +listeners. The speaker randomly selects a name in their +dictionary and all nodes in the group update their dictio- +nary according to the chosen agreement rule (except for +the committed nodes). 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PLOS ONE 6, +1–13 (2011). + +Supplementary Material for ”Hyper-cores promote localization and efficient seeding +in higher-order processes” +Marco Mancastroppa,1 Iacopo Iacopini,2 Giovanni Petri,3 and Alain Barrat1 +1Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, +Turing Center for Living Systems, Marseille, France +2Department of Network and Data Science, Central European University, 1100 Vienna, Austria +3CENTAI, Corso Inghilterra 3, 10138 Turin, Italy +In this Supplementary Material we present the same results as in the main text for all the considered data sets +and also further results. We first present in detail some of the statistical properties of the data sets and of the +static hypergraphs considered (Supplementary Section 1). In Supplementary Section 2 we present the results of +the (k, m)-core decomposition, showing how the (k, m)-cores and (k, m)-shells are populated as a function of k +and m, the functional form of the m-shell index Cm(i) for some nodes, the distributions of the hypercoreness and +s-coreness centralities and their correlations. In Supplementary Section 3 we present the results of the higher-order +non-linear contagion process [1], both in the SIS and SIR formulation. In Supplementary Section 4 we introduce +in details the threshold higher-order process [2], its numerical implementation and its results in relation to the +hyper-cores, both in the SIS and SIR formulation, as done for the higher-order non-linear contagion model. Finally +in Supplementary Section 5, the results of the higher-order naming-game process [3] are presented for both the +union and the unanimity rules. +Supplementary Section 1. +Properties of the data sets +The considered data sets describe interactions in several environments, mediated by different mechanisms, and +thus they differ in their fundamental statistical properties. This is summarized in Table I and Fig. 1: the number +of nodes and hyperedges vary among the data sets considered, the distribution Ψ(m) of the hyperedge sizes m, the +range of their sizes m ∈ [2, M] and the average hyperedge size ⟨m⟩ are different among the data sets. +data set +N +E +M +⟨m⟩ +LH10 +76 +1 102 +7 +3.4 +Thiers13 +327 +4 795 +7 +3.1 +InVS15 +217 +3 279 +10 +2.8 +SFHH +403 +6 398 +10 +2.7 +LyonSchool +242 +10 848 10 +4.0 +Mid1 +591 +61 521 13 +3.9 +Elem1 +339 +20 940 16 +4.7 +email-EU +979 +24 399 25 +3.5 +data set +N +E +M +⟨m⟩ +congress-bills +1 718 +83 105 +25 +8.8 +senate-committees +282 +302 +31 +17.6 +email-Enron +143 +1 459 +37 +3.1 +house-committees +1 290 +335 +82 +35.3 +music-review +1 106 +686 +83 +15.3 +senate-bills +294 +21 721 +99 +9.9 +algebra-questions +423 +980 +107 +7.6 +geometry-questions +580 +888 +230 +13.0 +Supplementary Table I: Some properties of the data sets. The tables give: the number of nodes N, the +number of hyperedges E, the maximum size of the hyperedges M and the average size of the hyperedges ⟨m⟩. +arXiv:2301.04235v1 [physics.soc-ph] 10 Jan 2023 + +2 +2 +4 +6 +102 +(m) +a +2 +4 +6 +101 +102 +103 +b +5 +10 +100 +101 +102 +103 +c +5 +10 +100 +101 +102 +103 +d +5 +10 +100 +101 +102 +103 +(m) +e +5 +10 +101 +102 +103 +104 +f +5 +10 +15 +101 +102 +103 +g +10 +20 +102 +103 +104 +h +10 +20 +104 +(m) +i +10 +20 +30 +100 +101 +j +0 +20 +40 +100 +101 +102 +103 +k +0 +50 +100 +101 +l +0 +50 +m +100 +101 +(m) +m +0 +50 +100 +m +100 +101 +102 +103 +n +0 +50 +100 +m +100 +101 +102 +o +0 +100 +200 +m +100 +101 +102 +p +Supplementary Figure 1: Hyperedge size distribution. We show the hyperedge size distribution Ψ(m), i.e. the +number of hyperedges of size m, for all the data sets: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), +SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills +(panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel +m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p). + +3 +Supplementary Section 2. +Hyper-core decomposition +1 +9 +17 +25 +33 +41 +49 +2 +3 +4 +5 +6 +7 +m +n(k, m) +a +100 +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +1 +8 +15 +22 +29 +36 +2 +3 +4 +5 +6 +7 +n(k, m) +b +10 +1 +100 +1 +9 +17 +25 +33 +41 +2 +3 +4 +5 +6 +7 +8 +9 +10 +n(k, m) +c +10 +1 +100 +1 +7 +13 +19 +25 +31 +2 +3 +4 +5 +6 +7 +8 +9 +10 +n(k, m) +d +10 +1 +100 +1 +28 +55 +82 +109 +136 +2 +3 +4 +5 +6 +7 +8 +9 +10 +m +e +10 +1 +100 +1 +47 +93 +139 +185 +231 +277 +2 +4 +6 +8 +10 +12 +f +10 +1 +100 +1 +48 +95 +142 +189 +236 +2 +4 +6 +8 +10 +12 +14 +16 +g +10 +1 +100 +1 +20 +39 +58 +77 +96 +2 +6 +10 +14 +18 +22 +h +10 +1 +100 +1 +183 +365 +547 +729 +911 +1093 +2 +6 +10 +14 +18 +22 +m +i +10 +1 +100 +1 +6 +11 +16 +21 +26 +31 +2 +7 +12 +17 +22 +27 +j +100 +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +1 +5 +9 +13 +17 +21 +25 +2 +8 +14 +20 +26 +32 +k +100 +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +1 +4 +7 +10 +13 +16 +19 +2 +16 +30 +44 +58 +72 +l +10 +1 +100 +1 +8 +15 +22 +29 +36 +k +2 +16 +30 +44 +58 +72 +m +m +10 +1 +100 +1 +199 +397 +595 +793 +991 +k +2 +18 +34 +50 +66 +82 +98 +n +100 +2 × 10 +1 +3 × 10 +1 +4 × 10 +1 +6 × 10 +1 +1 +11 +21 +31 +41 +51 +k +2 +20 +38 +56 +74 +92 +o +10 +1 +100 +1 +15 +29 +43 +57 +71 +k +2 +40 +78 +116 +154 +192 +230 +p +10 +1 +100 +Supplementary Figure 2: Hyper-core decomposition I. All panels show colormaps giving the relative size +n(k,m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a +function of m and k (white regions correspond to n(k,m) = 0). The following data sets are considered: LH10 +(panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 +(panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), +house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and +geometry-questions (panel p). + +4 +0 +20 +40 +0.2 +0.4 +0.6 +0.8 +1.0 +n(k, m) +a +m = 2.0 +m = 3.0 +m = 4.0 +m = 5.0 +0 +20 +40 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +b +m = 2.0 +m = 3.0 +m = 4.0 +m = 5.0 +0 +20 +40 +0.2 +0.4 +0.6 +0.8 +1.0 +c +m = 2.0 +m = 3.0 +m = 4.0 +m = 5.0 +0 +10 +20 +30 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +d +m = 2.0 +m = 3.0 +m = 4.0 +m = 5.0 +0 +100 +200 +0.2 +0.4 +0.6 +0.8 +1.0 +n(k, m) +e +m = 2.0 +m = 3.0 +m = 4.0 +m = 5.0 +0 +200 +400 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +f +m = 2.0 +m = 4.0 +m = 6.0 +m = 8.0 +0 +200 +400 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +g +m = 2.0 +m = 4.0 +m = 6.0 +m = 8.0 +0 +50 +100 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +h +m = 2.0 +m = 6.0 +m = 10.0 +m = 14.0 +0 +500 +1000 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +n(k, m) +i +m = 2.0 +m = 6.0 +m = 10.0 +m = 14.0 +0 +10 +20 +30 +0.2 +0.4 +0.6 +0.8 +1.0 +j +m = 2.0 +m = 12.0 +m = 17.0 +m = 22.0 +0 +10 +20 +0.2 +0.4 +0.6 +0.8 +1.0 +k +m = 2.0 +m = 4.0 +m = 6.0 +m = 8.0 +5 +10 +15 +0.2 +0.4 +0.6 +0.8 +1.0 +l +m = 2.0 +m = 15.0 +m = 41.0 +m = 54.0 +0 +20 +40 +k +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +n(k, m) +m +m = 2.0 +m = 10.0 +m = 18.0 +m = 32.0 +0 +500 +1000 +k +0.2 +0.4 +0.6 +0.8 +1.0 +n +m = 2.0 +m = 18.0 +m = 34.0 +m = 50.0 +0 +20 +40 +60 +k +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +o +m = 2.0 +m = 8.0 +m = 14.0 +m = 20.0 +0 +25 +50 +75 +k +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +p +m = 2.0 +m = 15.0 +m = 28.0 +m = 41.0 +Supplementary Figure 3: Hyper-core decomposition II. All panels show the relative size n(k,m) (number of +nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a function of k for +fixed values of m. The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), +SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills +(panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel +m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p). + +5 +1 +9 +17 +25 +33 +41 +49 +2 +3 +4 +5 +6 +7 +m +s(k, m) +a +10 +1 +1 +8 +15 +22 +29 +36 +2 +3 +4 +5 +6 +7 +s(k, m) +b +10 +2 +10 +1 +1 +9 +17 +25 +33 +41 +2 +3 +4 +5 +6 +7 +8 +9 +10 +s(k, m) +c +10 +2 +10 +1 +1 +7 +13 +19 +25 +31 +2 +3 +4 +5 +6 +7 +8 +9 +10 +s(k, m) +d +10 +2 +10 +1 +1 +28 +55 +82 +109 +136 +2 +3 +4 +5 +6 +7 +8 +9 +10 +m +e +10 +2 +10 +1 +1 +47 +93 +139 +185 +231 +277 +2 +4 +6 +8 +10 +12 +f +10 +2 +10 +1 +1 +48 +95 +142 +189 +236 +2 +4 +6 +8 +10 +12 +14 +16 +g +10 +2 +10 +1 +1 +20 +39 +58 +77 +96 +2 +6 +10 +14 +18 +22 +h +10 +2 +10 +1 +1 +183 +365 +547 +729 +911 +1093 +2 +6 +10 +14 +18 +22 +m +i +10 +3 +10 +2 +10 +1 +1 +6 +11 +16 +21 +26 +31 +2 +7 +12 +17 +22 +27 +j +10 +2 +10 +1 +1 +5 +9 +13 +17 +21 +25 +2 +8 +14 +20 +26 +32 +k +10 +2 +10 +1 +1 +4 +7 +10 +13 +16 +19 +2 +16 +30 +44 +58 +72 +l +10 +1 +1 +8 +15 +22 +29 +36 +k +2 +16 +30 +44 +58 +72 +m +m +10 +3 +10 +2 +10 +1 +1 +199 +397 +595 +793 +991 +k +2 +18 +34 +50 +66 +82 +98 +n +10 +2 +10 +1 +1 +11 +21 +31 +41 +51 +k +2 +20 +38 +56 +74 +92 +o +10 +2 +10 +1 +1 +15 +29 +43 +57 +71 +k +2 +40 +78 +116 +154 +192 +230 +p +10 +2 +10 +1 +Supplementary Figure 4: (k, m)-shells. All panels show colormaps giving the relative size s(k,m) (number of +nodes in the hyper-shell, divided by the total number of nodes N) of the (k, m)-shell as a function of m and k +(white regions correspond to s(k,m) = 0). The following data sets are considered: LH10 (panel a), Thiers13 (panel +b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel +h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), +music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p). + +6 +2 +4 +6 +0.00 +0.25 +0.50 +0.75 +1.00 +Cm(i)/km +max +a +R = 0.08 +R = 0.7 +R = 3.1 +R = 6.0 +2 +4 +6 +0.00 +0.25 +0.50 +0.75 +1.00 +b +R = 0.05 +R = 1.6 +R = 3.0 +R = 6.0 +2.5 +5.0 +7.5 +10.0 +0.00 +0.25 +0.50 +0.75 +1.00 +c +R = 0.02 +R = 0.8 +R = 1.1 +R = 9.0 +2.5 +5.0 +7.5 +10.0 +0.00 +0.25 +0.50 +0.75 +1.00 +d +R = 0.03 +R = 1.7 +R = 2.4 +R = 8.0 +2.5 +5.0 +7.5 +10.0 +0.00 +0.25 +0.50 +0.75 +1.00 +Cm(i)/km +max +e +R = 0.4 +R = 3.1 +R = 3.8 +R = 6.2 +5 +10 +0.00 +0.25 +0.50 +0.75 +1.00 +f +R = 0.3 +R = 3.0 +R = 3.6 +R = 9.6 +5 +10 +15 +0.00 +0.25 +0.50 +0.75 +1.00 +g +R = 0.2 +R = 3.1 +R = 4.6 +R = 10.9 +10 +20 +0.00 +0.25 +0.50 +0.75 +1.00 +h +R = 0.009 +R = 1.0 +R = 5.8 +R = 22.2 +10 +20 +0.00 +0.25 +0.50 +0.75 +1.00 +Cm(i)/km +max +i +R = 0.003 +R = 7.6 +R = 14.4 +R = 24.0 +10 +20 +30 +0.00 +0.25 +0.50 +0.75 +1.00 +j +R = 1.0 +R = 13.9 +R = 22.3 +R = 30.0 +10 +20 +30 +0.00 +0.25 +0.50 +0.75 +1.00 +k +R = 0.1 +R = 6.3 +R = 27.5 +R = 36.0 +0 +25 +50 +75 +0.00 +0.25 +0.50 +0.75 +1.00 +l +R = 0.2 +R = 27.1 +R = 50.4 +R = 81.0 +0 +25 +50 +75 +m +0.00 +0.25 +0.50 +0.75 +1.00 +Cm(i)/km +max +m +R = 0.03 +R = 37.8 +R = 53.5 +R = 82.0 +0 +50 +100 +m +0.00 +0.25 +0.50 +0.75 +1.00 +n +R = 0.1 +R = 48.4 +R = 89.4 +R = 98.0 +0 +50 +100 +m +0.00 +0.25 +0.50 +0.75 +1.00 +o +R = 0.02 +R = 3.1 +R = 78.4 +R = 106.0 +0 +100 +200 +m +0.00 +0.25 +0.50 +0.75 +1.00 +p +R = 0.01 +R = 11.2 +R = 196.5 +R = 229.0 +Supplementary Figure 5: m-shell index. All panels show the normalized m-shell index function Cm(i)/km +max as a +function of m for four nodes: one node is selected randomly among the nodes in the class with highest +hyper-coreness R; one node is selected randomly among the nodes in the class with smallest hyper-coreness R; the +two remaining node are selected from intermediate hyper-coreness classes, so that the positions in the +hyper-coreness ranking of the four nodes are equispaced. The following data sets are considered: LH10 (panel a), +Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), +email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), +house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and +geometry-questions (panel p). + +7 +0 +50 +0 +2 +4 +6 +R(i) +a +0 +5 +R +0 +20 +P(R) +0 +200 +0 +2 +4 +6 +b +0 +5 +R +0 +50 +P(R) +0 +100 +200 +0 +2 +4 +6 +8 +c +0 +5 +R +0 +50 +P(R) +0 +200 +400 +0 +2 +4 +6 +8 +d +0 +5 +R +0 +100 +P(R) +0 +100 +200 +0 +2 +4 +6 +8 +R(i) +e +2.5 5.0 +R +0 +50 +P(R) +0 +250 +500 +0.0 +2.5 +5.0 +7.5 +10.0 +f +0 +10 +R +0 +200 +P(R) +0 +200 +0.0 +2.5 +5.0 +7.5 +10.0 +g +0 +10 +R +0 +100 +P(R) +0 +500 +1000 +0 +5 +10 +15 +20 +h +0 +20 +R +0 +250 +P(R) +0 +1000 +0 +10 +20 +30 +R(i) +i +0 +25 +R +0 +250 +P(R) +0 +200 +0 +10 +20 +30 +40 +j +0 +25 +R +0 +50 +P(R) +0 +100 +0 +20 +40 +k +0 +25 +R +0 +20 +P(R) +0 +500 +1000 +0 +25 +50 +75 +100 +l +0 +50 +R +0 +100 +P(R) +0 +500 +1000 +Rank(i) +0 +25 +50 +75 +100 +R(i) +m +0 +50 +R +0 +200 +P(R) +0 +200 +Rank(i) +0 +50 +100 +n +0 +100 +R +0 +100 +P(R) +0 +200 +400 +Rank(i) +0 +50 +100 +o +0 +100 +R +0 +100 +P(R) +0 +250 +500 +Rank(i) +0 +100 +200 +300 +p +0 +200 +R +0 +200 +P(R) +Supplementary Figure 6: Hyper-coreness centrality. In all panels the hyper-coreness R(i) is plotted as a +function of the corresponding node rank: the insets show the distribution P(R) of the hyper-coreness. The +following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), +LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), +senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), +senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p). + +8 +0 +50 +0 +50 +100 +150 +200 +S(i) +a +0 +200 +S +0 +10 +P(S) +0 +200 +0 +50 +100 +b +0 +100 +S +0 +50 +P(S) +0 +100 +200 +0 +50 +100 +150 +c +0 +100 +S +0 +100 +P(S) +0 +200 +400 +0 +50 +100 +d +0 +100 +S +0 +100 +P(S) +0 +100 +200 +0 +200 +400 +600 +800 +S(i) +e +250 500 +S +0 +50 +P(S) +0 +250 +500 +0 +500 +1000 +1500 +f +0 +1000 +S +0 +200 +P(S) +0 +200 +0 +500 +1000 +1500 +g +0 +1000 +S +0 +100 +P(S) +0 +500 +1000 +0 +250 +500 +750 +1000 +h +0 +1000 +S +0 +250 +P(S) +0 +1000 +0 +2000 +4000 +6000 +S(i) +i +0 +5000 +S +0 +200 +P(S) +0 +200 +0 +100 +200 +300 +400 +j +0 +150 300 +S +0 +50 +P(S) +0 +100 +0 +50 +100 +150 +k +0 +100 +S +0 +20 +P(S) +0 +500 +1000 +0 +200 +400 +l +0 +250 +S +0 +250 +P(S) +0 +500 +1000 +Rank(i) +0 +100 +200 +300 +400 +S(i) +m +0 +250 +S +0 +200 +P(S) +0 +200 +Rank(i) +0 +5000 +10000 +15000 +n +0 +10000 +S +0 +100 +P(S) +0 +200 +400 +Rank(i) +0 +100 +200 +300 +400 +o +0 +250 +S +0 +100 +P(S) +0 +250 +500 +Rank(i) +0 +250 +500 +750 +1000 +p +0 +1000 +S +0 +200 +P(S) +Supplementary Figure 7: s-coreness centrality. In all panels the s-coreness S(i) is plotted as a function of the +corresponding node rank: the insets give the distribution P(S) of the s-coreness. The following data sets are +considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 +(panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron +(panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) +and geometry-questions (panel p). + +9 +0 +100 +200 +0 +2 +4 +6 +R(i) +a += 0.98 += 0.92 +0 +50 +100 +0 +2 +4 +6 +b += 0.93 += 0.88 +0 +100 +0 +2 +4 +6 +8 +c += 0.94 += 0.83 +0 +50 +100 +0 +2 +4 +6 +8 +d += 0.77 += 0.75 +250 +500 +2 +4 +6 +R(i) +e += 0.73 += 0.51 +500 +1000 +0.0 +2.5 +5.0 +7.5 +10.0 +f += 0.74 += 0.74 +0 +1000 +0.0 +2.5 +5.0 +7.5 +10.0 +g += 0.82 += 0.69 +0 +500 +1000 +0 +5 +10 +15 +20 +h += 0.90 += 0.85 +0 +5000 +0 +10 +20 +R(i) +i += 0.92 += 0.83 +100 +200 +300 +0 +10 +20 +30 +j += 0.93 += 0.78 +0 +50 +100 +0 +10 +20 +30 +k += 0.53 += 0.53 +0 +200 +0 +20 +40 +60 +80 +l += 0.92 += 0.79 +0 +200 +S(i) +0 +20 +40 +60 +80 +R(i) +m += 0.74 += 0.58 +0 +10000 +S(i) +0 +25 +50 +75 +100 +n += 0.96 += 0.85 +0 +200 +S(i) +0 +25 +50 +75 +100 +o += 0.92 += 0.88 +0 +500 +1000 +S(i) +0 +100 +200 +p += 0.88 += 0.88 +Supplementary Figure 8: Hyper-coreness vs. s-coreness centralities. All panels show scatterplots of the +hyper-coreness R(i) vs. the s-coreness S(i) for all nodes: the text-box reports the Pearson correlation coefficient ρ +of R(i) and S(i) and the Kendall’s τ coefficient of the corresponding node rankings (in all cases the p-value for +both the coefficients is p ≪ 0.001). The following data sets are considered: LH10 (panel a), Thiers13 (panel b), +InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), +congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), +music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p). + +10 +Supplementary Section 3. +Higher-order non-linear contagion process +data set +ν +λ +LH10 +4.0 +5 × 10−4 +Thiers13 +3.0 +5 × 10−4 +InVS15 +4.0 +5 × 10−4 +SFHH +4.0 +5 × 10−4 +LyonSchool +2.0 +5 × 10−4 +Mid1 +3.0 +5 × 10−5 +Elem1 +1.5 +5 × 10−5 +email-EU +2.5 +5 × 10−5 +data set +ν +λ +congress-bills +2.0 +5 × 10−6 +senate-committees +1.25 +5 × 10−4 +email-Enron +2.0 +5 × 10−4 +house-committees +1.25 +5 × 10−4 +music-review +1.25 +5 × 10−4 +senate-bills +1.5 +5 × 10−6 +algebra-questions +1.25 +5 × 10−4 +geometry-questions +1.5 +5 × 10−5 +Supplementary Table II: Parameters for Figs. 9-11. The tables summarize the parameters of the higher-order +non-linear SIS contagion process considered for each data set in Figs. 9-11. +data set +ν +λ +LH10 +1.5 +0.010 +Thiers13 +4.0 +0.001 +InVS15 +4.0 +0.001 +SFHH +4.0 +0.010 +LyonSchool +4.0 +0.001 +Mid1 +4.0 +5 × 10−5 +Elem1 +4.0 +10−4 +email-EU +4.0 +5 × 10−5 +data set +ν +λ +congress-bills +1.5 +5 × 10−5 +senate-committees +4.0 +10−4 +email-Enron +4.0 +5 × 10−4 +house-committees +4.0 +5 × 10−5 +music-review +3.0 +5 × 10−4 +senate-bills +4.0 +5 × 10−5 +algebra-questions +4.0 +0.001 +geometry-questions +4.0 +5 × 10−4 +Supplementary Table III: Parameters for Figs. 12-14. The tables summarize the parameters of the +higher-order non-linear SIR contagion process considered for each data sets in Figs. 12-14. + +11 +1 +9 +17 +25 +33 +41 +49 +2 +3 +4 +5 +6 +7 +m +/T +a +0.70 +0.75 +0.80 +0.85 +0 +50 +n +0.7 +0.8 +0.9 +/T n +1 +8 +15 +22 +29 +36 +2 +3 +4 +5 +6 +7 +/T +b +0.4 +0.5 +0.6 +0.7 +0 +250 +n +0.4 +0.6 +/T n +1 +9 +17 +25 +33 +41 +2 +3 +4 +5 +6 +7 +8 +9 +10 +/T +c +0.6 +0.7 +0.8 +0 +200 +n +0.6 +0.8 +/T n +1 +7 +13 +19 +25 +31 +2 +3 +4 +5 +6 +7 +8 +9 +10 +/T +d +0.60 +0.65 +0.70 +0.75 +0.80 +0 +250 +n +0.6 +0.7 +0.8 +/T n +1 +28 +55 +82 +109 +136 +2 +3 +4 +5 +6 +7 +8 +9 +10 +m +e +0.78 +0.80 +0.82 +0.84 +0.86 +0.88 +0 +200 +n +0.80 +0.85 +0.90 +/T n +1 +47 +93 +139 +185 +231 +277 +2 +4 +6 +8 +10 +12 +f +0.74 +0.76 +0.78 +0.80 +0.82 +0 +500 +n +0.75 +0.80 +/T n +1 +48 +95 +142 +189 +236 +2 +4 +6 +8 +10 +12 +14 +16 +g +0.1 +0.2 +0.3 +0 +200 +n +0.1 +0.2 +0.3 +/T n +1 +20 +39 +58 +77 +96 +2 +6 +10 +14 +18 +22 +h +0.4 +0.5 +0.6 +0 +400 800 +n +0.4 +0.6 +/T n +1 +183 +365 +547 +729 +911 +1093 +2 +6 +10 +14 +18 +22 +m +i +0.14 +0.16 +0.18 +0.20 +0 +1000 +n +0.15 +0.20 +/T n +1 +6 +11 +16 +21 +26 +31 +2 +7 +12 +17 +22 +27 +j +0.60 +0.65 +0.70 +0.75 +0.80 +0 +200 +n +0.6 +0.7 +0.8 +/T n +1 +5 +9 +13 +17 +21 +25 +2 +8 +14 +20 +26 +32 +k +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0 +100 +n +0.6 +0.8 +/T n +1 +4 +7 +10 +13 +16 +19 +2 +16 +30 +44 +58 +72 +l +0.70 +0.75 +0.80 +0.85 +0 +1000 +n +0.7 +0.8 +/T n +1 +8 +15 +22 +29 +36 +k +2 +16 +30 +44 +58 +72 +m +m +0.6 +0.7 +0.8 +0 +1000 +n +0.6 +0.8 +/T n +S-rank +R-rank +1 +199 +397 +595 +793 +991 +k +2 +18 +34 +50 +66 +82 +98 +n +0.60 +0.65 +0.70 +0.75 +0.80 +0 +200 +n +0.6 +0.7 +0.8 +/T n +S-rank +R-rank +1 +11 +21 +31 +41 +51 +k +2 +20 +38 +56 +74 +92 +o +0.4 +0.5 +0.6 +0.7 +0.8 +0 +250 +n +0.4 +0.6 +0.8 +/T n +S-rank +R-rank +1 +15 +29 +43 +57 +71 +k +2 +40 +78 +116 +154 +192 +230 +p +0.4 +0.5 +0.6 +0.7 +0 +500 +n +0.4 +0.6 +0.8 +/T n +S-rank +R-rank +Supplementary Figure 9: Higher-order non-linear contagion process - SIS model - I. All panels give, as a +heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state +averaged over the nodes of the (k, m)-hyper-core. The insets represent τ/T averaged over the first n nodes +according to the coreness rankings as a function of n. All results are obtained by averaging the results of 103 +numerical simulations, with a single random seed of infection and with an observation window T = 103. The +following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), +Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), +music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered +for each data set are reported in Table II. + +12 +0 +20 +40 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +/T +a +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +0.4 +0.5 +0.6 +0.7 b +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 c +m = 2 +m = 3 +m = 4 +m = 5 +0 +10 +20 +30 +0.60 +0.65 +0.70 +0.75 +0.80 d +m = 2 +m = 3 +m = 4 +m = 5 +0 +100 +200 +0.82 +0.84 +0.86 +0.88 +/T +e +m = 2 +m = 3 +m = 4 +m = 5 +0 +200 +400 +0.74 +0.76 +0.78 +0.80 +0.82 +f +m = 2 +m = 4 +m = 6 +m = 8 +0 +200 +400 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 g +m = 2 +m = 4 +m = 6 +m = 8 +0 +50 +100 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 h +m = 2 +m = 6 +m = 10 +m = 14 +0 +500 +1000 +0.12 +0.14 +0.16 +0.18 +0.20 +/T +i +m = 2 +m = 6 +m = 10 +m = 14 +0 +10 +20 +30 +0.60 +0.65 +0.70 +0.75 +0.80 j +m = 2 +m = 12 +m = 17 +m = 22 +0 +10 +20 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 k +m = 2 +m = 4 +m = 6 +m = 8 +5 +10 +15 +0.70 +0.75 +0.80 +0.85 +l +m = 2 +m = 15 +m = 41 +m = 54 +0 +20 +40 +k +0.5 +0.6 +0.7 +0.8 +/T +m +m = 2 +m = 10 +m = 18 +m = 32 +0 +500 +1000 +k +0.60 +0.65 +0.70 +0.75 +0.80 +n +m = 2 +m = 18 +m = 34 +m = 50 +0 +20 +40 +60 +k +0.4 +0.5 +0.6 +0.7 +0.8 +o +m = 2 +m = 8 +m = 14 +m = 20 +0 +25 +50 +75 +k +0.4 +0.5 +0.6 +0.7 +p +m = 2 +m = 15 +m = 28 +m = 41 +Supplementary Figure 10: Higher-order non-linear contagion process - SIS model - II. In all panels the +average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the +(k, m)-hyper-core is shown as a function of k at fixed values of m. All results are obtained by averaging the results +of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103. +The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), +Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), +music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered +for each data set are reported in Table II. + +13 +2 +4 +6 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +/T +a +k = 1 +k = 10 +k = 20 +k = 30 +2 +4 +6 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 b +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 c +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +0.60 +0.65 +0.70 +0.75 +0.80 d +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +0.80 +0.82 +0.84 +0.86 +0.88 +/T +e +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +0.74 +0.76 +0.78 +0.80 +0.82 +f +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +15 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +g +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +0.30 +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 h +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +0.13 +0.14 +0.15 +0.16 +0.17 +/T +i +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +0.60 +0.65 +0.70 +0.75 +0.80 +j +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 k +k = 1 +k = 7 +k = 14 +k = 21 +0 +25 +50 +75 +0.70 +0.75 +0.80 +0.85 +l +k = 1 +k = 6 +k = 12 +k = 18 +0 +25 +50 +75 +m +0.5 +0.6 +0.7 +0.8 +/T +m +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +0.60 +0.65 +0.70 +0.75 +n +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +0.4 +0.5 +0.6 +0.7 +0.8 +o +k = 1 +k = 10 +k = 20 +k = 30 +0 +100 +200 +m +0.4 +0.5 +0.6 +0.7 +p +k = 1 +k = 10 +k = 20 +k = 30 +Supplementary Figure 11: Higher-order non-linear contagion process - SIS model - III. In all panels the +average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the +(k, m)-hyper-core is shown as a function of m at fixed values of k. All results are obtained by averaging the results +of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103. +The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), +Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), +music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered +for each data set are reported in Table II. + +14 +1 +9 +17 +25 +33 +41 +49 +2 +3 +4 +5 +6 +7 +m +R +a +60 +62 +64 +66 +68 +70 +0 +50 +n +60 +65 +70 +R +n +1 +8 +15 +22 +29 +36 +2 +3 +4 +5 +6 +7 +R +b +80 +100 +120 +140 +0 +250 +n +100 +150 +R +n +1 +9 +17 +25 +33 +41 +2 +3 +4 +5 +6 +7 +8 +9 +10 +R +c +40 +60 +80 +0 +200 +n +50 +100 +R +n +1 +7 +13 +19 +25 +31 +2 +3 +4 +5 +6 +7 +8 +9 +10 +R +d +340 +350 +360 +370 +380 +390 +0 +250 +n +350 +375 +R +n +1 +28 +55 +82 +109 +136 +2 +3 +4 +5 +6 +7 +8 +9 +10 +m +e +190 +200 +210 +220 +0 +200 +n +210 +220 +230 +R +n +1 +47 +93 +139 +185 +231 +277 +2 +4 +6 +8 +10 +12 +f +100 +150 +200 +250 +0 +500 +n +100 +200 +300 +R +n +1 +48 +95 +142 +189 +236 +2 +4 +6 +8 +10 +12 +14 +16 +g +100 +125 +150 +175 +200 +225 +0 +200 +n +150 +200 +R +n +1 +20 +39 +58 +77 +96 +2 +6 +10 +14 +18 +22 +h +50 +100 +150 +200 +250 +0 +400 800 +n +100 +200 +300 +R +n +1 +183 +365 +547 +729 +911 +1093 +2 +6 +10 +14 +18 +22 +m +i +900 +1000 +1100 +1200 +1300 +1400 +0 +1000 +n +1000 +1250 +R +n +1 +6 +11 +16 +21 +26 +31 +2 +7 +12 +17 +22 +27 +j +40 +50 +60 +70 +80 +0 +200 +n +40 +60 +80 +R +n +1 +5 +9 +13 +17 +21 +25 +2 +8 +14 +20 +26 +32 +k +40 +50 +60 +0 +100 +n +40 +60 +R +n +1 +4 +7 +10 +13 +16 +19 +2 +16 +30 +44 +58 +72 +l +100 +150 +200 +0 +1000 +n +100 +200 +300 +R +n +1 +8 +15 +22 +29 +36 +k +2 +16 +30 +44 +58 +72 +m +m +500 +600 +700 +800 +900 +0 +1000 +n +400 +600 +800 +R +n +S-rank +R-rank +1 +199 +397 +595 +793 +991 +k +2 +18 +34 +50 +66 +82 +98 +n +240 +250 +260 +270 +280 +0 +200 +n +240 +260 +280 +R +n +S-rank +R-rank +1 +11 +21 +31 +41 +51 +k +2 +20 +38 +56 +74 +92 +o +200 +250 +300 +350 +0 +250 +n +200 +300 +R +n +S-rank +R-rank +1 +15 +29 +43 +57 +71 +k +2 +40 +78 +116 +154 +192 +230 +p +350 +400 +450 +500 +550 +0 +500 +n +300 +400 +500 +R +n +S-rank +R-rank +Supplementary Figure 12: Higher-order non-linear contagion process - SIR model - I. All panels show, as +a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR +process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core). The insets +represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings. All results are +obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set +which is the result of 10 simulations). The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), +SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), +email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and +geometry-questions (p). The (λ, ν) values considered for each data set are reported in Table III. + +15 +0 +20 +40 +58 +60 +62 +64 +66 +68 +70 +R +a +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +80 +100 +120 +140 +160 b +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +40 +60 +80 +100 c +m = 2 +m = 3 +m = 4 +m = 5 +0 +10 +20 +30 +330 +340 +350 +360 +370 +380 +390 d +m = 2 +m = 3 +m = 4 +m = 5 +0 +100 +200 +205 +210 +215 +220 +225 +230 +R +e +m = 2 +m = 3 +m = 4 +m = 5 +0 +200 +400 +100 +150 +200 +250 +f +m = 2 +m = 4 +m = 6 +m = 8 +0 +200 +400 +140 +160 +180 +200 +220 +g +m = 2 +m = 4 +m = 6 +m = 8 +0 +50 +100 +50 +100 +150 +200 +250 +h +m = 2 +m = 6 +m = 10 +m = 14 +0 +500 +1000 +900 +1000 +1100 +1200 +1300 +1400 +R +i +m = 2 +m = 6 +m = 10 +m = 14 +0 +10 +20 +30 +40 +50 +60 +70 +80 j +m = 2 +m = 12 +m = 17 +m = 22 +0 +10 +20 +30 +40 +50 +60 +k +m = 2 +m = 4 +m = 6 +m = 8 +5 +10 +15 +100 +150 +200 +250 l +m = 2 +m = 15 +m = 41 +m = 54 +0 +20 +40 +k +400 +500 +600 +700 +800 +900 +R +m +m = 2 +m = 10 +m = 18 +m = 32 +0 +500 +1000 +k +240 +250 +260 +270 +280 +290 n +m = 2 +m = 18 +m = 34 +m = 50 +0 +20 +40 +60 +k +200 +250 +300 +350 +o +m = 2 +m = 8 +m = 14 +m = 20 +0 +25 +50 +75 +k +300 +350 +400 +450 +500 +550 p +m = 2 +m = 15 +m = 28 +m = 41 +Supplementary Figure 13: Higher-order non-linear contagion process - SIR model - II. In all panels the +average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the +(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m. All +results are obtained by averaging the results of 300 numerical simulations for each seed (except for the +congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a), +Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), +senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), +algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered for each data set are reported in +Table III. + +16 +2 +4 +6 +58 +60 +62 +64 +66 +68 +70 +R +a +k = 1 +k = 10 +k = 20 +k = 30 +2 +4 +6 +80 +90 +100 +110 +120 +130 +b +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +40 +60 +80 +100 c +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +330 +340 +350 +360 +370 +380 +390 d +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +190 +200 +210 +220 +R +e +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +80 +100 +120 +140 +160 +180 +200 +f +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +15 +100 +120 +140 +160 +180 +200 +220 +g +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +50 +100 +150 +200 +250 +h +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +850 +900 +950 +1000 +1050 +1100 +1150 +R +i +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +40 +50 +60 +70 +j +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +30 +40 +50 +60 +k +k = 1 +k = 7 +k = 14 +k = 21 +0 +25 +50 +75 +100 +150 +200 +250 +l +k = 1 +k = 6 +k = 12 +k = 18 +0 +25 +50 +75 +m +400 +500 +600 +700 +800 +900 +R +m +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +240 +250 +260 +270 +n +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +200 +250 +300 +350 +o +k = 1 +k = 10 +k = 20 +k = 30 +0 +100 +200 +m +300 +350 +400 +450 +500 +550 +p +k = 1 +k = 10 +k = 20 +k = 30 +Supplementary Figure 14: Higher-order non-linear contagion process - SIR model - III. In all panels the +average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the +(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k. All +results are obtained by averaging the results of 300 numerical simulations for each seed (except for the +congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a), +Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), +senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), +algebra-questions (o) and geometry-questions (p). The (λ, ν) values considered for each data set are reported in +Table III. + +17 +Supplementary Section 4. +Threshold higher-order contagion process +Here we consider another spreading process in which multi-body interactions drive the infection through a thresh- +old effect and group contagion [2, 4]: the threshold higher-order contagion process. We consider both the SIR and +SIS epidemic models on static hypergraphs: for each hyperedge of size m in which i individuals are in the state I, +if the fraction of infected individuals i/m is larger or equal to a threshold θ, i.e. if i ≥ ⌈θm⌉, a group infection is +activated at rate λ in which the susceptible nodes in the hyperedge become all infected. Note that if we consider a +single seed of infection: for θ ≤ 1/M the group infection is activated in all the hyperedges containing the seed; for +θ = 1/m the spreading is activated only in the hyperedges containing the seed that have size larger or equal to m; +for θ > 1/2 the spreading is inhibited since more than one infected node is required to activate the infection in all +hyperedges. I individuals recover independently at constant rate µ, becoming either S (SIS model) or R (SIR). +We perform numerical simulations of this process, for both SIS and SIR models, on empirical hypergraphs: the +simulation procedures are analogous to those described in the main text for the higher-order non-linear contagion +process (see Methods), since the two processes only differ in the infection mechanism. In the threshold higher-order +contagion, for each time-step ∆t, given a hyperedge of size m containing i infected nodes, if i ≥ ⌈θm⌉ a group +infection process is activated with probability λ and all susceptible nodes in the hyperedge are infected. Thus, in +each time-step each of the interaction groups respecting the condition i ≥ ⌈θm⌉ produces a group infection process +with probability λ. +Therefore, also in this case we quantify the ”spreading power” of each node considered separately as seed for the +SIR model and the nodes on which the epidemic is mainly localized in the steady state, i.e. the nodes that drive +and sustain the process, for the SIS model. In Figs. 15-20 are shown the results of these simulations. +data set +θ +λ +LH10 +0.03 +0.005 +Thiers13 +1/7 +0.001 +InVS15 +1/10 +0.001 +SFHH +1/10 +0.001 +LyonSchool +0.15 +0.001 +Mid1 +0.03 +0.001 +Elem1 +0.03 +0.001 +email-EU +0.03 +0.001 +data set +θ +λ +congress-bills +0.03 +0.001 +senate-committees +0.03 +0.01 +email-Enron +1/37 +0.01 +house-committees +0.03 +0.01 +music-review +0.03 +0.01 +senate-bills +1/99 +0.0001 +algebra-questions +1/107 +0.001 +geometry-questions +0.03 +0.001 +Supplementary Table IV: Parameters for Figs. 15-17. The tables summarize the parameters of the threshold +higher-order SIS contagion process considered for each data set in Figs. 15-17. +data set +θ +λ +LH10 +0.03 +0.01 +Thiers13 +1/7 +0.001 +InVS15 +1/10 +0.01 +SFHH +0.03 +0.01 +LyonSchool +0.15 +0.01 +Mid1 +1/13 +0.001 +Elem1 +0.03 +0.001 +email-EU +0.03 +0.001 +data set +θ +λ +congress-bills +0.03 +0.001 +senate-committees +0.15 +0.01 +email-Enron +0.3 +0.01 +house-committees +1/82 +0.01 +music-review +1/83 +0.01 +senate-bills +0.3 +0.001 +algebra-questions +1/107 +0.001 +geometry-questions +1/230 +0.001 +Supplementary Table V: Parameters for Figs. 18-20. The tables summarize the parameters of the threshold +higher-order SIR contagion process considered for each data set in Figs. 18-20. + +18 +1 +9 +17 +25 +33 +41 +49 +2 +3 +4 +5 +6 +7 +m +/T +a +0.75 +0.80 +0.85 +0 +50 +n +0.8 +0.9 +/T n +1 +8 +15 +22 +29 +36 +2 +3 +4 +5 +6 +7 +/T +b +0.04 +0.06 +0.08 +0 +250 +n +0.05 +0.10 +/T n +1 +9 +17 +25 +33 +41 +2 +3 +4 +5 +6 +7 +8 +9 +10 +/T +c +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 +0 +200 +n +0.05 +0.10 +0.15 +/T n +1 +7 +13 +19 +25 +31 +2 +3 +4 +5 +6 +7 +8 +9 +10 +/T +d +0.10 +0.15 +0.20 +0 +250 +n +0.1 +0.2 +/T n +1 +28 +55 +82 +109 +136 +2 +3 +4 +5 +6 +7 +8 +9 +10 +m +e +0.45 +0.50 +0.55 +0.60 +0.65 +0.70 +0 +200 +n +0.6 +0.7 +/T n +1 +47 +93 +139 +185 +231 +277 +2 +4 +6 +8 +10 +12 +f +0.74 +0.76 +0.78 +0.80 +0.82 +0 +500 +n +0.75 +0.80 +/T n +1 +48 +95 +142 +189 +236 +2 +4 +6 +8 +10 +12 +14 +16 +g +0.65 +0.70 +0.75 +0.80 +0 +200 +n +0.7 +0.8 +/T n +1 +20 +39 +58 +77 +96 +2 +6 +10 +14 +18 +22 +h +0.3 +0.4 +0.5 +0.6 +0.7 +0 +400 800 +n +0.4 +0.6 +/T n +1 +183 +365 +547 +729 +911 +1093 +2 +6 +10 +14 +18 +22 +m +i +0.65 +0.70 +0.75 +0.80 +0.85 +0 +1000 +n +0.7 +0.8 +0.9 +/T n +1 +6 +11 +16 +21 +26 +31 +2 +7 +12 +17 +22 +27 +j +0.55 +0.60 +0.65 +0.70 +0.75 +0 +200 +n +0.6 +0.7 +/T n +1 +5 +9 +13 +17 +21 +25 +2 +8 +14 +20 +26 +32 +k +0.65 +0.70 +0.75 +0.80 +0 +100 +n +0.6 +0.7 +0.8 +/T n +1 +4 +7 +10 +13 +16 +19 +2 +16 +30 +44 +58 +72 +l +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +0 +1000 +n +0.4 +0.6 +/T n +1 +8 +15 +22 +29 +36 +k +2 +16 +30 +44 +58 +72 +m +m +0.4 +0.5 +0.6 +0.7 +0.8 +0 +1000 +n +0.4 +0.6 +0.8 +/T n +S-rank +R-rank +1 +199 +397 +595 +793 +991 +k +2 +18 +34 +50 +66 +82 +98 +n +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +0 +200 +n +0.4 +0.6 +/T n +S-rank +R-rank +1 +11 +21 +31 +41 +51 +k +2 +20 +38 +56 +74 +92 +o +0.1 +0.2 +0.3 +0.4 +0 +250 +n +0.2 +0.4 +/T n +S-rank +R-rank +1 +15 +29 +43 +57 +71 +k +2 +40 +78 +116 +154 +192 +230 +p +0.2 +0.3 +0.4 +0.5 +0 +500 +n +0.2 +0.4 +/T n +S-rank +R-rank +Supplementary Figure 15: Threshold higher-order contagion process - SIS model - I. All panels give, as a +heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state +averaged over the nodes of the (k, m)-hyper-core. The insets represent τ/T averaged over the first n nodes +according to the coreness rankings as a function of n. All results are obtained by averaging the results of 103 +numerical simulations, with a single random seed of infection and with an observation window T = 103. The +following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), +Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), +music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p). The values (λ,θ) values +considered for each data set are summarized in Table IV. + +19 +0 +20 +40 +0.75 +0.80 +0.85 +0.90 +/T +a +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 b +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 c +m = 2 +m = 3 +m = 4 +m = 5 +0 +10 +20 +30 +0.10 +0.15 +0.20 +d +m = 2 +m = 3 +m = 4 +m = 5 +0 +100 +200 +0.575 +0.600 +0.625 +0.650 +0.675 +0.700 +/T +e +m = 2 +m = 3 +m = 4 +m = 5 +0 +200 +400 +0.74 +0.76 +0.78 +0.80 +0.82 +f +m = 2 +m = 4 +m = 6 +m = 8 +0 +200 +400 +0.650 +0.675 +0.700 +0.725 +0.750 +0.775 +0.800 g +m = 2 +m = 4 +m = 6 +m = 8 +0 +50 +100 +0.3 +0.4 +0.5 +0.6 +0.7 h +m = 2 +m = 6 +m = 10 +m = 14 +0 +500 +1000 +0.65 +0.70 +0.75 +0.80 +0.85 +0.90 +/T +i +m = 2 +m = 6 +m = 10 +m = 14 +0 +10 +20 +30 +0.55 +0.60 +0.65 +0.70 +0.75 +j +m = 2 +m = 12 +m = 17 +m = 22 +0 +10 +20 +0.65 +0.70 +0.75 +0.80 k +m = 2 +m = 4 +m = 6 +m = 8 +5 +10 +15 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 l +m = 2 +m = 15 +m = 41 +m = 54 +0 +20 +40 +k +0.4 +0.5 +0.6 +0.7 +0.8 +/T +m +m = 2 +m = 10 +m = 18 +m = 32 +0 +500 +1000 +k +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 n +m = 2 +m = 18 +m = 34 +m = 50 +0 +20 +40 +60 +k +0.1 +0.2 +0.3 +0.4 +0.5 o +m = 2 +m = 8 +m = 14 +m = 20 +0 +25 +50 +75 +k +0.1 +0.2 +0.3 +0.4 +p +m = 2 +m = 15 +m = 28 +m = 41 +Supplementary Figure 16: Threshold higher-order contagion process - SIS model - II. In all panels the +average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core +is shown as a function of k at fixed values of m. All results are obtained by averaging the results of 103 numerical +simulations, with a single random seed of infection and with an observation window T = 103. The following data +sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), +email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), +senate-bills (n), algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each +data set are summarized in Table IV. + +20 +2 +4 +6 +0.75 +0.80 +0.85 +/T +a +k = 1 +k = 10 +k = 20 +k = 30 +2 +4 +6 +0.04 +0.05 +0.06 +b +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +0.050 +0.075 +0.100 +0.125 +0.150 +0.175 c +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +0.05 +0.10 +0.15 +0.20 +d +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +0.45 +0.50 +0.55 +0.60 +0.65 +/T +e +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +0.74 +0.76 +0.78 +0.80 f +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +15 +0.65 +0.70 +0.75 +g +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +0.3 +0.4 +0.5 +0.6 +h +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +0.64 +0.66 +0.68 +0.70 +0.72 +0.74 +0.76 +/T +i +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +0.55 +0.60 +0.65 +0.70 +0.75 +j +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +0.60 +0.65 +0.70 +0.75 +0.80 k +k = 1 +k = 7 +k = 14 +k = 21 +0 +25 +50 +75 +0.40 +0.45 +0.50 +0.55 +0.60 +0.65 +l +k = 1 +k = 6 +k = 12 +k = 18 +0 +25 +50 +75 +m +0.4 +0.5 +0.6 +0.7 +0.8 +/T +m +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +n +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +0.1 +0.2 +0.3 +0.4 +o +k = 1 +k = 10 +k = 20 +k = 30 +0 +100 +200 +m +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +p +k = 1 +k = 10 +k = 20 +k = 30 +Supplementary Figure 17: Threshold higher-order contagion process - SIS model - III. In all panels the +average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core +is shown as a function of m at fixed values of k. All results are obtained by averaging the results of 103 numerical +simulations, with a single random seed of infection and with an observation window T = 103. The following data +sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), +email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), +senate-bills (n), algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each +data set are summarized in Table IV. + +21 +1 +9 +17 +25 +33 +41 +49 +2 +3 +4 +5 +6 +7 +m +R +a +50 +55 +60 +0 +50 +n +50 +60 +R +n +1 +8 +15 +22 +29 +36 +2 +3 +4 +5 +6 +7 +R +b +8 +10 +12 +0 +250 +n +7.5 +10.0 +12.5 +R +n +1 +9 +17 +25 +33 +41 +2 +3 +4 +5 +6 +7 +8 +9 +10 +R +c +160 +170 +180 +190 +0 +200 +n +160 +180 +R +n +1 +7 +13 +19 +25 +31 +2 +3 +4 +5 +6 +7 +8 +9 +10 +R +d +280 +300 +320 +340 +0 +250 +n +300 +350 +R +n +1 +28 +55 +82 +109 +136 +2 +3 +4 +5 +6 +7 +8 +9 +10 +m +e +220 +225 +230 +235 +240 +0 +200 +n +235 +240 +R +n +1 +47 +93 +139 +185 +231 +277 +2 +4 +6 +8 +10 +12 +f +480 +500 +520 +0 +500 +n +475 +500 +525 +R +n +1 +48 +95 +142 +189 +236 +2 +4 +6 +8 +10 +12 +14 +16 +g +220 +240 +260 +0 +200 +n +225 +250 +275 +R +n +1 +20 +39 +58 +77 +96 +2 +6 +10 +14 +18 +22 +h +200 +250 +300 +350 +0 +400 800 +n +200 +400 +R +n +1 +183 +365 +547 +729 +911 +1093 +2 +6 +10 +14 +18 +22 +m +i +1200 +1300 +1400 +1500 +1600 +0 +1000 +n +1200 +1400 +1600 +R +n +1 +6 +11 +16 +21 +26 +31 +2 +7 +12 +17 +22 +27 +j +10 +15 +20 +25 +0 +200 +n +20 +40 +R +n +1 +5 +9 +13 +17 +21 +25 +2 +8 +14 +20 +26 +32 +k +50 +55 +60 +65 +70 +0 +100 +n +60 +80 +R +n +1 +4 +7 +10 +13 +16 +19 +2 +16 +30 +44 +58 +72 +l +500 +600 +700 +800 +0 +1000 +n +600 +800 +R +n +1 +8 +15 +22 +29 +36 +k +2 +16 +30 +44 +58 +72 +m +m +400 +500 +600 +700 +800 +0 +1000 +n +400 +600 +800 +R +n +S-rank +R-rank +1 +199 +397 +595 +793 +991 +k +2 +18 +34 +50 +66 +82 +98 +n +40 +50 +60 +70 +80 +0 +200 +n +40 +60 +80 +R +n +S-rank +R-rank +1 +11 +21 +31 +41 +51 +k +2 +20 +38 +56 +74 +92 +o +20 +30 +40 +50 +60 +70 +0 +250 +n +25 +50 +75 +R +n +S-rank +R-rank +1 +15 +29 +43 +57 +71 +k +2 +40 +78 +116 +154 +192 +230 +p +50 +75 +100 +125 +150 +0 +500 +n +50 +100 +150 +R +n +S-rank +R-rank +Supplementary Figure 18: Threshold higher-order contagion process - SIR model - I. All panels show, as +a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR +process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core). The insets +represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings. All results are +obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set +which is the result of 10 simulations). The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), +SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), +email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and +geometry-questions (p). The values (λ,θ) values considered for each data set are summarized in Table V. + +22 +0 +20 +40 +50.0 +52.5 +55.0 +57.5 +60.0 +62.5 +65.0 +R +a +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +6 +8 +10 +12 +b +m = 2 +m = 3 +m = 4 +m = 5 +0 +20 +40 +160 +170 +180 +190 +c +m = 2 +m = 3 +m = 4 +m = 5 +0 +10 +20 +30 +280 +300 +320 +340 +d +m = 2 +m = 3 +m = 4 +m = 5 +0 +100 +200 +232 +234 +236 +238 +240 +R +e +m = 2 +m = 3 +m = 4 +m = 5 +0 +200 +400 +480 +500 +520 +f +m = 2 +m = 4 +m = 6 +m = 8 +0 +200 +400 +220 +230 +240 +250 +260 +270 +280 g +m = 2 +m = 4 +m = 6 +m = 8 +0 +50 +100 +150 +200 +250 +300 +350 +400 h +m = 2 +m = 6 +m = 10 +m = 14 +0 +500 +1000 +1200 +1300 +1400 +1500 +1600 +R +i +m = 2 +m = 6 +m = 10 +m = 14 +0 +10 +20 +30 +10 +15 +20 +25 +j +m = 2 +m = 12 +m = 17 +m = 22 +0 +10 +20 +50 +55 +60 +65 +70 +75 k +m = 2 +m = 4 +m = 6 +m = 8 +5 +10 +15 +500 +600 +700 +800 +l +m = 2 +m = 15 +m = 41 +m = 54 +0 +20 +40 +k +400 +500 +600 +700 +800 +R +m +m = 2 +m = 10 +m = 18 +m = 32 +0 +500 +1000 +k +40 +50 +60 +70 +80 n +m = 2 +m = 18 +m = 34 +m = 50 +0 +20 +40 +60 +k +20 +30 +40 +50 +60 +70 o +m = 2 +m = 8 +m = 14 +m = 20 +0 +25 +50 +75 +k +40 +60 +80 +100 +120 +140 p +m = 2 +m = 15 +m = 28 +m = 41 +Supplementary Figure 19: Threshold higher-order contagion process - SIR model - II. In all panels the +average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the +(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m. All +results are obtained by averaging the results of 300 numerical simulations for each seed (except for the +congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a), +Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), +senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), +algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each data set are +summarized in Table V. + +23 +2 +4 +6 +50.0 +52.5 +55.0 +57.5 +60.0 +62.5 +R +a +k = 1 +k = 10 +k = 20 +k = 30 +2 +4 +6 +6 +7 +8 +9 +10 +b +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +160 +170 +180 +190 +c +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +280 +300 +320 +340 +d +k = 1 +k = 10 +k = 20 +k = 30 +2.5 +5.0 +7.5 +10.0 +222.5 +225.0 +227.5 +230.0 +232.5 +235.0 +237.5 +R +e +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +470 +480 +490 +500 +510 +f +k = 1 +k = 10 +k = 20 +k = 30 +5 +10 +15 +210 +220 +230 +240 +250 +260 +270 +g +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +150 +200 +250 +300 +350 +h +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +1150 +1200 +1250 +1300 +1350 +R +i +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +10 +15 +20 +25 +j +k = 1 +k = 10 +k = 20 +k = 30 +10 +20 +30 +50 +55 +60 +65 +70 +k +k = 1 +k = 7 +k = 14 +k = 21 +0 +25 +50 +75 +500 +600 +700 +800 +l +k = 1 +k = 6 +k = 12 +k = 18 +0 +25 +50 +75 +m +400 +500 +600 +700 +800 +R +m +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +37.5 +40.0 +42.5 +45.0 +47.5 +50.0 +n +k = 1 +k = 10 +k = 20 +k = 30 +0 +50 +100 +m +20 +30 +40 +50 +60 +o +k = 1 +k = 10 +k = 20 +k = 30 +0 +100 +200 +m +40 +60 +80 +100 +p +k = 1 +k = 10 +k = 20 +k = 30 +Supplementary Figure 20: Threshold higher-order contagion process - SIR model - III. In all panels the +average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the +(k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k. All +results are obtained by averaging the results of 300 numerical simulations for each seed (except for the +congress-bills data set which is the result of 10 simulations). The following data sets are considered: LH10 (a), +Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), +senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), +algebra-questions (o) and geometry-questions (p). The values (λ,θ) values considered for each data set are +summarized in Table V. + +24 +Supplementary Section 5. +Higher-order naming-game (NG) process +data set +β +p +tmax +T +InVS15 +0.41 +1.8 × 10−2 +105 +104 +Mid1 +0.59 +1.5 × 10−2 +105 +104 +email-EU +0.52 +9.2 × 10−3 +5 × 105 +5 × 104 +congress-bills +0.59 +2.4 × 10−2 +5 × 105 +5 × 104 +house-committees +0.55 +2.5 × 10−2 +5 × 105 +5 × 104 +music-review +0.52 +9.0 × 10−3 +105 +104 +Supplementary Table VI: Parameters for Fig. 21 - Union rule. The table summarizes the main parameters of +the higher-order naming-game process considered for the temporal dynamics of Fig. 21 in the various data sets +(union rule). +data set +Arandom +As−core +Ahyper−core +InVS15 +24.3% +54.8% +54.8% +Mid1 +14.5% +25.5% +23.7% +email-EU +37.0% +45.9% +56.4% +congress-bills +40.8% +47.1% +49.7% +house-committees +63.0% +64.0% +64.6% +music-review +51.6% +55.8% +59.5% +Supplementary Table VII: Minority takeover areas for Fig. 21 - Union rule. The table reports the area Ax +of the explored parameter space in which the minority take-over, i.e. n∗ +A = 1, takes place for the different data sets +of Fig. 21 (union rule) and for the different strategies of committed seeding x ∈ {random, s − core, hyper − core}. +data set +β +p +tmax +T +InVS15 +0.38 +1.4 × 10−2 +105 +104 +Mid1 +0.38 +2.7 × 10−2 +105 +104 +email-EU +0.41 +1.7 × 10−2 +5 × 105 +5 × 104 +congress-bills +0.48 +2.3 × 10−2 +5 × 105 +5 × 104 +house-committees +0.41 +3.0 × 10−3 +5 × 105 +5 × 104 +music-review +0.52 +1.0 × 10−2 +105 +104 +Supplementary Table VIII: Parameters for Fig. 22 - Unanimity rule. The table summarizes the main +parameters of the higher-order naming-game process considered for the temporal dynamics of Fig. 22 in the +various data sets (unanimity rule). + +25 +data set +Arandom +As−core +Ahyper−core +InVS15 +8.6% +35.2% +32.9% +Mid1 +1.0% +2.5% +2.5% +email-EU +7.3% +8.6% +14.7% +congress-bills +7.9% +16.3% +41.5% +house-committees +54.6% +63.3% +64.4% +music-review +33.9% +56.6% +62.6% +Supplementary Table IX: Minority takeover areas for Fig. 22 - Unanimity rule. The table reports the area +Ax of the explored parameter space in which the minority take-over, i.e. n∗ +A = 1, takes place for the different data +sets of Fig. 22 (unanimity rule) and for the different strategies of committed seeding +x ∈ {random, s − core, hyper − core}. + +26 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +p +a +10 +2 +InVS15 +Random +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +b +10 +2 +s-core +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +c +10 +2 +n * +A +hyper-core +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +d +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +p +e +10 +2 +Mid1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +f +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +g +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +h +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +i +10 +2 +email-EU +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +j +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +k +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +l +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +m +10 +2 +congress-bills +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +n +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +o +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +p +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +q +10 +2 +house-committees +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +r +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +s +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +t +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +u +10 +2 +music-review +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +v +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +w +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +x +Random +R-rank +s-rank +Supplementary Figure 21: +Higher-order NG process - Union rule. In the first three panels of each row the +stationary fraction n∗ +A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p +and the agreement probability β through a heat-map. We consider the union rule and the following data sets: InVS15 (first +row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews +(sixth row). For each row, the committed nodes are selected through: the random seeding strategy in the first panel; the +top s-coreness seeding strategy in the second panel; the top hyper-coreness seeding strategy in the third panel. In the +fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VI (cross +markers in the heatmaps). The minority take-over, i.e. n∗ +A = 1, takes place over an area A of the explored parameter space: +in Table VII we report its value for the different strategies and data sets considered. All simulations are run until the +absorbing state with n∗ +A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗ +A is +obtained by averaging over 100 values sampled in the last T time-steps (see Table VI for the tmax and T values for each +data set). The results refer to the median values obtained over 200 simulations. + +27 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +p +a +10 +2 +InVS15 +Random +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +b +10 +2 +s-core +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +3.2 +c +10 +2 +n * +A +hyper-core +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +d +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +p +e +10 +2 +Mid1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +f +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +2.6 +2.9 +g +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +h +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +i +10 +2 +email-EU +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +j +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +k +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +l +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +m +10 +2 +congress-bills +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +n +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +o +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +p +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +q +10 +2 +house-committees +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +r +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +s +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +t +Random +R-rank +s-rank +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +p +u +10 +2 +music-review +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +v +10 +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.1 +0.4 +0.7 +1.0 +1.3 +1.6 +1.9 +2.2 +2.5 +2.8 +w +10 +2 +n * +A +1/N +0.2 +0.4 +0.6 +0.8 +1.0 +102 +103 +104 +105 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +nA(t) +x +Random +R-rank +s-rank +Supplementary Figure 22: +Higher-order NG process - Unanimity rule. In the first three panels of each row the +fraction n∗ +A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the +agreement probability β through a heat-map. We consider the unanimity rule and the following data sets: InVS15 (first +row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews +(sixth row). For each row, the committed nodes are selected through: the random seeding strategy in the first panel; the +top s-coreness seeding strategy in the second panel; the top hyper-coreness seeding strategy in the third panel. In the +fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VIII (cross +markers in the heatmaps). The minority take-over, i.e. n∗ +A = 1, takes place over an area A of the explored parameter space: +in Table IX we report its value for the different strategies and data sets considered. All simulations are run until the +absorbing state with n∗ +A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗ +A is +obtained by averaging over 100 values sampled in the last T time-steps (see Table VIII for the tmax and T values for each +data set). The results refer to the median values obtained over 200 simulations. + +28 +[1] St-Onge, G. et al. +Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs. +Communications Physics 5, 25 (2022). +[2] de Arruda, G. F., Petri, G., Rodriguez, P. M. & Moreno, Y. Multistability, intermittency and hybrid transitions in social +contagion models on hypergraphs. arXiv preprint - arXiv:2112.04273 (2021). +[3] Iacopini, I., Petri, G., Baronchelli, A. & Barrat, A. +Group interactions modulate critical mass dynamics in social +convention. Communications Physics 5, 64 (2022). +[4] de Arruda, G. F., Petri, G. & Moreno, Y. Social contagion models on hypergraphs. Phys. Rev. Research 2, 023032 (2020). + diff --git a/FNE2T4oBgHgl3EQf-Anw/content/tmp_files/load_file.txt b/FNE2T4oBgHgl3EQf-Anw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d723a42f3c3792505117818cf6306bb518e687a --- /dev/null +++ b/FNE2T4oBgHgl3EQf-Anw/content/tmp_files/load_file.txt @@ -0,0 +1,5505 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf,len=5504 +page_content='Hyper-cores promote localization and efficient seeding in higher-order processes Marco Mancastroppa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 Iacopo Iacopini,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 Giovanni Petri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 and Alain Barrat1 1Aix Marseille Univ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Universit´e de Toulon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' CPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Turing Center for Living Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' France 2Department of Network and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Central European University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 1100 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Austria 3CENTAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Corso Inghilterra 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 10138 Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Italy Going beyond networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' in order to include higher-order interactions involving groups of elements of arbitrary sizes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' has been recognized as a major step in reaching a better description of many complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the resulting hypergraph representation, tools to identify particularly cohesive structures and central nodes are still scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We propose here to decompose a hypergraph in hyper- cores, defined as subsets of nodes connected by at least a certain number of hyperedges (groups of nodes) of at least a certain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We illustrate this procedure on empirical data sets described by hypergraphs, showing how this suggests a novel notion of centrality for nodes in hypergraphs, the hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We assess the role of the hyper-cores and of nodes with large hyper-coreness values in several dynamical processes based on higher-order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We show that such nodes have large spreading power and that spreading processes are localized in hyper-cores with large connectedness along groups of large sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the emergence of social conventions moreover, very few committed individuals with high hyper-coreness can rapidly overturn a majority convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Our work opens multiple research avenues, from fingerprinting and comparing empirical data sets to model validation and study of temporally varying hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' INTRODUCTION Network theory provides a powerful framework for de- scribing a wide range of complex systems composed of elements interacting in pairs [1–4]: over the years, this theory has developed numerous concepts and techniques to characterize the structure of complex networks at var- ious scales, from the single element (node or link) to groups of nodes to the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Moreover, net- works are the support of dynamical processes of various types, from spreading to synchronization phenomena [3], thus understanding how network’s features impact such processes, or which parts of a network play the most im- portant role, is of primordial relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Several concepts and results in this respect are now well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For instance, hubs, nodes with a very large number of con- nections (degree), are known to influence processes such as spreading or opinion dynamics, because of their ten- dency to be reached easily and their ability to transmit to many other nodes [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The statistics of the individual number of connections of nodes are however not a rich enough characterization: the existence of well-connected groups of nodes might indeed be even more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In this direction, the tendency of hubs –observed in real- world networks– to be connected to each other far above chance is quantified by the rich-club coefficient [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' A more systematic way to decompose a network into a hi- erarchy of subgraphs of increasing connectedness is given instead by the k-core decomposition [6–9]: the k-core of a network is by definition the maximal subgraph such that all its nodes have degree (number of neighbours in the subgraph) at least k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This decomposition provides a fingerprint of the network’s structure [8, 10, 11] and grad- ually focuses on more central and densely interconnected parts of the network that were shown to play a crucial role in spreading processes [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In fact, the coreness of a node, defined as the largest value of k such that the node belongs to the corresponding k-core, gives an alter- native measure of centrality and largely determines the impact of a spreading process initiated (seeded) in that node [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Given the relevance of this decomposition, it has also been extended to weighted networks [15], via the s-core decomposition [16] (s representing the strength of a node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', the sum of the weights of its adjacent links), to temporally evolving networks [17, 18], and to multi- layer networks [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Despite their convenience, network representations are limited to systems composed of only binary interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' However, over the last few years, it has become clear that many real systems include interactions between groups of units [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Examples range from group conver- sations among friends [22] to research teams and co- authorship of scientific articles [23], from neural systems [24] to interactions between multiple species in ecologi- cal ones [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Analogously, considering a purely dyadic network substrate for the unfolding of processes such as consensus formation or (social) contagion could put a limit on the ability to describe key mechanisms that are at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For instance, reinforcement mechanisms – in which two or more people can convince others in a group conversation– cannot be naturally accounted for when considering only dyadic interactions [26–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In these cases, systems and processes can be effectively rep- resented within the framework of hypergraphs, a “higher- order” generalization of networks in which nodes can in- teract in hyperedges, groups of arbitrary size [21, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Higher-order interactions give rise to novel structures [32– 34] and phenomena [20, 35], highlighting the need for new characterization tools able to detect hierarchies and rele- vant subparts of all these systems that are better repre- sented by hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Here, we contribute to this endeavour by proposing the decomposition of a hypergraph in (k, m)-hyper-cores, which we define as a series of subhypergraphs of increas- ing connectivity k, ensured by hyperedges of increasing sizes m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We apply this decomposition to a wide range of data sets, representing systems of different nature, iden- tifying non-trivial mesoscopic higher-order structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In so doing, we put forward the hyper-coreness, a new cen- trality measure for nodes in hypergraphs based on their inclusion in the hyper-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Finally, and crucially, we investigate the role of the newly defined hyper-cores arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='04235v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='soc-ph] 10 Jan 2023 2 and of the nodes with largest hyper-coreness in spread- ing and consensus processes based on group interactions [28, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We show that spreading processes tend to be localized on hyper-cores associated to large k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We then study the performance of hyper-coreness- based strategies as opposed to both random and strength- based ones [16] when it comes to identifying influential nodes that sustain and drive higher-order processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We find that hyper-coreness can be effectively used to max- imise the total outbreak size in non-linear spreading pro- cesses [36] and help committed minorities reach the tip- ping point leading to a systemic takeover in social con- vention games [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hyper-core decomposition and hyper-coreness We define the hyper-cores, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the higher-order cores of a hypergraph, through a systematic decomposition of a hypergraph in a double hierarchy of nested sub- hypergraphs of increasing connectedness and hyperedge sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Let us consider a (static) hypergraph H = (V, E), where V is the set of its N = |V| nodes and E is the set of its hyperedges [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We recall that a hyperedge e = {i1, i2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', im} is a set of m nodes, which can thus represent a group interaction between these nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We denote by M = maxe∈E |e| the largest hyperedge size in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Each node i ∈ V can be characterized by a vector of degrees d(i) = [d2(i), d3(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', dm(i), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', dM(i)] whose component dm(i) denotes the m-hyper-degree of the node i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', the number of distinct hyperedges of size m to which it belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We denote by Dm(i) = � p≥m dp(i) the number of distinct hyperedges of size at least m to which i belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We define the (k, m)-hyper-core as the maximum sub- hypergraph J induced by the set of nodes A ⊆ V and with hyperedges of size at least m, such that ∀ i ∈ A, DJ m(i) ≥ k, where DJ m(i) denotes the number of dis- tinct hyperedges of size at least m in which i is involved within the subhypergraph J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In other terms, all the nodes in the (k, m)-hyper-core belong to at least k hyperedges of size at least m, within the hyper-core itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The set of hyperedges of the subhypergraph J , induced by the set A ⊆ V, is defined by S = {e ∩ A|e ∈ E ∧ |e ∩ A| ≥ m} [39] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', a hyperedge of S is a subset of a hyperedge of E, of size at least m and containing only nodes of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' To obtain the (k, m)-hyper-core of a hypergraph, one can first remove from E all hyperedges of size smaller than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' One then removes recursively from V all nodes i with Dm(i) < k, until all the nodes in the remaining subhy- pergraph are involved in at least k hyperedges of size at least m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Note that this process does not correspond only to the removal of nodes with Dm(i) < k in the original hypergraph H: indeed, each time a node is removed, the sizes of the hyperedges to which it belongs decrease by one unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Thus, the removal of a node can induce the removal of some of the hyperedges to which it belongs, if their size becomes less than m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 1 we illustrate the process on an example hypergraph and highlight some of its (k, m)-hyper-cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' As k and m increases, the (k, m)-hyper-cores progres- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Sketch of the (k, m)-hyper-core decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We show a hypergraph and highlight some of its (k, m)-hyper- cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Note the inclusions as k or m increase: the (1, 2)- hyper-core contains the (1, 3)-hyper-core, which contains the (2, 3)-hyper-core;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' similarly the (1, 2)-hyper-core contains the (2, 2)-hyper-core which contains the (2, 3)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' On the other hand, the (1, 3)-hyper-core and the (2, 2)-hyper-core share some nodes but neither is included in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The green nodes belong to the (1, 2)-hyper-core but neither to the (1, 3)- nor the (2, 2)- ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The blue nodes belong to the (1, 3)-hyper-core but are excluded from the (2, 3) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Or- ange nodes belong to the (2, 2)-hyper-core but are excluded from the (2, 3) one because they belong only to hyperedges of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (1, 4)-core and (1, 5)-core contain all the nodes involved respectively in at least one interaction with m ≥ 4 and m ≥ 5 (for simplicity these cores are not highlighted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (k, 2)-cores and (k, 3)-cores with k ≥ 3, and the (k, 4)- cores and (k, 5)-cores with k ≥ 2 are all empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Notice that the node i does not belong to the (2, 3)-core even if D3(i) = 2 because of the recursive and interaction downgrading mecha- nisms of the decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' in the (1, 3)-core and (2, 3)-core the pairwise interactions ei ∀i ∈ [1, 5] are excluded, thus the (1, 3)-core is composed of two disjoint subhypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' sively identify groups of nodes increasingly connected with each other through interactions of increasing or- der (the (k, m)-hyper-core includes the (k, m + 1)- and (k + 1, m)-hyper-cores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We define the m-shell index Cm(i) of a node i as the value of k such that i belongs to the (k, m)-hyper-core but not to the (k + 1, m)-hyper- core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (k, m)-shell S(k,m) can then be defined as the set of all nodes whose shell index Cm(i) at size m is k, and we denote by km max the maximum value of k such that the shell S(k,m) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The ratio Cm(i)/km max thus quantifies how well-connected node i is in the hy- pergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' As it is a function of m, different nodes will have different functions with potentially different func- tional shapes, which makes it difficult to compare and rank them (see the Supplemental Material, SM, for some examples of Cm(i) functional shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We thus define for each node i its hyper-coreness R(i) as: R(i) = M � m=2 g(m)Cm(i)/km max, (1) where g(m) is an arbitrary weight function, which can 3 weigh differently the various possible sizes of higher-order interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hereafter, for simplicity we will fix g(m) = 1: in this case, by definition R ∈ [0, M −1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Other choices could be considered, for example to emphasise hyperedges of larger or smaller sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The R hyper-coreness gives thus a summary of a node centrality with respect to the various (k, m)-hyper-cores, taking into account how central the node is for all orders of interactions and making it possible to rank the nodes of the hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hyper-core decomposition of empirical hypergraphs To illustrate the decomposition processes along (k, m)- hyper-cores, we rely on a number of empirical hyper- graphs, obtained from publicly available data sets, that describe a variety of systems of agents interacting in dif- ferent environments, both through online media and face- to-face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In particular, we consider data sets of face-to-face interactions provided by the SocioPatterns collaboration [40–42] and by the Contacts among Utah’s School-age Population (CUSP) project [43], collected in contexts ranging from workplaces to schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We also use data sets of email communication (email-EU, email-Enron [44– 46]) and of other types of online interactions, namely on- line reviews of products (music-review [46, 47]) or online opinion exchange on specific topics in scientific forums [46, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We moreover consider data describing commit- tees membership (house-committees, senate-committees [46, 49, 50]) and bills sponsorship (congress-bills, senate- bills [46, 49, 51, 52]) in the US Congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' These data sets cover a wide range of system sizes and have also very different interaction size distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We provide a detailed description of each data set in the Methods and in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the following, we give results on the music-review, email-EU, house-committees and congress- bills data sets while we refer to the SM for the other data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Figure 2 shows the results of the hyper-core decompo- sition on two data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The relative size n(k,m) of the (k, m)-hyper-cores exhibit distinct behaviors as a func- tion of k and m, identifying structural differences between data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In some cases, the decrease with k is rather smooth (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2a and SM), showing that most shells are populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In other cases abrupt drops and plateaus can be observed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2c and SM), corresponding to alter- natively empty and densely populated (k, m)-shells (see also SM for figures showing the sizes of the (k, m)-shells vs k and m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' These differences indicate that the (k, m)- hyper-cores could be used to provide a fingerprint of hy- pergraphs, just as the k-core decomposition provides a fingerprint of networks [8, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Also the distributions of hyper-coreness values R differ across data sets, as illus- trated in the rank-order plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2b,d and in the SM for all data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' While some data sets have an almost uniform distribution of values, others feature few nodes with high hyper-coreness and many nodes with medium hyper-coreness, or vice-versa –many nodes having low or high coreness and few with medium values (see SM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We also show in the SM some typical examples of the nor- malized m-shell index function Cm(i) as a function of m for various nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' to highlight the diversity of these func- 1 16 31 46 61 76 91 106 k 2 4 6 8 10 12 14 16 18 20 22 24 m n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) a email-EU 0 250 500 750 1000 Rank(i) 0 5 10 15 20 R(i) b 1 6 11 16 21 26 31 36 k 2 12 22 32 42 52 62 72 82 m n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) c music-review 0 250 500 750 1000 Rank(i) 0 20 40 60 80 R(i) d 10 1 100 0 50 100 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n(k, m) m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 500 1000 S(i) 0 10 20 R(i) 10 1 100 0 20 40 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n(k, m) m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 200 S(i) 0 40 80 R(i) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hyper-core decomposition of empirical hyper- graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Panels a and c show colormaps giving the relative size n(k,m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a func- tion of k and m (white regions correspond to n(k,m) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the insets n(k,m) is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In panels b and d the hyper-coreness R(i) is plotted as a function of the corresponding node rank;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the insets give scat- terplots of the hyper-coreness R(i) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the s-coreness, S(i), for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In panels a and b we consider the email-EU data set: R(i) and S(i) have a Pearson correlation coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='90 (p-value p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001) and the corresponding rank- ings have a Kendall’s τ coefficient of τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' in panels c and d we consider the music-review data set: R(i) and S(i) have a Pearson correlation coefficient of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001) and the corresponding rankings a Kendall’s τ coefficient of τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='58 (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' tions and the need to define a summary index such as the hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We finally compare in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2b,d the hyper- coreness with the centrality of nodes obtained by disre- garding the higher-order character of the interactions and projecting the hypergraph H onto a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' To this aim, we transform each hyperedge in a network clique, and each edge (i, j) of the resulting network is weighted by the number of distinct hyperedges in H involving both i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We then perform the s-core decomposition of this weighted network and assign its s-coreness S(i) to each node i [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' S(i) and R(i) are positively correlated, but they do not provide exactly the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In particular the hyper-coreness enhances the information given by the s-coreness by providing an internal hierar- chy within the nodes of maximal s-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This is evi- dent from the scatter plots, as nodes presenting the same s-coreness values correspond to values of hyper-coreness that can span across a broad range (y axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Having illustrated the relevance of the newly defined cores on empirical hypergraphs, we now move to study the role of these substructures in dynamical processes taking place on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In particular, we are go- 4 ing to investigate whether the (k, m)-hyper-cores and the hyper-coreness centrality can be used to identify nodes and structures relevant for spreading and consensus pro- cesses whose mechanisms are explicitly defined on hyper- edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Higher-order contagion processes localize in hyper-cores Networks have been widely used to describe the sub- strate on which contagion processes take place, such as the spread of a pathogen or information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In standard diffusion modeling approaches, nodes represent individ- uals that at any time can be in one of several possible states, such as S (susceptible), I (infectious) or R (re- covered);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' S nodes are typically thought to become I at rate β when they share a link with an infectious (I) in- dividual, while infected (I) nodes recover spontaneously at rate µ, either becoming again susceptible (S), in what is usually called the SIS model [53], or becoming recov- ered (R) in the so-called SIR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Recently, several models have been proposed to take into account possible higher-order mechanisms, that amount to reinforcement mechanisms affecting the contagion probability due to the simultaneous exposure to multiple sources of infections in group interactions [26, 36, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For instance, in a so- cial contagion process, the probability that an individual is convinced upon separate exposures to two “infectious” neighbours can be reinforced if these exposures occur dur- ing a group discussion featuring the three individuals al- together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Here, we show that hyper-cores play a crucial role in the dynamics of higher-order spreading processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In or- der to do this, we consider the recently proposed higher- order non-linear contagion [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In this model, each sus- ceptible node in a hyperedge of size m in which there are i infected individuals becomes infectious with rate λiν, where ν controls the non-linearity of the process (for ν = 1 the usual linear contagion is recovered, while for ν > 1 non-linearities are introduced) and λ ∈ [0, 1] (see Methods for more details on the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Infected in- dividuals (I) recover independently at constant rate µ, becoming either susceptible S (SIS model) or R (SIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The higher-order nature of contagion produces novel effects on the epidemic phenomenology, including abrupt transitions with bistability in the SIS phase diagram, in- termittent regimes [37, 54], and a mesoscopic localization of the infection on large hyperedges [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Against this background, the connectivity properties of hyper-cores we highlighted so far suggest that cores might play an even stronger role in such localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' To investigate this point, we perform numerical simula- tions of the higher-order non-linear SIS model on empiri- cal hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The system is initialized with one single seed of infection (a randomly chosen node in state I) in an otherwise fully susceptible population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We let the process evolve (see Methods for simulation details) until a steady state is reached in which the number of infectious individ- uals fluctuates (we consider parameter values such that the system remains active and the epidemic does not die out rapidly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We then consider a finite time-window T and measure for each node j the time τ(j) that it spends 1 6 11 16 21 26 31 36 k 2 12 22 32 42 52 62 72 82 m /T a music-review SIS 1 6 11 16 21 26 31 36 k 2 12 22 32 42 52 62 72 82 m R b SIR 1 3 5 7 9 1113151719 k 2 12 22 32 42 52 62 72 82 m /T c house-committees 1 3 5 7 9 1113151719 k 2 12 22 32 42 52 62 72 82 m R d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 0 500 1000 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 /T n S-rank R-rank 500 600 700 800 900 0 1000 n 400 600 800 R n S-rank R-rank 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='850 0 500 1000 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 /T n S-rank R-rank 100 125 150 175 200 225 0 500 1000 n 100 200 300 R n S-rank R-rank FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hyper-cores for seeding and localization in higher-order non-linear contagion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For the SIS model, panels a and c give the heatmap of the average fraction of time ⟨τ/T⟩ of infected nodes in the steady state as a function of k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Averages are computed over all the nodes of each (k, m)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets represent ⟨τ/T⟩ averaged over the first n nodes according to the coreness rank- ings as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For the SIR model, panels b and d show the heatmap of the average final size of the epidemic ⟨R∞⟩ as a function of k and m, where the process is seeded in a single node belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets represent ⟨R∞⟩ as a function of n averaged over the first n nodes according to coreness rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Panels a and b: music-review data set with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25, λ = 5 × 10−4 (a) and ν = 3, λ = 5 × 10−4 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Panels c and d: house- committees data set with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25, λ = 5 × 10−4 (c) and ν = 4, λ = 5 × 10−5 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' in the I state during that window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This allows to iden- tify the nodes on which the epidemic is mainly localized in the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the nodes that drive and sustain the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Figure 3 reports results of simulations performed on two data sets (the music-review and house-committees data sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Similar results are shown in the SM for the other considered data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Panels 3a and 3c show that nodes in (k, m)-hyper-cores with either increasing k or m tend to be more often infectious, as the values of τ(j)/T averaged over all nodes of each (k, m)-hyper-core increase with k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This implies that the process is more local- ized in the (k, m)-hyper-cores with large k (which favors connectedness, hence mutual reachability) and m (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', large hyperedges where large values of i can be obtained yielding large infection rates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets of the pan- els moreover show the average of τ/T over the n nodes with highest hyper-coreness R or highest s-coreness S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The nodes with highest coreness tend to be more often in the infectious state, and this tendency is stronger for the 5 hyper-coreness than for the s-coreness: among the nodes with largest value of s-coreness, the hyper-coreness al- lows to distinguish which ones are most involved in the higher-order spreading processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Moreover, in the SM we show that a similar phenomenology is obtained with a different model of contagion involving higher-order mech- anisms [37, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' High hyper-coreness seeds increase total outbreak size Nodes belonging to large interaction groups have also been shown to be optimal seeds of higher-order non-linear contagions in terms of spreading speed at the beginning of an SIS outbreak [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Which nodes have the largest spreading power in the long run, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', in terms of final size reached by a SIR process [12], remains however an open question for higher-order spreading processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We thus consider the higher-order non-linear SIR model, in which the dynamics, starting from a single seed, evolves until no individual is in the state I anymore (only nodes in states S or R remain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' To quantify the “spreading power” of each node j considered separately as seed, we average the final epidemic size R∞(j), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', the number of nodes in state R at the end of the process, over 300 stochastic runs for each seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Figure 3b and 3d show that this average final epidemic size ⟨R∞(j)⟩, averaged over all nodes of each (k, m)-hyper-core, increase with k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets also show that the nodes with higher hyper-coreness lead to larger epidemics, determining a hierarchy even among the nodes with highest s-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In summary, nodes with higher connectedness along groups of larger sizes can seed more efficiently, and the hyper-coreness provides a good identification of the nodes with highest spreading power in higher-order non-linear contagion processes (similar results are shown in the SM for another higher-order contagion model [37, 54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hypercore seeding facilitates systemic takeover by minority norms Group interactions can also play an important role in the formation of a consensus and the emergence of shared conventions in a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' According to critical mass theory, regular individuals might then benefit –towards addressing societal challenges– from the presence of a committed minority that aims at overturning the sta- tus quo [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Recently, it has been shown that groups can modulate this takeover [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' An important issue in this respect concerns the best “seeding” strategy –where should the committed minority start from in order to best achieve the takeover?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Here we show how hyper-coreness can provide an answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We consider the well-known naming-game (NG) model [38], which describes how a shared convention can emerge in a population of locally interacting agents [57, 58], and has been shown to account for the outcome of controlled experiments of social coordination [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the minimal version of this model, recently modified to take group in- teractions into account [28], individuals are represented by the N nodes of a static hypergraph, and each node is endowed with a dictionary that can contain at most two names (representing conventions or norms), A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' At each time-step a hyperedge is chosen randomly and a speaker is randomly selected within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The speaker randomly chooses a name from its dictionary and com- municates it to the other hyperedge members (the listen- ers), who can agree or not on the proposed name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' To determine the possibility of an agreement within the hy- peredge, we consider two distinct alternatives [28]: (i) the union rule, for which an agreement can be reached if at least one of the listeners has the proposed name in its dictionary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' (ii) the unanimity rule, for which the agree- ment can be reached only if all the nodes in the group have the proposed name in their dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' A parame- ter β ∈ [0, 1] modulates the social influence by controlling the propensity of the listeners to actually accept the local consensus: the agreement in the group becomes effective only with probability β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In this case, all nodes in the hy- peredge add the accepted name to their dictionary, if it was not already present, deleting all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' If instead no agreement is reached, the listeners simply add the name given by the speaker to their dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Crucially, we include in the population a committed minority of Np individuals who do not obey the aforemen- tioned rules whenever they are listeners, but they instead stick to their norm, a single name A, and their dictionary is never updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We initiate the process with the rest of the population, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the majority, having only the name B in their dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The system can evolve towards different regimes of co-existence of the two names or of dominance of one name over the other, depending on β, on the considered rule, and on the relative size of the mi- nority p = Np/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In particular, the committed minority can overcome the majority, with the whole population eventually converging on A, for a range of intermediate values of β and for large enough p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' When committed individuals are chosen at random in the population, this range increases when the hypergraph contains hyperedges of larger sizes [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This naturally raises the question of whether the committed minority might also benefit from belonging to specific substructures of a given hypergraph, such as hyper-cores with large connectedness and group sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Here we investigate this issue through numerical simu- lations of the higher-order NG process on empirical static hypergraphs for varying values of the parameter β and of the fraction p of committed individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In our numeri- cal experiments committed individuals are selected with different seeding strategies: (i) at random from the entire population (random);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' (ii) as the Np ones with the high- est hyper-coreness R (top hyper-coreness);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' (iii) as the Np ones with the highest s-coreness (top s-coreness) in the projected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In each experiment we measure the frac- tion nA of nodes holding only A in their dictionary (both committed or not), and focus on its large time limit n∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This limit can be either 1, if the population reaches the absorbing state in which all nodes agree on A, or, if the absorbing state is not reached before tmax time-steps, we average over 100 values of nA(t) sampled from the last T time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Figure 4 reports the simulation results for two empir- ical data sets, congress-bills (a-d) and the email-EU (e- h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The results for the other data sets can be found in 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 b 10 2 s-coreness 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 g 10 2 n * A 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 nA(t) d Random R-rank s-rank 103 104 105 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 nA(t) h Random R-rank s-rank 1/N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 1/N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Comparison of seeding strategies for committed minorities in a higher-order naming-game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the heatmaps a-c, e-g the stationary fraction n∗ A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the agreement probability β (steps correspond to the fact that p varies by increments of 1/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' a-c: congress-bills data set with unanimity rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' e-g: email-EU data set with union rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Committed nodes are selected through different strategies, in particular random seeding (a,e), top s-coreness (b,f), and top hyper-coreness (c,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Panels d,h show the temporal evolution of nA(t) according to the different seeding strategies and for fixed values of β and p (cross markers in the heatmaps), (d): (β, p) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='48, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 × 10−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' (h): (β, p) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 × 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The minority takeover, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' n∗ A = 1, takes place for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9% of the explored parameter space in panel a, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3% in b, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5% in c, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0% in e, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9% in f, and 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4% in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All simulations are run until the absorbing state n∗ A = 1 is reached or the dynamics has evolved for tmax = 5 × 105 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The stationary fraction n∗ A is obtained by averaging over 100 values sampled in the last T = 5 × 104 time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Results refer to the median values obtained over 200 simulations for each pair of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For the random strategy, we recover the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' [28]: for low values of β, a co-existence state of A and B is observed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' at low fraction of committed and large β values , the majority remains B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Contrarily, at intermediate values of β, the minority takes over and the whole population converges on A (in a way favored by the union rule w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the unanimity rule).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Non-random seed- ing strategies yield the same phenomenology but enhance the range of parameters in which the minority overturns the majority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This is especially the case when we place committed individuals in the most central nodes accord- ing to their hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In this case, a tiny fraction of committed is able to take over on a wide range of β values, while for lower β values, a co-existence regime is always observed –due to the small propensity to ac- cept a local consensus [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For example, with the top hyper-coreness strategy a fraction p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='51 × 10−2 in the congress-bills data set with unanimity rule is enough to allow the minority takeover over a range of β values whose extension is ∆β ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This cannot be achieved with the other two seeding strategies, for which below p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 × 10−2 only ∆β ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 can be reached (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Analogously, in the email-EU data set with the union rule a fraction p = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 × 10−3 is enough to ob- tain the minority dominance over ∆β ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 when seeded according the top hyper-coreness strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In this case, with the top s-coreness and the random strategies the same result is obtained only for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='33 × 10−2 and p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 × 10−2 respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' To further quantify the differences among these strategies we can also calculate the value of critical mass pc necessary to bring the system to the tipping point while keeping β con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the congress-bills data set with unanimity rule and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='62 for instance, the critical mass for the top hyper-coreness strategy is pc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 × 10−3 as compared to pc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='68 × 10−2 and pc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='04 × 10−2 obtained with the random and the top s-coreness strategies respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4a-c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' similarly, in the email-EU data set with union rule and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='83, these values are respectively pc = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 × 10−3, pc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='53 × 10−2, pc = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 × 10−3 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4e-g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The hyper-coreness centrality is thus particularly effec- tive in identifying nodes with a crucial role in higher-order NG processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Indeed, nodes belonging to (k, m)-hyper- cores with large values of k and m, if committed, can convince many others through their simultaneous pres- ence in several large groups, and this can be efficiently sustained by their large connectedness, favouring conver- gence on their convention even outside the committed minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In addition, the temporal evolution diplayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4d,h illustrate how, even when all seeding strategies lead to the agreement on the convention initially sup- 7 ported by the minority, the convergence is much faster for the hyper-coreness seeding strategy, followed by the s-coreness and the random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' DISCUSSION We have considered here a systematic procedure to ex- tract, from a given hypergraph, structures of increasing connectedness along increasing group sizes: the (k, m)- hyper-cores, in which each node is connected to the other by at least k hyperedges (representing higher-order in- teractions) of sizes at least m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Using the maximal con- nectedness values of each node we define a new concept of centrality in hypergraphs: a node hyper-coreness summa- rizes its relative depth in the hierarchies of hyper-cores at all orders (interaction sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Applying these concepts to empirical data describing a variety of higher-order sys- tems, we have shown how the (k, m)-hyper-cores pro- vide a fingerprint of empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Crucially, we have also highlighted how hyper-cores with increas- ing k and m play important roles in several dynamic processes with higher-order mechanisms unfolding upon hypergraphs, such as contagion processes and consensus formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The hyper-coreness centrality in particular identifies nodes with high spreading power and on which stationary contagion processes tend to localize;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' moreover nodes with high hyper-coreness, if belonging to a commit- ted minority, can be particularly efficient at overturning a majority convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Our work opens the door to several research directions in the expanding field of hypergraphs structure and dy- namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' It can provide an additional systematic charac- terization of both empirical and model hypergraphs, and thus potentially a model validation tool as well as a com- parison method between hypergraphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' by computing distances between the (k, m)-hypercore profiles of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For specific systems where additional properties of the nodes are known, the shell indices and hyper-coreness values of nodes could be compared in more detail to pro- vide insights into their relative positions and roles in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Moreover, while here we focused on static hypergraphs, many such systems evolve in time [60, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hyper-cores and hyper-coreness could be used to investigate the evo- lution of the higher-order interactions at multiple scales, from the global evolution of the structure described by hyper-core sizes, to the shell indices and hyper-coreness of individual nodes from one period to the next [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' An interesting case study in this direction could be for in- stance the evolution of the hyper-core positions of scien- tists in co-authorship “networks”, which are in fact by construction evolving hypergraphs [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' MATERIALS AND METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Data description and preprocessing Several data sets we considered are publicly available in the form of static hypergraphs, thus they do not require any preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' These data sets describe: email communications: within a European institu- tion (email-EU [44]), and within Enron, between a core-set of workers (email-Enron [45, 46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Each node corresponds to an email address and a hy- peredge includes the sender and all receivers of an email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' interactions in legislative bills in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Congress (congress-bills) and in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Senate (senate-bills) [46, 49, 51, 52]: each node corresponds to a member of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Congress or Senate and a hyperedge involves sponsors and co-sponsors of legislative bills discussed in the Congress or Senate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' interactions in committees in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' House of Representatives (house-committees) and in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Senate (senate-committees) [46, 49, 50]: each node corresponds to a member of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' House of Rep- resentatives or Senate and each hyperedge involves nodes that share membership in a committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' online interactions (3 data sets): exchanges be- tween users of MathOverflow on algebra top- ics (algebra-questions) or on geometry topics (geometry-questions), in which each node corre- sponds to a user of MathOverflow and each hyper- edge involves those users who have answered a spe- cific question belonging to the topic of algebra or ge- ometry [46, 48];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' interactions between Amazon users on music (music-review [46, 47]), in which each node corresponds to an Amazon user and each hyperedge involves users who have reviewed a specific product belonging to the category of blues music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Moreover, we built static hypergraphs from several data sets of time-resolved face-to-face human interac- tions, as in [26, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The data sets are provided by the SocioPatterns collaboration [40–42] and by the Contacts among Utah’s School-age Population (CUSP) project [43] and describe interactions between individuals in sev- eral contexts : a hospital (LH10 [62]), a workplace (InVS15 [41, 63]), a conference (SFHH [41]), a high-school (Thiers13 [64]), two primary-schools (LyonSchool [65], Elem1 [43]) and a middle-school (Mid1 [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For these data sets we carried out an aggregation procedure to ob- tain static hypergraphs: (i) we aggregate the data over time windows of 15 minutes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' (ii) we identify the cliques in each time window, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' groups of nodes forming a fully connected cluster, (iii) we identify in each temporal win- dow the maximum cliques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' cliques not completely contained in a larger clique, and promote them to a hy- peredge status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Overall, the data sets considered describe interactions in several different environments, mediated by different mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' They correspond to a wide variety of sta- tistical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' data set size, hyperedges size dis- tributions), as shown in the SM where these statistical properties of the data sets are reported in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Models and stochastic simulations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Higher-order non-linear contagion We performed stochastic numerical simulations of the higher-order non-linear contagion model on each empir- ical static hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The simulations are performed with discrete time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The S → I infection mecha- nism is the same for the SIR and the SIS models: for each time-step ∆t, given a hyperedge of size m contain- ing i infected nodes, each of the susceptible nodes in it can be infected with probability (1 − e−λiν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Therefore, the probability that a node j is infected in a time-step ∆t is: pj = 1 − � e∈E(j) e−λiν e , (2) where E(j) denotes the set of hyperedges in which the node j is involved and ie is the number of infected nodes in the hyperedge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Each infected node heals (return- ing susceptible in SIS or gaining immunity in SIR) with probability µ in each time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the SIS process the population is initialized with a single infectious seed randomly selected in the population and the process is iterated until the system reaches a steady state with a fluctuating number of infectious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' An observation time window T is then considered and the time τ spent in the infectious state is estimated for all nodes over that time-window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The results are averaged over 103 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the SIR process the population is initialized with a single infectious seed j and the dynamic process is iter- ated until no more infectious nodes are present: the final epidemic size R∞(j) obtained by seeding the infection in j is defined as the final number of nodes in the R state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The results are averaged over 300 simulations for each infection seed j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Higher-order NG process We also performed numerical simulations of the higher- order NG process on the empirical hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The system with N nodes is initialized by fixing Np nodes as belonging to the committed minority (equivalently, with a fraction p = Np/N of committed nodes), with only the name A in their dictionary, and setting the dictionaries of all the other nodes of the majority with only the name B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The committed nodes are selected following one of the three seeding strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' randomly from the whole population or as the Np nodes with highest s-coreness or hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' If several nodes have the same coreness value, the committed nodes are randomly selected within the coreness class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The simulations are performed in discrete time-steps: at each time-step a hyperedge is randomly selected (ac- tivation of the group) and within it a node is randomly chosen as the speaker, while the other nodes behave as listeners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The speaker randomly selects a name in their dictionary and all nodes in the group update their dictio- nary according to the chosen agreement rule (except for the committed nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The process is iterated until the system reaches the absorbing state where all nodes have only the name A in their dictionary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' nA(t) = n∗ A = 1, or until the system has evolved for tmax time-steps: in this last case the stationary fraction of nodes with the name A in their dictionary n∗ A is obtained by averaging nA(t) over 100 values sampled in the last T = 50, 000 time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The results refer to the median values ob- tained over 200 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' ACKNOWLEDGEMENTS M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' acknowledge support from the Agence Nationale de la Recherche (ANR) project DATAREDUX (ANR-19-CE46-0008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 Giovanni Petri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 and Alain Barrat1 1Aix Marseille Univ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Universit´e de Toulon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' CPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Turing Center for Living Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' France 2Department of Network and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Central European University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 1100 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Austria 3CENTAI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Corso Inghilterra 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 10138 Turin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Italy In this Supplementary Material we present the same results as in the main text for all the considered data sets and also further results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We first present in detail some of the statistical properties of the data sets and of the static hypergraphs considered (Supplementary Section 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In Supplementary Section 2 we present the results of the (k, m)-core decomposition, showing how the (k, m)-cores and (k, m)-shells are populated as a function of k and m, the functional form of the m-shell index Cm(i) for some nodes, the distributions of the hypercoreness and s-coreness centralities and their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In Supplementary Section 3 we present the results of the higher-order non-linear contagion process [1], both in the SIS and SIR formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In Supplementary Section 4 we introduce in details the threshold higher-order process [2], its numerical implementation and its results in relation to the hyper-cores, both in the SIS and SIR formulation, as done for the higher-order non-linear contagion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Finally in Supplementary Section 5, the results of the higher-order naming-game process [3] are presented for both the union and the unanimity rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Supplementary Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Properties of the data sets The considered data sets describe interactions in several environments, mediated by different mechanisms, and thus they differ in their fundamental statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' This is summarized in Table I and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 1: the number of nodes and hyperedges vary among the data sets considered, the distribution Ψ(m) of the hyperedge sizes m, the range of their sizes m ∈ [2, M] and the average hyperedge size ⟨m⟩ are different among the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' data set N E M ⟨m⟩ LH10 76 1 102 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 Thiers13 327 4 795 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 InVS15 217 3 279 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 SFHH 403 6 398 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 LyonSchool 242 10 848 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 Mid1 591 61 521 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 Elem1 339 20 940 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 email-EU 979 24 399 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 data set N E M ⟨m⟩ congress-bills 1 718 83 105 25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 senate-committees 282 302 31 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 email-Enron 143 1 459 37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 house-committees 1 290 335 82 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 music-review 1 106 686 83 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 senate-bills 294 21 721 99 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 algebra-questions 423 980 107 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 geometry-questions 580 888 230 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 Supplementary Table I: Some properties of the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The tables give: the number of nodes N, the number of hyperedges E, the maximum size of the hyperedges M and the average size of the hyperedges ⟨m⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='04235v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='soc-ph] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 1: Hyperedge size distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We show the hyperedge size distribution Ψ(m), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the number of hyperedges of size m, for all the data sets: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 3 Supplementary Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Hyper-core decomposition 1 9 17 25 33 41 49 2 3 4 5 6 7 m n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) a 100 2 × 10 1 3 × 10 1 4 × 10 1 6 × 10 1 1 8 15 22 29 36 2 3 4 5 6 7 n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) b 10 1 100 1 9 17 25 33 41 2 3 4 5 6 7 8 9 10 n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) c 10 1 100 1 7 13 19 25 31 2 3 4 5 6 7 8 9 10 n(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='51 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='92 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='71 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='78 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='116 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='154 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 2: Hyper-core decomposition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show colormaps giving the relative size n(k,m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a function of m and k (white regions correspond to n(k,m) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 4 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n(k, m) a m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 b m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 c m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 d m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n(k, m) e m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 f m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 g m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 h m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n(k, m) i m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 j m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 k m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 l m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 20 40 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n(k, m) m m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 500 1000 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 20 40 60 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 o m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 25 50 75 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 p m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 m = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 Supplementary Figure 3: Hyper-core decomposition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show the relative size n(k,m) (number of nodes in the hyper-core, divided by the total number of nodes N) of the (k, m)-hyper-core as a function of k for fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 5 1 9 17 25 33 41 49 2 3 4 5 6 7 m s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) a 10 1 1 8 15 22 29 36 2 3 4 5 6 7 s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) b 10 2 10 1 1 9 17 25 33 41 2 3 4 5 6 7 8 9 10 s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) c 10 2 10 1 1 7 13 19 25 31 2 3 4 5 6 7 8 9 10 s(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='109 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='136 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 4: (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' m)-shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show colormaps giving the relative size s(k,m) (number of nodes in the hyper-shell, divided by the total number of nodes N) of the (k, m)-shell as a function of m and k (white regions correspond to s(k,m) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 6 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 Cm(i)/km max a R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='08 R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 b R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='05 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 c R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='02 R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 d R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='03 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 R = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 Cm(i)/km max e R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 f R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 R = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 g R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 R = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 h R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='009 R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 R = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 R = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 Cm(i)/km max i R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='003 R = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 R = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 j R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 R = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 R = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 R = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 k R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 R = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 R = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 25 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 l R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 R = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 25 50 75 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 Cm(i)/km max m R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='03 R = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 R = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 R = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 50 100 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 n R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 50 100 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 o R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='02 R = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 R = 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 R = 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0 100 200 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='00 p R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='01 R = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 R = 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 R = 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 Supplementary Figure 5: m-shell index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show the normalized m-shell index function Cm(i)/km max as a function of m for four nodes: one node is selected randomly among the nodes in the class with highest hyper-coreness R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' one node is selected randomly among the nodes in the class with smallest hyper-coreness R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the two remaining node are selected from intermediate hyper-coreness classes, so that the positions in the hyper-coreness ranking of the four nodes are equispaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 7 0 50 0 2 4 6 R(i) a 0 5 R 0 20 P(R) 0 200 0 2 4 6 b 0 5 R 0 50 P(R) 0 100 200 0 2 4 6 8 c 0 5 R 0 50 P(R) 0 200 400 0 2 4 6 8 d 0 5 R 0 100 P(R) 0 100 200 0 2 4 6 8 R(i) e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 R 0 50 P(R) 0 250 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 f 0 10 R 0 200 P(R) 0 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': 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the hyper-coreness R(i) is plotted as a function of the corresponding node rank: the insets show the distribution P(R) of the hyper-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Rank(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='P(S) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Rank(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='P(S) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Rank(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='P(S) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Rank(i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='P(S) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 7: s-coreness centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the s-coreness S(i) is plotted as a function of the corresponding node rank: the insets give the distribution P(S) of the s-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 9 0 100 200 0 2 4 6 R(i) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='98 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='92 0 50 100 0 2 4 6 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='93 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='88 0 100 0 2 4 6 8 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='94 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='83 0 50 100 0 2 4 6 8 d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='77 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 250 500 2 4 6 R(i) e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='73 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='51 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 0 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='82 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='69 0 500 1000 0 5 10 15 20 h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='90 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 0 5000 0 10 20 R(i) i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='92 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='83 100 200 300 0 10 20 30 j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='93 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='78 0 50 100 0 10 20 30 k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='53 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='53 0 200 0 20 40 60 80 l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='92 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='79 0 200 S(i) 0 20 40 60 80 R(i) m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='58 0 10000 S(i) 0 25 50 75 100 n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='96 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 0 200 S(i) 0 25 50 75 100 o = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='92 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='88 0 500 1000 S(i) 0 100 200 p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='88 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='88 Supplementary Figure 8: Hyper-coreness vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' s-coreness centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show scatterplots of the hyper-coreness R(i) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the s-coreness S(i) for all nodes: the text-box reports the Pearson correlation coefficient ρ of R(i) and S(i) and the Kendall’s τ coefficient of the corresponding node rankings (in all cases the p-value for both the coefficients is p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (panel a), Thiers13 (panel b), InVS15 (panel c), SFHH (panel d), LyonSchool (panel e), Mid1 (panel f), Elem1 (panel g), email-EU (panel h), congress-bills (panel i), senate-committees (panel j), email-Enron (panel k), house-committees (panel l), music-review (panel m), senate-bills (panel n), algebra-questions (panel o) and geometry-questions (panel p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 10 Supplementary Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Higher-order non-linear contagion process data set ν λ LH10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 Thiers13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 InVS15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 SFHH 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 LyonSchool 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 Mid1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−5 Elem1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5 × 10−5 email-EU 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5 × 10−5 data set ν λ congress-bills 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−6 senate-committees 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 5 × 10−4 email-Enron 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 house-committees 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 5 × 10−4 music-review 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 5 × 10−4 senate-bills 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5 × 10−6 algebra-questions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 5 × 10−4 geometry-questions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5 × 10−5 Supplementary Table II: Parameters for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 9-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The tables summarize the parameters of the higher-order non-linear SIS contagion process considered for each data set in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 9-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' data set ν λ LH10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='010 Thiers13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001 InVS15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001 SFHH 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='010 LyonSchool 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001 Mid1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−5 Elem1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 10−4 email-EU 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−5 data set ν λ congress-bills 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5 × 10−5 senate-committees 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 10−4 email-Enron 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 house-committees 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−5 music-review 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 senate-bills 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−5 algebra-questions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='001 geometry-questions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 5 × 10−4 Supplementary Table III: Parameters for Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 12-14.' metadata={'source': 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154 192 230 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0 500 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 /T n S-rank R-rank Supplementary Figure 9: Higher-order non-linear contagion process - SIS model - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels give, as a heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the (k, m)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets represent τ/T averaged over the first n nodes according to the coreness rankings as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (λ, ν) values considered for each data set are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 12 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 b m = 2 m = 3 m = 4 m = 5 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 c m = 2 m = 3 m = 4 m = 5 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='82 f m = 2 m = 4 m = 6 m = 8 0 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='35 g m = 2 m = 4 m = 6 m = 8 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 h m = 2 m = 6 m = 10 m = 14 0 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 /T i m = 2 m = 6 m = 10 m = 14 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 j m = 2 m = 12 m = 17 m = 22 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 k m = 2 m = 4 m = 6 m = 8 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 l m = 2 m = 15 m = 41 m = 54 0 20 40 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 /T m m = 2 m = 10 m = 18 m = 32 0 500 1000 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 n m = 2 m = 18 m = 34 m = 50 0 20 40 60 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 o m = 2 m = 8 m = 14 m = 20 0 25 50 75 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 p m = 2 m = 15 m = 28 m = 41 Supplementary Figure 10: Higher-order non-linear contagion process - SIS model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the (k, m)-hyper-core is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (λ, ν) values considered for each data set are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 13 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 b k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 c k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 d k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='82 f k = 1 k = 10 k = 20 k = 30 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 g k = 1 k = 10 k = 20 k = 30 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 h k = 1 k = 10 k = 20 k = 30 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='17 /T i k = 1 k = 10 k = 20 k = 30 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 j k = 1 k = 10 k = 20 k = 30 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 k k = 1 k = 7 k = 14 k = 21 0 25 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 l k = 1 k = 6 k = 12 k = 18 0 25 50 75 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 /T m k = 1 k = 10 k = 20 k = 30 0 50 100 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 n k = 1 k = 10 k = 20 k = 30 0 50 100 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 o k = 1 k = 10 k = 20 k = 30 0 100 200 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 p k = 1 k = 10 k = 20 k = 30 Supplementary Figure 11: Higher-order non-linear contagion process - SIS model - III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the (k, m)-hyper-core is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (λ, ν) values considered for each data set are reported in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='17 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='S-rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R-rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 12: Higher-order non-linear contagion process - SIR model - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show, as a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (λ, ν) values considered for each data set are reported in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='62 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='160 b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='330 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='360 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='370 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='380 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='390 d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='205 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='210 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='215 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='225 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='160 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='180 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='280 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='290 n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='550 p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 13: Higher-order non-linear contagion process - SIR model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (λ, ν) values considered for each data set are reported in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 16 2 4 6 58 60 62 64 66 68 70 R a k = 1 k = 10 k = 20 k = 30 2 4 6 80 90 100 110 120 130 b k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': 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360 370 380 390 d k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='190 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='160 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 14: Higher-order non-linear contagion process - SIR model - III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The (λ, ν) values considered for each data set are reported in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 17 Supplementary Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Threshold higher-order contagion process Here we consider another spreading process in which multi-body interactions drive the infection through a thresh- old effect and group contagion [2, 4]: the threshold higher-order contagion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We consider both the SIR and SIS epidemic models on static hypergraphs: for each hyperedge of size m in which i individuals are in the state I, if the fraction of infected individuals i/m is larger or equal to a threshold θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' if i ≥ ⌈θm⌉, a group infection is activated at rate λ in which the susceptible nodes in the hyperedge become all infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Note that if we consider a single seed of infection: for θ ≤ 1/M the group infection is activated in all the hyperedges containing the seed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' for θ = 1/m the spreading is activated only in the hyperedges containing the seed that have size larger or equal to m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' for θ > 1/2 the spreading is inhibited since more than one infected node is required to activate the infection in all hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' I individuals recover independently at constant rate µ, becoming either S (SIS model) or R (SIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We perform numerical simulations of this process, for both SIS and SIR models, on empirical hypergraphs: the simulation procedures are analogous to those described in the main text for the higher-order non-linear contagion process (see Methods), since the two processes only differ in the infection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the threshold higher-order contagion, for each time-step ∆t, given a hyperedge of size m containing i infected nodes, if i ≥ ⌈θm⌉ a group infection process is activated with probability λ and all susceptible nodes in the hyperedge are infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Thus, in each time-step each of the interaction groups respecting the condition i ≥ ⌈θm⌉ produces a group infection process with probability λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Therefore, also in this case we quantify the ”spreading power” of each node considered separately as seed for the SIR model and the nodes on which the epidemic is mainly localized in the steady state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the nodes that drive and sustain the process, for the SIS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 15-20 are shown the results of these simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' data set θ λ LH10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='03 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0 1000 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 /T n 1 8 15 22 29 36 k 2 16 30 44 58 72 m m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 0 1000 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 /T n S-rank R-rank 1 199 397 595 793 991 k 2 18 34 50 66 82 98 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': 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+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0 250 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 /T n S-rank R-rank 1 15 29 43 57 71 k 2 40 78 116 154 192 230 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0 500 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 /T n S-rank R-rank Supplementary Figure 15: Threshold higher-order contagion process - SIS model - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels give, as a heat-map as a function of k and m, the average fraction ⟨τ/T⟩ of time being infected in the SIS steady state averaged over the nodes of the (k, m)-hyper-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets represent τ/T averaged over the first n nodes according to the coreness rankings as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The values (λ,θ) values considered for each data set are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 19 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='90 /T a m = 2 m = 3 m = 4 m = 5 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 b m = 2 m = 3 m = 4 m = 5 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='175 c m = 2 m = 3 m = 4 m = 5 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 d m = 2 m = 3 m = 4 m = 5 0 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='700 /T e m = 2 m = 3 m = 4 m = 5 0 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='82 f m = 2 m = 4 m = 6 m = 8 0 200 400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='800 g m = 2 m = 4 m = 6 m = 8 0 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 h m = 2 m = 6 m = 10 m = 14 0 500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='90 /T i m = 2 m = 6 m = 10 m = 14 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 j m = 2 m = 12 m = 17 m = 22 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 k m = 2 m = 4 m = 6 m = 8 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 l m = 2 m = 15 m = 41 m = 54 0 20 40 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 /T m m = 2 m = 10 m = 18 m = 32 0 500 1000 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 n m = 2 m = 18 m = 34 m = 50 0 20 40 60 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 o m = 2 m = 8 m = 14 m = 20 0 25 50 75 k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 p m = 2 m = 15 m = 28 m = 41 Supplementary Figure 16: Threshold higher-order contagion process - SIS model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The values (λ,θ) values considered for each data set are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 20 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='85 /T a k = 1 k = 10 k = 20 k = 30 2 4 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='06 b k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='175 c k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 d k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 /T e k = 1 k = 10 k = 20 k = 30 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 f k = 1 k = 10 k = 20 k = 30 5 10 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 g k = 1 k = 10 k = 20 k = 30 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 h k = 1 k = 10 k = 20 k = 30 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='76 /T i k = 1 k = 10 k = 20 k = 30 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 j k = 1 k = 10 k = 20 k = 30 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 k k = 1 k = 7 k = 14 k = 21 0 25 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 l k = 1 k = 6 k = 12 k = 18 0 25 50 75 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 /T m k = 1 k = 10 k = 20 k = 30 0 50 100 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='42 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average fraction ⟨τ/T⟩ of time being infected in the steady state averaged over the nodes of the (k, m)-hyper-core is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 103 numerical simulations, with a single random seed of infection and with an observation window T = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The values (λ,θ) values considered for each data set are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 21 1 9 17 25 33 41 49 2 3 4 5 6 7 m R a 50 55 60 0 50 n 50 60 R n 1 8 15 22 29 36 2 3 4 5 6 7 R b 8 10 12 0 250 n 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R-rank ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 18: Threshold higher-order contagion process - SIR model - I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All panels show, as a function of k and m through a heat-map, the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The insets represent, as a function of n, R∞ averaged over the first n nodes according to coreness rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The values (λ,θ) values considered for each data set are summarized in Table V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='280 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='320 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='234 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='236 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='238 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='480 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='520 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='280 g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 h ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='65 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 34 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='70 o ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='140 p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m = 41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='Supplementary Figure 19: Threshold higher-order contagion process - SIR model - II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of k at fixed values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The values (λ,θ) values considered for each data set are summarized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 23 2 4 6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 R a k = 1 k = 10 k = 20 k = 30 2 4 6 6 7 8 9 10 b k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 160 170 180 190 c k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 280 300 320 340 d k = 1 k = 10 k = 20 k = 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='470 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='480 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='490 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='510 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='k = 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='210 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='220 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='230 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='240 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='260 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='g ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 n k = 1 k = 10 k = 20 k = 30 0 50 100 m 20 30 40 50 60 o k = 1 k = 10 k = 20 k = 30 0 100 200 m 40 60 80 100 p k = 1 k = 10 k = 20 k = 30 Supplementary Figure 20: Threshold higher-order contagion process - SIR model - III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In all panels the average epidemic final-size ⟨R∞⟩ produced by seeding the SIR process in a single seed belonging to the (k, m)-hyper-core (averaged over all nodes of the hyper-core) is shown as a function of m at fixed values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All results are obtained by averaging the results of 300 numerical simulations for each seed (except for the congress-bills data set which is the result of 10 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The following data sets are considered: LH10 (a), Thiers13 (b), InVS15 (c), SFHH (d), LyonSchool (e), Mid1 (f), Elem1 (g), email-EU (h), congress-bills (i), senate-committees (j), email-Enron (k), house-committees (l), music-review (m), senate-bills (n), algebra-questions (o) and geometry-questions (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The values (λ,θ) values considered for each data set are summarized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 24 Supplementary Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Higher-order naming-game (NG) process data set β p tmax T InVS15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='41 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 × 10−2 105 104 Mid1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 × 10−2 105 104 email-EU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 × 10−3 5 × 105 5 × 104 congress-bills 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='59 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 × 10−2 5 × 105 5 × 104 house-committees 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='5 × 10−2 5 × 105 5 × 104 music-review 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='52 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 × 10−3 105 104 Supplementary Table VI: Parameters for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 21 - Union rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The table summarizes the main parameters of the higher-order naming-game process considered for the temporal dynamics of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 21 in the various data sets (union rule).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 nA(t) x Random R-rank s-rank Supplementary Figure 21: Higher-order NG process - Union rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the first three panels of each row the stationary fraction n∗ A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the agreement probability β through a heat-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We consider the union rule and the following data sets: InVS15 (first row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews (sixth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For each row, the committed nodes are selected through: the random seeding strategy in the first panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the top s-coreness seeding strategy in the second panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the top hyper-coreness seeding strategy in the third panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VI (cross markers in the heatmaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The minority take-over, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' n∗ A = 1, takes place over an area A of the explored parameter space: in Table VII we report its value for the different strategies and data sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All simulations are run until the absorbing state with n∗ A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗ A is obtained by averaging over 100 values sampled in the last T time-steps (see Table VI for the tmax and T values for each data set).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 102 103 104 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='0 nA(t) x Random R-rank s-rank Supplementary Figure 22: Higher-order NG process - Unanimity rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the first three panels of each row the fraction n∗ A of nodes supporting only the name A is shown as a function of the fraction of committed nodes p and the agreement probability β through a heat-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' We consider the unanimity rule and the following data sets: InVS15 (first row), Mid1 (second row), email-EU (third row), congress-bills (fourth row), house-committees (fifth row), music-reviews (sixth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' For each row, the committed nodes are selected through: the random seeding strategy in the first panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the top s-coreness seeding strategy in the second panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' the top hyper-coreness seeding strategy in the third panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' In the fourth panel we show the temporal dynamics of nA(t) for fixed β and p, whose values are reported in Table VIII (cross markers in the heatmaps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The minority take-over, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' n∗ A = 1, takes place over an area A of the explored parameter space: in Table IX we report its value for the different strategies and data sets considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' All simulations are run until the absorbing state with n∗ A = 1 is reached or the dynamics has evolved for tmax time steps and the stationary fraction n∗ A is obtained by averaging over 100 values sampled in the last T time-steps (see Table VIII for the tmax and T values for each data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' The results refer to the median values obtained over 200 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' 28 [1] St-Onge, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Influential groups for seeding and sustaining nonlinear contagion in heterogeneous hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Communications Physics 5, 25 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' [2] de Arruda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', Petri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', Rodriguez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' & Moreno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Multistability, intermittency and hybrid transitions in social contagion models on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' arXiv preprint - arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content='04273 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' [3] Iacopini, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', Petri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', Baronchelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' & Barrat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Group interactions modulate critical mass dynamics in social convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Communications Physics 5, 64 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' [4] de Arruda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=', Petri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' & Moreno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Social contagion models on hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} +page_content=' Research 2, 023032 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE2T4oBgHgl3EQf-Anw/content/2301.04235v1.pdf'} diff --git a/GNE4T4oBgHgl3EQfgA3B/vector_store/index.faiss b/GNE4T4oBgHgl3EQfgA3B/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..36eefc5e0b19e7666668708e3e0ea193a27b76e1 --- /dev/null +++ b/GNE4T4oBgHgl3EQfgA3B/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bbb77c7839bc762bc16c959fdcf648d744e7c84018d0dd3dfb432506c7aa1f21 +size 21626925 diff --git 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Consumer Acceptance +of the Use of Artificial Intelligence in Online +Shopping: +Evidence +From +Hungary. +Amfiteatru +Economic, 23(56), pp.155-173. + +DOI: 10.24818/EA/2021/56/155 + +Article History +Received: 30 September 2020 +Revised: 7 November 2020 +Accepted: 26 December 2020 + + +Abstract +The rapid development of technology has drastically changed the way consumers do their +shopping. The volume of global online commerce has significantly been increasing partly +due to the recent COVID-19 crisis that has accelerated the expansion of e-commerce. +A growing number of webshops integrate Artificial Intelligence (AI), state-of-the-art +technology into their stores to improve customer experience, satisfaction and loyalty. +However, little research has been done to verify the process of how consumers adopt and +use AI-powered webshops. Using the technology acceptance model (TAM) as a theoretical +background, this study addresses the question of trust and consumer acceptance of +Artificial Intelligence in online retail. An online survey in Hungary was conducted to build +a database of 439 respondents for this study. To analyse data, structural equation modelling +(SEM) was used. After the respecification of the initial theoretical model, a nested model, +which was also based on TAM, was developed and tested. The widely used TAM was +found to be a suitable theoretical model for investigating consumer acceptance of the use of +Artificial Intelligence in online shopping. Trust was found to be one of the key factors +influencing consumer attitudes towards Artificial Intelligence. Perceived usefulness as the +other key factor in attitudes and behavioural intention was found to be more important than +the perceived ease of use. These findings offer valuable implications for webshop owners to +increase customer acceptance. + +Keywords: consumer acceptance, artificial intelligence, online shopping, AI-powered +webshops, technology acceptance model, trust, perceived usefulness, perceived ease of use, +attitudes, behavioural intention, Hungary + +JEL Classification: L81, M31, O30 + + + +* Corresponding author, Szabolcs Nagy – e-mail: nagy.szabolcs@uni-miskolc.hu + +AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +156 +Amfiteatru Economic +Introduction +The rapid development of digital technology has changed online shopping (Daley, 2018). In +recent years, the use of Artificial Intelligence (AI) in online commerce has been increased since +AI is an excellent tool to meet rapidly changing consumer demand and to increase sales +efficiency. The global spending by retailers on AI services is expected to quadruple and reach +$12 billion by 2023, and over 325000 retailers will adopt AI technology (Maynard, 2019). +Smidt and Power (2020) claimed that online product research has significantly increased +over the past years. USA's largest online retailer, Amazon, is the exemplary case of how +to effectively integrate AI into online retail. Besides the rich assortment, fast delivery +and competitive prices, a more localised shopping journey can be created. Thus Amazon +can use location-specific pricing and send destination-specific messages to its +customers, who will pay in their local currency (Barmada, 2020). +Novel marketing techniques supported by new technologies, including the use of AI systems +spark the proliferation of new marketing methods to effectively reach target consumers and to +offer enhanced consumer experiences (Pusztahelyi, 2020). Pursuant to Asling (2017), the use +of AI in online shopping makes customer-centric search and a new level of personalisation +possible resulting in a more efficient sales process. Information technology (IT) has changed +the nature of company-customer relationships (Rust and Huang, 2014). However, any +technology-driven transformation is based on trust (Pricewaterhouse Coopers, 2018). +Online retailers need more in-depth insight into how consumers perceive and accept the use of +AI in webshops and how much they trust them. They also need to know how to use AI most +effectively to increase online spending and online purchase frequency since the importance of +time and cost efficiency in shopping has recently become more and more critical. In this +regard, online shopping means a convenient way for customers to buy the desired products. +So far, only a few researchers have addressed the question of trust and consumer +acceptance of AI in online retail. Based on the technology acceptance model (TAM), this +study aims to fill this research gap and proposes an integrated theoretical framework of +consumers' acceptance of AI-powered webshops. Further objectives of this paper are to +investigate the relationships between the elements of TAM; to analyse the effects of trust, +perceived usefulness and perceived ease of use on attitudes and behavioural intention. +After reviewing the use of AI in online shopping, this paper discusses the role of trust in +online shopping and presents the technology acceptance model. The next section deals with +the research methodology, including the research questions, hypotheses and the sample. In +the results and discussion section, the validity and reliability of the model, as well as the +model fit are presented. Hypothesis testing, detailed analysis of the relationships between +the elements of the nested model, and comparison of the results with the previous research +findings are also discussed here before the conclusions sections. + +1. Literature review +According to IBM's U.S. Retail Index, the COVID-19 has speeded up the change from +traditional shopping to online purchasing by circa five years (Haller, Lee and Cheung, +2020). Due to the pandemic situation, there is an increased demand for AI in the retail +industry (Meticulous Market Research, 2020). + +Artificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +157 +1.1. The use of AI in online shopping +AI systems are a set of software and hardware that can be used to continuously assess and +analyse data to characterise environmental factors and to determine decisions and actions +(European Commission, 2018). Prior research mainly focused on the advantages of the use +of AI in online settings and failed to address how consumers accept AI in online retail. +According to utility theory, this new technology helps consumers to find and choose the +best product alternatives, while decreases the search cost and search time (Pricewaterhouse +Coopers, 2018), thus increasing utility (Stigler, 1961; Bakos, 1977; Stigler and Becker, +1977; André, et al. 2017; Lynch and Ariely, 2000). AI filters the information for each target +customer and provides what exactly is needed (Paschen, Wilson and Ferreira, 2020). AI +supports automating business processes, gains insight through data analysis, and engages +with customers and employees (Davenport and Ronanki, 2018). +Artificial intelligence is widely used to increase the efficiency of marketing (Kwong, Jiang, and +Luo, 2016) and retail (Weber and Schütte, 2019) and to automate marketing (Dumitriu and +Popescu, 2020). AI-powered online stores provide their customers with automated assistance +during the consumer journey (Yoo, Lee and Park, 2010; Pantano and Pizzi, 2020). It is a great +advantage, especially for the elder people, who are averse to technical innovations. +Consumers' online information search and product selection habits can be better understood by +AI to offer a more personalised shopping route (Rust and Huang, 2014). It is a great +opportunity for online shops to analyse the profile of existing and potential customers and +thereby suggest tailor-made marketing offerings for them (Onete, Constantinescu and Filip, +2008). AI also makes the contact with both the customers and the employees continuous and +interactive. Frequently asked questions (FAQs) regarding the products, product-use and +ordering process can be automated by a chatbot. New sales models use automated algorithms +to recommend unique, personalised marketing offerings, thus increasing customer satisfaction +and engagement. To sum up the advantages, AI systems operate automatically and analyse big +data in real-time to interpret and shape consumer behavioural patterns to offer products and +services in a personalised way, thus enhancing the shopping experience. +However, AI systems also have some disadvantages. They work most effectively with big data; +therefore, the implementation of AI systems requires huge investments (Roetzer, 2017). +1.2. The role of trust in online shopping +Trust is of great importance in online commerce. According to Kim, Ferrin and Rao (2008), +consumer confidence has a positive effect on a consumer's intention to buy. The higher the +consumer trust in an online shop is, the more likely the consumer will be to go through the +buying process. Trust is especially crucial when the customer perceives a financial risk. +Thatcher et al. (2013) identified two types of trust: general and specific trust. General trust +concerns the e-commerce environment, consumer beliefs about and attitudes towards it. +Specific trust is related to the shopping experience in a specific virtual store. Confidence can be +enhanced through interactive communication between the retailer and the buyer by using +appropriate product descriptions and images to reduce the perceived risk. As stated in Cătoiu et +al. (2014) there is a strong negative correlation between perceived risks and trust. According to +Reichheld and Schefter (2000, p. 107), “price does not rule the Web; trust does”. + +AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +158 +Amfiteatru Economic +Aranyossy and Magisztrák (2016) found that a higher level of e-commerce trust was +associated with more frequent online shopping. However, when shopping online, customers +do not necessarily notice that a website uses AI tools (Daley, 2018). +All things considered, AI marks a new era in online sales. However, continuous +technological development such as the use of AI-powered websites divides society, as there +are those who accept novelty while others reject it. + +1.3. Technology Acceptance Model (TAM) +Consumers' adaptation to new technologies can be explained by several models. Dhagarra, +Goswami and Kumar (2020) summarised them as follows: (1) Theory of Reasoned Action +(TRA) by Fishbein and Ajzen (1975); (2) Theory of Planned Behaviour (TPB) by Ajzen +(1985); (3) Technology Acceptance Model (TAM) by Davis (1986); (4) Innovation +Diffusion Theory (IDT) by Rajagopal (2002); (5) Technology Readiness Index (TRI) by +Parasuraman, (2000); and (6) Unified Theory of Acceptance and Use of Technology +(UTAUT) by Venkatesh, et al. (2003). +Technology acceptance model (TAM), an extension of (TRA), is one of the most widely- +used theoretical models (Venkatesh, 2000) to explain why an IT user accepts or rejects +information technology and to predict IT user behaviour (Legris, Ingham, and Collerette, +2003). The original TAM contains six elements: external variables, perceived usefulness, +perceived ease of use, attitude, behavioural intention to use and actual use. According to +TAM, external variables have a direct influence on perceived usefulness (PU) and +perceived ease of use (PEU), i.e. the two cognitive belief components. Perceived ease of +use directly influences PU and attitude, whereas perceived usefulness has a direct impact on +attitude and behavioural intention to use, which affects actual use (Figure no. 1). + +Figure no. 1. The original technology acceptance model (TAM) +Source: Davis, 1986. +Ha and Stoel (2008) examined the factors affecting customer acceptance of online shopping +and found that perceived ease of use, perceived trust and perceived shopping enjoyment +had the greatest impact on customer acceptance. Ease of use, trust and shopping enjoyment +had a significant impact on perceived usefulness; trust, shopping enjoyment, and usefulness + +Perceived +Usefulness +External +Attitude +Behavioral +Variables +Towards +Intentionto +Actual Use +Use +Use +Perceived +Easeof UseArtificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +159 +had a significant effect on attitude towards online shopping. They also found that attitude +and perceived usefulness had an influential role in consumer intention to purchase online. +According to Vijayasarathy (2004), there is a positive association between consumer attitude +towards online shopping and the beliefs concerning usefulness, compatibility, security and ease +of use. Also, the intention to purchase online is strongly influenced by consumer beliefs about +online shopping, self-efficacy and attitude. Surprisingly, no positive relationship between +purchasing intention and consumer beliefs about the usefulness of online shopping was +reported (Vijayasarathy, 2004). Gefen, Karahanna and Straub (2003) found that perceived +usefulness and perceived ease of use influence consumer repurchase intention. +It must be noted that Schepman and Rodway (2020) expressed some criticisms about the +applicability of TAM to measure attitudes towards AI. According to them, it is the online +retailers that can decide to integrate AI into webshops, and consumers have no choice but to +use it when shopping online in such stores. Therefore, traditional technology acceptance +models might not be ideal to measure attitudes towards AI. However, we are convinced that +consumers still have the free will to decide whether to use new technology, i.e. to shop +online in an AI-powered webshop, or not. + +2. Methodology and research questions +2.1. Methodology +The constructs and the measurement instruments presented in Table no. 1 were developed +based on the literature review, and according to the Technology Acceptance Model. +Variables with asterisk and in italics were adapted from Park (2009), the others were +adapted from Hu and O'Brien (2016). However, each variable was modified by the authors +to make it possible to measure the perceived role of AI in online shopping. +For data collection, a questionnaire made up of 26 questions (variables) was used (Table +no. 1). Additionally, six demographics variables - gender, education, age, occupation, place +of residence and internet subscription - were also included in the survey. All measurement +instruments were listed in Table no. 1 but the demographics variables were measured on a +seven-point Likert-scale ranging from strongly disagree (1) strongly agree (7). +In the very first section of the questionnaire, respondents were provided with a detailed +explanation of AI-powered webshops and shopping apps, which are online stores where +shopping is supported by artificial intelligence. AI-powered webshops present personalised +product/service offerings based on previous search patterns and purchases that we made +before, and automatically display products that AI chooses for us. Also, AI offers similar +products to those that were originally viewed but were not available in the right size +(product recommendation based on visual similarity). Another typical sign of an AI- +powered webshop is that when the customer is leaving the web store, AI warns about the +products left in the cart, to complete the purchase. AI-powered webshops often use +chatbots, i.e. a virtual assistant is available if the customer has any questions, and visual +(image-based) search is also possible: after uploading a product picture, AI recommends +the most similar ones to that. Virtual changing rooms, voice recognition and automatic +search completion are also available in AI-powered webshops such as Amazon, e-Bay, +Alibaba, AliExpress, GearBest, eMAG.hu, PCland.hu, Ecipo, Bonprix, Answear, Reserved, +Fashiondays, Fashionup, Spartoo, Orsay, to mention just a few. + +AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +160 +Amfiteatru Economic +Table no. 1. Constructs and measurement instruments +Construct +Definition +Measurement Instruments +Perceived +Usefulness +(PU) +The degree to which a +consumer believes that AI +used in online shopping +would make his or her +purchases more effective. +PU1. The use of AI in retail (shopping ads and +webshops) allows me to find the best deals. +PU2. The use of AI in retail enhances my +effectiveness in purchasing. +PU3. The use of AI in retail is useful to me. +PU4 The use of AI in retail saves time for me. * +Perceived +Ease of Use +(PEU) +The degree to which a +consumer believes that +using AI in webshops will +be free of effort. +PEU1. AI-powered shopping apps and webshops +are easy to use. +PEU2. Shopping does not require a lot of my +mental efforts if supported by AI (alternatives are +offered by AI). +PEU3. Shopping is not so complicated if AI offers +products to me. +PEU4 Learning how to use AI-powered shopping +apps and webshops is easy for me. * +PEU5 It is easy to become skilful at using AI- +powered shopping apps and webshops* +Experience +(EXP) +The consumers' +knowledge about and the +experience with +purchasing in an AI- +powered webshop. +EXP1. I'm experienced in online shopping. +EXP2. I have already used AI-powered applications +(chatbots, etc.) +Trust +(TRUST) +The subjective probability +with which people believe +that AI works for their +best interest. +T1. I am convinced that AI in retail is used to +provide customers with the best offerings. +T2. I trust in apps and webshops that use AI. +Subjective +Norm +(SN) +The degree to which a +consumer perceives that +most people who are +important to him or her +think he or she should or +should not make +purchases in AI-powered +webshops. +SN1. People who influence my behaviour would +prefer me to use AI-powered shopping apps and +webshops. +SN2. I like using AI-powered webshops and +shopping apps based on the similarity of my values +and the social values underlying its use. * +Task +Relevance +(TR) +The degree to which a +consumer believes that +AI-powered webshops are +applicable to his or her +shopping task. +TR1 I think AI can be used effectively in webshops +and shopping apps. +Compen- +sation +(COMP) +The degree to which a +consumer believes that he +or she has the ability to +make purchases in AI- +powered webshops. +I would prefer AI-powered shopping apps and +webshops… +C1. if there was no one around to visit physical +shops/shopping malls with. +C2. if I had less time. +C3. if I had a built-in help facility for assistance +when needed. +Perceived +Quality +PQ +The degree of how good a +consumer perceives the +quality of a product in AI- +powered webshops. +PQ1 AI finds/offers better products for me than I +could. + +Artificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +161 +Construct +Definition +Measurement Instruments +Perceived +Enjoyment +PE +The extent to which +shopping in AI-powered +webshops is perceived +to be enjoyable. +PE1 Shopping is more fun, enjoyable when AI +helps me to find the best-suited products. +Attitude +ATT +The consumer's attitude +towards shopping in AI- +powered webshops. +ATT1 Shopping in a webshop/shopping app that is +powered by AI is a good idea +ATT2 Shopping in a webshop/shopping app that is +powered by AI is a wise idea +ATT3 I am positive towards webshop/shopping app +that is powered by AI +Behavioural +Intention +BI +A consumer's behavioural +intention to do the +shopping in AI-powered +webshops. +BI1 I intend to visit webshops and to use shopping +apps that are powered by AI more frequently. +BI2 I'm willing to spend more on products offered +by webshops and apps powered by AI +Sources: Adapted from Hu and O'Brien, 2016; *Park, 2009. +An online survey in Google Form was conducted to collect data in July and August 2020 in +Hungary. Because of the Theory Acceptance Model, previous online shopping experience +with AI-powered webshops was the one and only eligibility criterion for respondents to +participate in this study. Convenience sampling method was used to reach the maximum +number of respondents. Data was migrated from Google Form to MS Excel, SPSS 24 and +AMOS, and was checked for coding accuracy. The database was complete and contained +no missing data. Descriptive statistical analyses were done in SPSS. AMOS was employed +to test the hypotheses and the theoretical model by structural equation modelling (SEM). + +2.2. Research questions and hypotheses +Based on the literature review, this study aims to address the following research questions +respectively: + R1: Can the technology acceptance model (TAM) be used for investigating consumer +acceptance of the use of artificial intelligence in online shopping? + R2: If so, what are the key factors influencing behavioural intention to visit AI- +powered webshops and apps? +Based on the Technology Acceptance Model, an initial theoretical model was developed +(Figure no. 2). The arrows that link constructs (latent variables such as COMP, EXP, +TRUST, SN, PEU, PU, ATT, BI) represent hypothesised causal relationships (hypotheses) +in the direction of arrows. One of the objectives of this study is to test those hypotheses. +Error terms for all observed indicators are indicated by e1 to e35, respectively. + + +AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +162 +Amfiteatru Economic + +Figure no. 2. The initial theoretical model + +2.3. The sample +A sample size of 200 is an appropriate minimum for SEM in AMOS (Marsh, Balla, and +MacDonald, 1988), and a minimum of 10-20 subjects per parameter estimates in the model +are optimal (Schumacker and Lomax, 2010). Therefore, the ideal sample size is between +380 and 760, considering the number of parameter estimates (38) in the initial model. The +actual sample size of 439 respondents fits into this category. +Of the sample of 439 respondents, 62.2% were female, 37.8% male. Their average age was +32.2 years. 60,8% of the respondents had tertiary education, 38% had secondary education, +and 1.2% had primary education. Most respondents resided in county seats (47.6%); the +rest lived in other towns/cities (24.1%), villages (17.8%) and the capital (10.5%). Most +respondents were subscribed to both mobile and wired internet services (84.3%), while +7.7% had only mobile internet, and 7.1% had only wired internet services. Only 0.9% of +respondents had not got any subscription to internet services (wired or mobile). There is no +data available on the distribution of the e-shoppers in Hungary, therefore, it is impossible to +tell if this sample reflects the characteristics of the e-shoppers’ population in Hungary. + +3. Results and discussion +The initial model (Figure no. 2), which proved to be too complex and did not fit the current +data (CMIN/DF=7.72; p=.00; GFI=,693; CFI=.723; RMSEA=.124; HOELTER 0.5= 65), +was absolutely rejected. Therefore, it was not appropriate to interpret any individual + +C2 +COMP +PEU1PEU2PEU3PEU4PEU5 +PEU +e24 +EXP +EXP +EXP +ATT1ATT2ATT3 +BI2 +TRUST +SN +SN +PU +P2 +P3 +PU4Artificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +163 +parameter estimates, and further model modifications were required to obtain a better- +fitting model. Respecification of the initial model led to a nested model that fitted well and +is discussed further. During the respecification, the alternative model approach was used +(Malkanthie, 2015). To test the model, the same data set was used. Several modified +models were developed, and out of the theoretically justifiable models, the model with the +best data fit was selected (Figure no. 3) as suggested by Mueller and Hancock (2008). +The respecification process was started with testing the measurement model by a series of +Principal Component Analysis (PCA). Variables with factor loadings under 0.7 were +deleted. A rule of thumb in confirmatory factor analysis suggests that variables with factor +loadings under |0.7| must be dropped (Malkanthie, 2015). As a result, only one external +variable, which is related to trust (T2), remained in the model (Table no. 2). Perceived +Usefulness (PU) was measured by three variables (PU1, PU2 and PU3), whereas Perceived +Ease of Use was made up of two variables (PEU2 and PEU3), and Behavioural Intention +became unidimensional (B1). The attitude was composed of three variables (ATT1, ATT2 +and ATT3). The nested model, which is theoretically consistent with the research goals, +contains eight hypotheses: + H1: Attitude has a positive effect on behavioural intention. + H2: Perceived usefulness positively affects behavioural intention. + H3: Perceived usefulness has a positive effect on attitude. + H4: Perceived ease of use positively influences attitude. + H5: Perceived ease of use positively influences perceived usefulness. + H6: Perceived ease of use has a positive impact on trust. + H7: Trust has a positive effect on perceived usefulness. + H8: Trust positively influences attitude. + +Figure no. 3. The nested model + + +PU1PU2PU3 +PU +H7 +H3 +H2 +er +H8 +H1 +T2 +ATT +BI1 +H5 +H4 +H6 +ATT3 +TATT2 +[ATT1 +PEU +PEU2PEU3AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +164 +Amfiteatru Economic +3.1. Validity +To investigate the extent to which a set of items reflect the theoretical latent-construct they +are designed to measure, both convergent and discriminant validity were checked. +Convergent validity suggests that the variables of a factor that are theoretically related are +expected to correlate highly. According to the Fornell-Larcker criterion for convergent +validity, the Average Variance Extracted (AVE) should be greater than 0.5. According to +the Hair, et al. (1998) criteria, AVE should be greater than 0.5, standardised factor loading +of all items should be above 0.5, and composite reliability should be above 0.7. +In the nested measurement model, each factor loading was above .84 (Table no. 2). +Table no. 2. Summary of means, standard deviations, normality, +validity and reliability measures +Cons- +truct Measurement Instrument Mean STD Z Skew +Z +Kurt +Loa- +ding +α +AVE +CR +Perceived +Usefulness +PU1. The use of AI in +retail (shopping ads and +webshops) allows me to +find the best deals. +4.68 +1.53 +-4.06 +-1.40 +0.85 +0.91 +0.76 +0.91 +PU2. The use of AI in +retail enhances my +effectiveness in +purchasing. +4.67 +1.63 +-4.56 +-1.87 +0.89 +PU3. The use of AI in +retail is useful to me. +4.73 +1.69 +-4.16 +-2.70 +0.89 +Perceived Ease +of Use +PEU2. Shopping does not +require a lot of my mental +efforts if supported by AI +(alternatives are offered by +AI). +5.15 +1.62 +-6.38 +-0.59 +0.90 +0.88 +0.81 +0.9 +PEU3. Shopping is not so +complicated if AI offers +products to me. +5.06 +1.64 +-6.44 +-0.68 +0.90 +Trust +T2. I trust in apps and +webshops that use AI. +4.11 +1.62 +-2.00 +-2.90 +1.00 +1 +n.a. +n.a. +Attitude +ATT1 Shopping in a +webshop/shopping app that +is powered by AI is a good +idea +5.02 +1.63 +-4.99 +-1.58 +0.90 +0.9 +0.79 +0.92 +ATT2 Shopping in a +webshop/shopping app that +is powered by AI is a wise +idea +4.23 +1.62 +-1.60 +-2.39 +0.86 +ATT3 I am positive +towards webshop/shopping +app that is powered by AI +4.72 +1.70 +-4.11 +-1.87 +0.90 + +Artificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +165 +Cons- +truct Measurement Instrument Mean STD Z Skew +Z +Kurt +Loa- +ding +α +AVE +CR +Behavioural +Intention +BI1 I intend to visit +webshops and use +shopping apps that are +powered by AI more +frequently. +3.35 +1.78 +2.13 +-3.93 +1.0 +1 +n.a. +n.a. +Notes: STD=Standard Deviation, Z Skew=Z score for skewness, Z Kurt=Z score for +Kurtosis, α=Cronbach's alpha, AVE=Average Variance Extracted, CR=Composite +Reliability, N=439. + +Moreover, all AVE scores were also well above the threshold level (AVE (ATT)=0.79; +AVE (PU)=.76 and AVE (PEU)=0.81), and all CR scores exceeded 0.7 (CR (PU)=.91; CR +(PEU)=0.90 and CR (ATT)=0.92). Therefore, the model meets both the Fornell-Larcker +(1981) criterion and the Hair et al. (1998) criteria for convergent validity, so the internal +consistency of the model is acceptable. +To assess discriminant validity, i.e. the extent to which a construct is truly distinct to other +constructs, AVEs were compared with squared inter-construct correlations (SIC). AVE +scores higher than SIC scores indicate that discriminant validity is acceptable (ATT +AVE=0.79, SIC1=0.61 and SIC2=0.32; PU AVE=0.76, SIC1=0.40 and SIC2=0.61; PEU +AVE=0.81, SIC1=0.40 and SIC2=0.32). Discriminant validity was also confirmed by +investigating correlations among the constructs. Since there were no correlations above .85, +which is a threshold limit of poor discriminant validity in structural equation modelling +(David, 1998), results also confirmed adequate discriminant validity (PEU*T2=0.52; +PEU*PU=0.64; +PEU*ATT=0.57; +PEU*BI1=0.43; +T2*PU=0.73; +T2*ATT=0.74; +T2*BI1=0.53; PU*ATT=0.78; PU*BI1=0.64; ATT*BI1=0.66). +3.2. Reliability +To test the accuracy and consistency of the nested model, three reliability tests were used: +Cronbach's alpha (α), the Average Variance Extracted index (AVE) and Composite +Reliability (CR). The threshold value for an acceptable Cronbach's alpha is .70 (Cronbach, +1951). The measurement model is acceptable if all estimates are significant and above 0.5 +or 0.7 ideally; AVEs for all constructs are above 0.5 (Forner and Larcker, 1981); and +finally, CRs for all constructs are above 0.7 (Malkanthie, 2015). Table no. 2 shows that the +calculated Cronbach's alphas of all constructs were at least .87 or higher, and the AVE +scores were also higher than 0.76, as well as the CRs were above 0.9; therefore, the +reliability of the measurement model is optimal. +3.3. Model fit +Absolute- and relative model fits were tested. Each absolute measure was significant and +indicated a good fit. Although Chi-square statistics are sensitive to large sample size and +assume a multivariate normal distribution (Kelloway, 1998), even those measures were +acceptable. However, other model fit indexes are better to consider as criteria. Therefore, +the goodness-of-fit index (GFI), the adjusted goodness-of-fit index (AGFI), the root mean +squared error of approximation (RMSEA) and the standardised root mean squared residual + +AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +166 +Amfiteatru Economic +(SRMR) were also examined. All of them indicated a good absolute model fit. (Absolute +measures: Chi square=34.154 (DF=29); Probability level=0.23; CMIN/DF=1.18; +GFI=0.98; AGFI=0.96; RMSEA=0.02; SRMR=0.04). As far as the relative model fit is +concerned, TLI or NNFI, GFI, AGFI, NFI, IFI, CFI and Critical N (CN or HOELTER) +were calculated. All but CN range from zero to one. Values exceeding .9 show an +acceptable fit, above .95 a good fit (Bentler and Bonnet, 1980). CN (HOELTER), which +favours large samples over small ones (Bollen, 1990), is an improved method for +investigating model fit (Hoelter, 1983). CN should be above 200 to indicate a good model +fit. (Relative measures: TLI/NNFI=0.98; GFI=0.98; AGFI=0.96; NFI=0.93; IFI=0.99; +CFI=0.99 and HOELTER (CN)=546). The results of the absolute and relative model fit test +confirmed that the structural model is acceptable and suitable for the analysis and +interpretation of the parameter estimates. Therefore, it can be concluded that the technology +acceptance model is suitable for investigating consumer acceptance of the use of artificial +intelligence in online shopping, which is the answer to the first research question (R1). +3.4. Hypothesis testing and estimates +Because of the non-normality of the variables in the nested model, the asymptotically +distribution-free (ADF) method was used to estimate parameters in AMOS. ADF calculates +the asymptotically unbiased estimates of the chi-square goodness-of-fit test, the parameter +estimates, and the standard errors. The limitation of ADF is that it needs a large sample size +(Bian, 2012), which criterion was met in this study (N=439). Skewness and Kurtosis z- +values of the variables were out of the range of the normal distribution that is -2 and +2 +(George and Mallery, 2010). Moreover, the p values of the variables were significant +(p=.000) in the Shapiro-Wilk and Kolmogorov-Smirnov tests, which also confirmed non- +normality. +To address the second research question (R2) and to determine the key factors influencing +behavioural intention to use AI-powered webshops and apps, hypotheses were tested in the +structural model (Table no. 3). +Table no. 3. Direct, indirect, total effects and hypothesis testing +Hypothesis +Relationship +P +St. direct eff. +St. indirect eff. +St. total eff. +Result +H1 +BI1 ← ATT +*** +0.41 +0.00 +0.41 +accepted +H2 +BI1 ← PU +*** +0.32 +0.19 +0.51 +accepted +H3 +ATT ← PU +*** +0.48 +0.00 +0.48 +accepted +H4 +ATT←PEU +0.1 +0.09 +0.48 +0.57 +rejected +H5 +PU ← PEU +*** +0.35 +0.28 +0.64 +accepted +H6 +T2 ← PEU +*** +0.52 +0.00 +0.52 +accepted +H7 +PU ← T2 +*** +0.55 +0.00 +0.55 +accepted +H8 +ATT ← T2 +*** +0.35 +0.26 +0.61 +accepted +The arrows linking constructs represent hypotheses in the direction of arrows in the nested +model (Figure no. 3 and Figure no. 4). Asterisks signal statistically significant relations +between constructs. Gamma estimates were calculated from exogenous construct to +endogenous construct, and beta estimates between two endogenous constructs. Figure no. 4 +shows the standardised estimates, loadings and residuals regarding the relationships + +Artificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +167 +between constructs and observed indicators. A hypothesis was accepted if the presence of a +statistically significant relationship in the predicted direction was confirmed. +As Table no. 3 shows, all hypotheses were accepted except for H4. So, the present findings, +except for the relationship between perceived ease of use and attitude, are consistent with +the Technology Acceptance Model proposed by Davis (1986). Surprisingly, perceived ease +of use (PEU) was found to have no direct, significant effect on attitude (ATT), which is not +in agreement with the original TAM (H4 rejected). This discrepancy could be attributed to +the fact that shopping is not too complicated in AI-powered webshops, and it does not +require too much mental effort. However, this slightly unexpected result coincides with the +findings of a previous research by Ha and Stoel (2008), who examined the effect of PEU on +attitude towards online shopping. +In this study, with H5 and H6 accepted, perceived ease of use (PEU) was found to have a +significant, direct, positive impact on both the perceived usefulness (PU) and trust (T2). It +suggests that the easier it is for a consumer to use an AI-powered webshop, the higher level +of customer trust and perceived usefulness can be expected. Consumers trust in AI-powered +shopping apps and stores that are easy to use, and consider those that are too complicated +less useful. Similar results were obtained by Ha and Stoel (2008), who focused on +consumers' acceptance of e-shopping. Gefen, Karahanna and Straub (2003) also found that +perceived ease of use positively affected the perceived usefulness of a B2C website and the +trust in an e-vendor. + +Figure no. 4. Parameter estimates of the nested model +Trust in AI-powered webshops has a central role in forming attitudes and perceived +usefulness. Similar to what Gefen, Karahanna and Straub (2003), and Ha and Stoel (2008) +found, trust directly influenced perceived usefulness (H7 accepted). Moreover, trust also +impacted attitude (H8 accepted), in line with the research findings of Ha and Stoel (2008). + +,72 +,78 +,78 +PU1 +PU2 +PU3 ++ +,85 +488 +89 +62 +PU +32 +,55 +48 +68 +,47 +,35 +ATT +,41 +T2 +BI1 +,35 +,90 +52 +,86 +,90 +,09 +81 +,74 +,81 +00 +ATT3 +ATT2 +ATT1 +PEU +e11 +90 +,90 +,81 +,82 +PEU2 +PEU3 +eg +e10AE +Consumer Acceptance of the Use of Artificial Intelligence +in Online Shopping: Evidence From Hungary + +168 +Amfiteatru Economic +The strongest direct effect was found between trust and perceived usefulness (H7 +accepted). It suggests that the more we trust in Artificial Intelligence during the online +shopping journey, the more likely it is that we consider AI-powered apps and webshops +useful. Besides, a higher level of trust forms a more positive attitude towards shopping in +such webshops. Perceived usefulness has a central role in this model as it (PU) significantly +impacted attitude (H3 accepted) and behavioural intention (H2 accepted). The more useful +we find the use of artificial intelligence in online shopping believing that it allows us to +grab the best deals, the more likely we are to consider it a wise decision to do the shopping +in AI-powered webshops and apps more frequently. Not surprisingly, attitude towards AI- +powered webshops and apps was found to have a strong, significant, positive direct impact +on behavioural intention (H1 accepted). It suggests that forming consumers’ attitude plays a +vital role in increasing the traffic of AI-powered webshops and apps (Figure no. 4). +Although there was no significant direct relationship between perceived ease of use and +attitude, the indirect effect of PEU on attitude (PEU->ATT=0.48) was quite strong, similar +to its indirect impact on behavioural intention (PEU->BI1=0.43). Also, trust was found to +indirectly influence behavioural intention (T2->BI1=0.42). It suggests that if shopping +requires much mental effort and seems to be complicated in AI-powered webshops and +apps, consumers tend to form stronger negative attitudes towards them and also tend to +trust them less, which will result in weaker consumer intention to visit such webshops. +In the nested model perceived usefulness had the highest total effect on behavioural +intention. Therefore, AI-powered webshops and apps are advised to increase the level of +perceived usefulness to succeed by enabling customers to maximise purchase effectiveness +to grab the best deals, i.e. the ideal product with the highest utility. + +Conclusions +This research extends our knowledge of consumer acceptance of the use of artificial +intelligence in online shopping in many aspects. The widely used technology acceptance +model (TAM) was proved to be suitable for investigating consumer acceptance of the use +of artificial intelligence in online shopping. +As expected, it was confirmed in the nested model that the key factors influencing +consumer’ behavioural intention to use AI-powered webshops and apps are trust, perceived +usefulness, perceived ease of use and attitudes. In contrast to the original TAM (Davis, +1986), the direct relationship between perceived ease of use and attitudes was insignificant. +Nevertheless, it does not mean that user-friendliness of a webshop is not crucial as +perceived ease of use indirectly affects attitude and the behavioural intention. Instead, user- +friendliness and flawless operation of an artificial intelligence-powered website are the +prerequisites for market success. +Building trust has a central role in consumer acceptance of the use of artificial intelligence +in online shopping. If consumers do not trust in an AI-powered webshop/app, they tend to +consider it less useful and form a negative attitude towards it, which will result in less +online traffic. Also, AI must provide online consumers with tailor-made offerings to grab +the best deals, i.e. products with the highest value; and it is expected to shorten the product +search time to enhance shopping effectiveness. Not surprisingly, the favourable attitude + +Artificial Intelligence in Wholesale and Retail +AE + +Vol. 23 • No. 56 • February 2021 +169 +towards AI-powered webshops leads to more frequent online traffic in such electronic +stores. +Considering the strong positive impact of the recent COVID-19 crisis on e-commerce, the +use of artificial intelligence in online shopping is expected to expand further. According to +Bloomberg (2020) the pandemic lockdowns have a dual effect on consumer behaviour on +the development of AI. Nowadays, it is more important than ever to create a personalised +customer journey, to meet customers' demand and to provide a greater online shopping +experience. In these efforts, artificial intelligence can be a very effective tool, which was +confirmed by the research findings of this paper. +This study has several practical applications. It is useful for webshop owners and online +marketing managers to understand how consumers adapt to the new technology, i.e. the use +of artificial intelligence in online shopping. It is also beneficial to academics and +researchers who are interested in the adaptation of the Technology Acceptance Model in +online shopping. Those who are interested in the role of trust in consumer choices in the +online environment will also benefit from this study. +As far as the future research directions are concerned, it would be advisable to repeat this +study in a multi-cultural context. It might also be useful to test the model of the Technology +Readiness Index proposed by Parasuraman (2000) and to compare the results presented +here with the new findings. + +Acknowledgements +“The described article/presentation/study was carried out as part of the EFOP-3.6.1-16- +2016-00011 “Younger and Renewing University – Innovative Knowledge City – +institutional development of the University of Miskolc aiming at intelligent specialisation” +project implemented in the framework of the Szechenyi 2020 program. The realization of +this project is supported by the European Union, co-financed by the European Social +Fund.” + +References +Ajzen, I., 1985. The Theory of Planned Behavior. Organisational Behavior and Human +Decision Processes, 50, pp.179-211. +André, Q., Carmon, Z., Wertenbroch, K., Crum, A., Frank, D., Goldstein, W., Huber, J., +Boven, L., Weber, B. and Yang, H., 2017. Consumer Choice and Autonomy in the Age +of Artificial Intelligence and Big Data. 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Journal of Retailing and Consumer Services, 17(2), +pp.89-96. + diff --git a/INAzT4oBgHgl3EQfU_y9/content/tmp_files/load_file.txt b/INAzT4oBgHgl3EQfU_y9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8757a41a4fbaed3177156b066412167d1a4f9af --- /dev/null +++ b/INAzT4oBgHgl3EQfU_y9/content/tmp_files/load_file.txt @@ -0,0 +1,1060 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf,len=1059 +page_content='Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 155 CONSUMER ACCEPTANCE OF THE USE OF ARTIFICIAL INTELLIGENCE IN ONLINE SHOPPING: EVIDENCE FROM HUNGARY Szabolcs Nagy1* and Noémi Hajdú2 1)2) University of Miskolc, Miskolc, Hungary Please cite this article as: Nagy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' and Hadjú, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Amfiteatru Economic, 23(56), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='155-173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='24818/EA/2021/56/155 Article History Received: 30 September 2020 Revised: 7 November 2020 Accepted: 26 December 2020 Abstract The rapid development of technology has drastically changed the way consumers do their shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The volume of global online commerce has significantly been increasing partly due to the recent COVID-19 crisis that has accelerated the expansion of e-commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' A growing number of webshops integrate Artificial Intelligence (AI), state-of-the-art technology into their stores to improve customer experience, satisfaction and loyalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, little research has been done to verify the process of how consumers adopt and use AI-powered webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Using the technology acceptance model (TAM) as a theoretical background, this study addresses the question of trust and consumer acceptance of Artificial Intelligence in online retail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' An online survey in Hungary was conducted to build a database of 439 respondents for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' To analyse data, structural equation modelling (SEM) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' After the respecification of the initial theoretical model, a nested model, which was also based on TAM, was developed and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The widely used TAM was found to be a suitable theoretical model for investigating consumer acceptance of the use of Artificial Intelligence in online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Trust was found to be one of the key factors influencing consumer attitudes towards Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Perceived usefulness as the other key factor in attitudes and behavioural intention was found to be more important than the perceived ease of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' These findings offer valuable implications for webshop owners to increase customer acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Keywords: consumer acceptance, artificial intelligence, online shopping, AI-powered webshops, technology acceptance model, trust, perceived usefulness, perceived ease of use, attitudes, behavioural intention, Hungary JEL Classification: L81, M31, O30 Corresponding author, Szabolcs Nagy – e-mail: nagy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='szabolcs@uni-miskolc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='hu AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 156 Amfiteatru Economic Introduction The rapid development of digital technology has changed online shopping (Daley, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In recent years, the use of Artificial Intelligence (AI) in online commerce has been increased since AI is an excellent tool to meet rapidly changing consumer demand and to increase sales efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The global spending by retailers on AI services is expected to quadruple and reach $12 billion by 2023, and over 325000 retailers will adopt AI technology (Maynard, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Smidt and Power (2020) claimed that online product research has significantly increased over the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" USA's largest online retailer, Amazon, is the exemplary case of how to effectively integrate AI into online retail." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Besides the rich assortment, fast delivery and competitive prices, a more localised shopping journey can be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Thus Amazon can use location-specific pricing and send destination-specific messages to its customers, who will pay in their local currency (Barmada, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Novel marketing techniques supported by new technologies, including the use of AI systems spark the proliferation of new marketing methods to effectively reach target consumers and to offer enhanced consumer experiences (Pusztahelyi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Pursuant to Asling (2017), the use of AI in online shopping makes customer-centric search and a new level of personalisation possible resulting in a more efficient sales process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Information technology (IT) has changed the nature of company-customer relationships (Rust and Huang, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, any technology-driven transformation is based on trust (Pricewaterhouse Coopers, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Online retailers need more in-depth insight into how consumers perceive and accept the use of AI in webshops and how much they trust them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' They also need to know how to use AI most effectively to increase online spending and online purchase frequency since the importance of time and cost efficiency in shopping has recently become more and more critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In this regard, online shopping means a convenient way for customers to buy the desired products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' So far, only a few researchers have addressed the question of trust and consumer acceptance of AI in online retail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Based on the technology acceptance model (TAM), this study aims to fill this research gap and proposes an integrated theoretical framework of consumers' acceptance of AI-powered webshops." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Further objectives of this paper are to investigate the relationships between the elements of TAM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' to analyse the effects of trust, perceived usefulness and perceived ease of use on attitudes and behavioural intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' After reviewing the use of AI in online shopping, this paper discusses the role of trust in online shopping and presents the technology acceptance model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The next section deals with the research methodology, including the research questions, hypotheses and the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In the results and discussion section, the validity and reliability of the model, as well as the model fit are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Hypothesis testing, detailed analysis of the relationships between the elements of the nested model, and comparison of the results with the previous research findings are also discussed here before the conclusions sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Literature review According to IBM's U." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Retail Index, the COVID-19 has speeded up the change from traditional shopping to online purchasing by circa five years (Haller, Lee and Cheung, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Due to the pandemic situation, there is an increased demand for AI in the retail industry (Meticulous Market Research, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 157 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in online shopping AI systems are a set of software and hardware that can be used to continuously assess and analyse data to characterise environmental factors and to determine decisions and actions (European Commission, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Prior research mainly focused on the advantages of the use of AI in online settings and failed to address how consumers accept AI in online retail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to utility theory, this new technology helps consumers to find and choose the best product alternatives, while decreases the search cost and search time (Pricewaterhouse Coopers, 2018), thus increasing utility (Stigler, 1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Bakos, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Stigler and Becker, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' André, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Lynch and Ariely, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI filters the information for each target customer and provides what exactly is needed (Paschen, Wilson and Ferreira, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI supports automating business processes, gains insight through data analysis, and engages with customers and employees (Davenport and Ronanki, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Artificial intelligence is widely used to increase the efficiency of marketing (Kwong, Jiang, and Luo, 2016) and retail (Weber and Schütte, 2019) and to automate marketing (Dumitriu and Popescu, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI-powered online stores provide their customers with automated assistance during the consumer journey (Yoo, Lee and Park, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Pantano and Pizzi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It is a great advantage, especially for the elder people, who are averse to technical innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Consumers' online information search and product selection habits can be better understood by AI to offer a more personalised shopping route (Rust and Huang, 2014)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It is a great opportunity for online shops to analyse the profile of existing and potential customers and thereby suggest tailor-made marketing offerings for them (Onete, Constantinescu and Filip, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI also makes the contact with both the customers and the employees continuous and interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Frequently asked questions (FAQs) regarding the products, product-use and ordering process can be automated by a chatbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' New sales models use automated algorithms to recommend unique, personalised marketing offerings, thus increasing customer satisfaction and engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' To sum up the advantages, AI systems operate automatically and analyse big data in real-time to interpret and shape consumer behavioural patterns to offer products and services in a personalised way, thus enhancing the shopping experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, AI systems also have some disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' They work most effectively with big data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' therefore, the implementation of AI systems requires huge investments (Roetzer, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The role of trust in online shopping Trust is of great importance in online commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" According to Kim, Ferrin and Rao (2008), consumer confidence has a positive effect on a consumer's intention to buy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The higher the consumer trust in an online shop is, the more likely the consumer will be to go through the buying process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Trust is especially crucial when the customer perceives a financial risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Thatcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (2013) identified two types of trust: general and specific trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' General trust concerns the e-commerce environment, consumer beliefs about and attitudes towards it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Specific trust is related to the shopping experience in a specific virtual store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Confidence can be enhanced through interactive communication between the retailer and the buyer by using appropriate product descriptions and images to reduce the perceived risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' As stated in Cătoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (2014) there is a strong negative correlation between perceived risks and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to Reichheld and Schefter (2000, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 107), “price does not rule the Web;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' trust does”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 158 Amfiteatru Economic Aranyossy and Magisztrák (2016) found that a higher level of e-commerce trust was associated with more frequent online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, when shopping online, customers do not necessarily notice that a website uses AI tools (Daley, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' All things considered, AI marks a new era in online sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, continuous technological development such as the use of AI-powered websites divides society, as there are those who accept novelty while others reject it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Technology Acceptance Model (TAM) Consumers' adaptation to new technologies can be explained by several models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Dhagarra, Goswami and Kumar (2020) summarised them as follows: (1) Theory of Reasoned Action (TRA) by Fishbein and Ajzen (1975);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (2) Theory of Planned Behaviour (TPB) by Ajzen (1985);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (3) Technology Acceptance Model (TAM) by Davis (1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (4) Innovation Diffusion Theory (IDT) by Rajagopal (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (5) Technology Readiness Index (TRI) by Parasuraman, (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' and (6) Unified Theory of Acceptance and Use of Technology (UTAUT) by Venkatesh, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Technology acceptance model (TAM), an extension of (TRA), is one of the most widely- used theoretical models (Venkatesh, 2000) to explain why an IT user accepts or rejects information technology and to predict IT user behaviour (Legris, Ingham, and Collerette, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The original TAM contains six elements: external variables, perceived usefulness, perceived ease of use, attitude, behavioural intention to use and actual use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to TAM, external variables have a direct influence on perceived usefulness (PU) and perceived ease of use (PEU), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' the two cognitive belief components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Perceived ease of use directly influences PU and attitude, whereas perceived usefulness has a direct impact on attitude and behavioural intention to use, which affects actual use (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The original technology acceptance model (TAM) Source: Davis, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Ha and Stoel (2008) examined the factors affecting customer acceptance of online shopping and found that perceived ease of use, perceived trust and perceived shopping enjoyment had the greatest impact on customer acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Ease of use, trust and shopping enjoyment had a significant impact on perceived usefulness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' trust, shopping enjoyment, and usefulness Perceived Usefulness External Attitude Behavioral Variables Towards Intentionto Actual Use Use Use Perceived Easeof UseArtificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 159 had a significant effect on attitude towards online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' They also found that attitude and perceived usefulness had an influential role in consumer intention to purchase online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to Vijayasarathy (2004), there is a positive association between consumer attitude towards online shopping and the beliefs concerning usefulness, compatibility, security and ease of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Also, the intention to purchase online is strongly influenced by consumer beliefs about online shopping, self-efficacy and attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Surprisingly, no positive relationship between purchasing intention and consumer beliefs about the usefulness of online shopping was reported (Vijayasarathy, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Gefen, Karahanna and Straub (2003) found that perceived usefulness and perceived ease of use influence consumer repurchase intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It must be noted that Schepman and Rodway (2020) expressed some criticisms about the applicability of TAM to measure attitudes towards AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to them, it is the online retailers that can decide to integrate AI into webshops, and consumers have no choice but to use it when shopping online in such stores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, traditional technology acceptance models might not be ideal to measure attitudes towards AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, we are convinced that consumers still have the free will to decide whether to use new technology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' to shop online in an AI-powered webshop, or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Methodology and research questions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Methodology The constructs and the measurement instruments presented in Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1 were developed based on the literature review, and according to the Technology Acceptance Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Variables with asterisk and in italics were adapted from Park (2009), the others were adapted from Hu and O'Brien (2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, each variable was modified by the authors to make it possible to measure the perceived role of AI in online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' For data collection, a questionnaire made up of 26 questions (variables) was used (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Additionally, six demographics variables - gender, education, age, occupation, place of residence and internet subscription - were also included in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' All measurement instruments were listed in Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1 but the demographics variables were measured on a seven-point Likert-scale ranging from strongly disagree (1) strongly agree (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In the very first section of the questionnaire, respondents were provided with a detailed explanation of AI-powered webshops and shopping apps, which are online stores where shopping is supported by artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI-powered webshops present personalised product/service offerings based on previous search patterns and purchases that we made before, and automatically display products that AI chooses for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Also, AI offers similar products to those that were originally viewed but were not available in the right size (product recommendation based on visual similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Another typical sign of an AI- powered webshop is that when the customer is leaving the web store, AI warns about the products left in the cart, to complete the purchase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI-powered webshops often use chatbots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' a virtual assistant is available if the customer has any questions, and visual (image-based) search is also possible: after uploading a product picture, AI recommends the most similar ones to that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Virtual changing rooms, voice recognition and automatic search completion are also available in AI-powered webshops such as Amazon, e-Bay, Alibaba, AliExpress, GearBest, eMAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='hu, PCland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='hu, Ecipo, Bonprix, Answear, Reserved, Fashiondays, Fashionup, Spartoo, Orsay, to mention just a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 160 Amfiteatru Economic Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Constructs and measurement instruments Construct Definition Measurement Instruments Perceived Usefulness (PU) The degree to which a consumer believes that AI used in online shopping would make his or her purchases more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in retail (shopping ads and webshops) allows me to find the best deals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in retail enhances my effectiveness in purchasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in retail is useful to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU4 The use of AI in retail saves time for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' * Perceived Ease of Use (PEU) The degree to which a consumer believes that using AI in webshops will be free of effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AI-powered shopping apps and webshops are easy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Shopping does not require a lot of my mental efforts if supported by AI (alternatives are offered by AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Shopping is not so complicated if AI offers products to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU4 Learning how to use AI-powered shopping apps and webshops is easy for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" * PEU5 It is easy to become skilful at using AI- powered shopping apps and webshops* Experience (EXP) The consumers' knowledge about and the experience with purchasing in an AI- powered webshop." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' EXP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" I'm experienced in online shopping." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' EXP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' I have already used AI-powered applications (chatbots, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=') Trust (TRUST) The subjective probability with which people believe that AI works for their best interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' I am convinced that AI in retail is used to provide customers with the best offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' I trust in apps and webshops that use AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Subjective Norm (SN) The degree to which a consumer perceives that most people who are important to him or her think he or she should or should not make purchases in AI-powered webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' SN1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' People who influence my behaviour would prefer me to use AI-powered shopping apps and webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' SN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' I like using AI-powered webshops and shopping apps based on the similarity of my values and the social values underlying its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' * Task Relevance (TR) The degree to which a consumer believes that AI-powered webshops are applicable to his or her shopping task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' TR1 I think AI can be used effectively in webshops and shopping apps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Compen- sation (COMP) The degree to which a consumer believes that he or she has the ability to make purchases in AI- powered webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' I would prefer AI-powered shopping apps and webshops… C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' if there was no one around to visit physical shops/shopping malls with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' if I had less time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' if I had a built-in help facility for assistance when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Perceived Quality PQ The degree of how good a consumer perceives the quality of a product in AI- powered webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PQ1 AI finds/offers better products for me than I could.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 161 Construct Definition Measurement Instruments Perceived Enjoyment PE The extent to which shopping in AI-powered webshops is perceived to be enjoyable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PE1 Shopping is more fun, enjoyable when AI helps me to find the best-suited products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Attitude ATT The consumer's attitude towards shopping in AI- powered webshops." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" ATT1 Shopping in a webshop/shopping app that is powered by AI is a good idea ATT2 Shopping in a webshop/shopping app that is powered by AI is a wise idea ATT3 I am positive towards webshop/shopping app that is powered by AI Behavioural Intention BI A consumer's behavioural intention to do the shopping in AI-powered webshops." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' BI1 I intend to visit webshops and to use shopping apps that are powered by AI more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" BI2 I'm willing to spend more on products offered by webshops and apps powered by AI Sources: Adapted from Hu and O'Brien, 2016;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' *Park, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' An online survey in Google Form was conducted to collect data in July and August 2020 in Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Because of the Theory Acceptance Model, previous online shopping experience with AI-powered webshops was the one and only eligibility criterion for respondents to participate in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Convenience sampling method was used to reach the maximum number of respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Data was migrated from Google Form to MS Excel, SPSS 24 and AMOS, and was checked for coding accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The database was complete and contained no missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Descriptive statistical analyses were done in SPSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AMOS was employed to test the hypotheses and the theoretical model by structural equation modelling (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Research questions and hypotheses Based on the literature review, this study aims to address the following research questions respectively: R1: Can the technology acceptance model (TAM) be used for investigating consumer acceptance of the use of artificial intelligence in online shopping?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' R2: If so, what are the key factors influencing behavioural intention to visit AI- powered webshops and apps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Based on the Technology Acceptance Model, an initial theoretical model was developed (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The arrows that link constructs (latent variables such as COMP, EXP, TRUST, SN, PEU, PU, ATT, BI) represent hypothesised causal relationships (hypotheses) in the direction of arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' One of the objectives of this study is to test those hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Error terms for all observed indicators are indicated by e1 to e35, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 162 Amfiteatru Economic Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The initial theoretical model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The sample A sample size of 200 is an appropriate minimum for SEM in AMOS (Marsh, Balla, and MacDonald, 1988), and a minimum of 10-20 subjects per parameter estimates in the model are optimal (Schumacker and Lomax, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, the ideal sample size is between 380 and 760, considering the number of parameter estimates (38) in the initial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The actual sample size of 439 respondents fits into this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Of the sample of 439 respondents, 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='2% were female, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='8% male.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Their average age was 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='2 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 60,8% of the respondents had tertiary education, 38% had secondary education, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='2% had primary education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Most respondents resided in county seats (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='6%);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' the rest lived in other towns/cities (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1%), villages (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='8%) and the capital (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Most respondents were subscribed to both mobile and wired internet services (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='3%), while 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7% had only mobile internet, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1% had only wired internet services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='9% of respondents had not got any subscription to internet services (wired or mobile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' There is no data available on the distribution of the e-shoppers in Hungary, therefore, it is impossible to tell if this sample reflects the characteristics of the e-shoppers’ population in Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Results and discussion The initial model (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2), which proved to be too complex and did not fit the current data (CMIN/DF=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='72;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' GFI=,693;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' CFI=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='723;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' RMSEA=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='124;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' HOELTER 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5= 65), was absolutely rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, it was not appropriate to interpret any individual C2 COMP PEU1PEU2PEU3PEU4PEU5 PEU e24 EXP EXP EXP ATT1ATT2ATT3 BI2 TRUST SN SN PU P2 P3 PU4Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 163 parameter estimates, and further model modifications were required to obtain a better- fitting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Respecification of the initial model led to a nested model that fitted well and is discussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' During the respecification, the alternative model approach was used (Malkanthie, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' To test the model, the same data set was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Several modified models were developed, and out of the theoretically justifiable models, the model with the best data fit was selected (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3) as suggested by Mueller and Hancock (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The respecification process was started with testing the measurement model by a series of Principal Component Analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Variables with factor loadings under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7 were deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' A rule of thumb in confirmatory factor analysis suggests that variables with factor loadings under |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7| must be dropped (Malkanthie, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' As a result, only one external variable, which is related to trust (T2), remained in the model (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Perceived Usefulness (PU) was measured by three variables (PU1, PU2 and PU3), whereas Perceived Ease of Use was made up of two variables (PEU2 and PEU3), and Behavioural Intention became unidimensional (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The attitude was composed of three variables (ATT1, ATT2 and ATT3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The nested model, which is theoretically consistent with the research goals, contains eight hypotheses: H1: Attitude has a positive effect on behavioural intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H2: Perceived usefulness positively affects behavioural intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H3: Perceived usefulness has a positive effect on attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H4: Perceived ease of use positively influences attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H5: Perceived ease of use positively influences perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H6: Perceived ease of use has a positive impact on trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H7: Trust has a positive effect on perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' H8: Trust positively influences attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The nested model PU1PU2PU3 PU H7 H3 H2 er H8 H1 T2 ATT BI1 H5 H4 H6 ATT3 TATT2 [ATT1 PEU PEU2PEU3AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 164 Amfiteatru Economic 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Validity To investigate the extent to which a set of items reflect the theoretical latent-construct they are designed to measure, both convergent and discriminant validity were checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Convergent validity suggests that the variables of a factor that are theoretically related are expected to correlate highly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to the Fornell-Larcker criterion for convergent validity, the Average Variance Extracted (AVE) should be greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to the Hair, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (1998) criteria, AVE should be greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5, standardised factor loading of all items should be above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5, and composite reliability should be above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In the nested measurement model, each factor loading was above .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='84 (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Summary of means, standard deviations, normality, validity and reliability measures Cons- truct Measurement Instrument Mean STD Z Skew Z Kurt Loa- ding α AVE CR Perceived Usefulness PU1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in retail (shopping ads and webshops) allows me to find the best deals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='91 PU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in retail enhances my effectiveness in purchasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='89 PU3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The use of AI in retail is useful to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='89 Perceived Ease of Use PEU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Shopping does not require a lot of my mental efforts if supported by AI (alternatives are offered by AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='9 PEU3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Shopping is not so complicated if AI offers products to me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='64 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='90 Trust T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' I trust in apps and webshops that use AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Attitude ATT1 Shopping in a webshop/shopping app that is powered by AI is a good idea 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='92 ATT2 Shopping in a webshop/shopping app that is powered by AI is a wise idea 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='86 ATT3 I am positive towards webshop/shopping app that is powered by AI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='90 Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 165 Cons- truct Measurement Instrument Mean STD Z Skew Z Kurt Loa- ding α AVE CR Behavioural Intention BI1 I intend to visit webshops and use shopping apps that are powered by AI more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='93 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='0 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Notes: STD=Standard Deviation, Z Skew=Z score for skewness, Z Kurt=Z score for Kurtosis, α=Cronbach's alpha, AVE=Average Variance Extracted, CR=Composite Reliability, N=439." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Moreover, all AVE scores were also well above the threshold level (AVE (ATT)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='79;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AVE (PU)=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='76 and AVE (PEU)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='81), and all CR scores exceeded 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7 (CR (PU)=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' CR (PEU)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='90 and CR (ATT)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, the model meets both the Fornell-Larcker (1981) criterion and the Hair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (1998) criteria for convergent validity, so the internal consistency of the model is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' To assess discriminant validity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' the extent to which a construct is truly distinct to other constructs, AVEs were compared with squared inter-construct correlations (SIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AVE scores higher than SIC scores indicate that discriminant validity is acceptable (ATT AVE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='79, SIC1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='61 and SIC2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU AVE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='76, SIC1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='40 and SIC2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='61;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU AVE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='81, SIC1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='40 and SIC2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Discriminant validity was also confirmed by investigating correlations among the constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Since there were no correlations above .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='85, which is a threshold limit of poor discriminant validity in structural equation modelling (David, 1998), results also confirmed adequate discriminant validity (PEU*T2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='52;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU*PU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU*ATT=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='57;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PEU*BI1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='43;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' T2*PU=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='73;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' T2*ATT=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='74;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' T2*BI1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='53;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU*ATT=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='78;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' PU*BI1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='64;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' ATT*BI1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Reliability To test the accuracy and consistency of the nested model, three reliability tests were used: Cronbach's alpha (α), the Average Variance Extracted index (AVE) and Composite Reliability (CR)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" The threshold value for an acceptable Cronbach's alpha is ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='70 (Cronbach, 1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The measurement model is acceptable if all estimates are significant and above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7 ideally;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AVEs for all constructs are above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='5 (Forner and Larcker, 1981);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' and finally, CRs for all constructs are above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='7 (Malkanthie, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" 2 shows that the calculated Cronbach's alphas of all constructs were at least ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='87 or higher, and the AVE scores were also higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='76, as well as the CRs were above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' therefore, the reliability of the measurement model is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Model fit Absolute- and relative model fits were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Each absolute measure was significant and indicated a good fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Although Chi-square statistics are sensitive to large sample size and assume a multivariate normal distribution (Kelloway, 1998), even those measures were acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, other model fit indexes are better to consider as criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, the goodness-of-fit index (GFI), the adjusted goodness-of-fit index (AGFI), the root mean squared error of approximation (RMSEA) and the standardised root mean squared residual AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 166 Amfiteatru Economic (SRMR) were also examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' All of them indicated a good absolute model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (Absolute measures: Chi square=34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='154 (DF=29);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Probability level=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' CMIN/DF=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' GFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='98;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AGFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='96;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' RMSEA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='02;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' SRMR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' As far as the relative model fit is concerned, TLI or NNFI, GFI, AGFI, NFI, IFI, CFI and Critical N (CN or HOELTER) were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' All but CN range from zero to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Values exceeding .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='9 show an acceptable fit, above .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='95 a good fit (Bentler and Bonnet, 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' CN (HOELTER), which favours large samples over small ones (Bollen, 1990), is an improved method for investigating model fit (Hoelter, 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' CN should be above 200 to indicate a good model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' (Relative measures: TLI/NNFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='98;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' GFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='98;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' AGFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='96;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' NFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='93;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' IFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' CFI=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='99 and HOELTER (CN)=546).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The results of the absolute and relative model fit test confirmed that the structural model is acceptable and suitable for the analysis and interpretation of the parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, it can be concluded that the technology acceptance model is suitable for investigating consumer acceptance of the use of artificial intelligence in online shopping, which is the answer to the first research question (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Hypothesis testing and estimates Because of the non-normality of the variables in the nested model, the asymptotically distribution-free (ADF) method was used to estimate parameters in AMOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' ADF calculates the asymptotically unbiased estimates of the chi-square goodness-of-fit test, the parameter estimates, and the standard errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The limitation of ADF is that it needs a large sample size (Bian, 2012), which criterion was met in this study (N=439).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Skewness and Kurtosis z- values of the variables were out of the range of the normal distribution that is -2 and +2 (George and Mallery, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Moreover, the p values of the variables were significant (p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='000) in the Shapiro-Wilk and Kolmogorov-Smirnov tests, which also confirmed non- normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' To address the second research question (R2) and to determine the key factors influencing behavioural intention to use AI-powered webshops and apps, hypotheses were tested in the structural model (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Direct, indirect, total effects and hypothesis testing Hypothesis Relationship P St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' direct eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' indirect eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' total eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Result H1 BI1 ← ATT *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='41 accepted H2 BI1 ← PU *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='51 accepted H3 ATT ← PU *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='48 accepted H4 ATT←PEU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='57 rejected H5 PU ← PEU *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='64 accepted H6 T2 ← PEU *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='52 accepted H7 PU ← T2 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='55 accepted H8 ATT ← T2 *** 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='61 accepted The arrows linking constructs represent hypotheses in the direction of arrows in the nested model (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3 and Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Asterisks signal statistically significant relations between constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Gamma estimates were calculated from exogenous construct to endogenous construct, and beta estimates between two endogenous constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4 shows the standardised estimates, loadings and residuals regarding the relationships Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 167 between constructs and observed indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' A hypothesis was accepted if the presence of a statistically significant relationship in the predicted direction was confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' As Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 3 shows, all hypotheses were accepted except for H4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' So, the present findings, except for the relationship between perceived ease of use and attitude, are consistent with the Technology Acceptance Model proposed by Davis (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Surprisingly, perceived ease of use (PEU) was found to have no direct, significant effect on attitude (ATT), which is not in agreement with the original TAM (H4 rejected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' This discrepancy could be attributed to the fact that shopping is not too complicated in AI-powered webshops, and it does not require too much mental effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' However, this slightly unexpected result coincides with the findings of a previous research by Ha and Stoel (2008), who examined the effect of PEU on attitude towards online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In this study, with H5 and H6 accepted, perceived ease of use (PEU) was found to have a significant, direct, positive impact on both the perceived usefulness (PU) and trust (T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It suggests that the easier it is for a consumer to use an AI-powered webshop, the higher level of customer trust and perceived usefulness can be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Consumers trust in AI-powered shopping apps and stores that are easy to use, and consider those that are too complicated less useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Similar results were obtained by Ha and Stoel (2008), who focused on consumers' acceptance of e-shopping." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Gefen, Karahanna and Straub (2003) also found that perceived ease of use positively affected the perceived usefulness of a B2C website and the trust in an e-vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Parameter estimates of the nested model Trust in AI-powered webshops has a central role in forming attitudes and perceived usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Similar to what Gefen, Karahanna and Straub (2003), and Ha and Stoel (2008) found, trust directly influenced perceived usefulness (H7 accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Moreover, trust also impacted attitude (H8 accepted), in line with the research findings of Ha and Stoel (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' ,72 ,78 ,78 PU1 PU2 PU3 + ,85 488 89 62 PU 32 ,55 48 68 ,47 ,35 ATT ,41 T2 BI1 ,35 ,90 52 ,86 ,90 ,09 81 ,74 ,81 00 ATT3 ATT2 ATT1 PEU e11 90 ,90 ,81 ,82 PEU2 PEU3 eg e10AE Consumer Acceptance of the Use of Artificial Intelligence in Online Shopping: Evidence From Hungary 168 Amfiteatru Economic The strongest direct effect was found between trust and perceived usefulness (H7 accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It suggests that the more we trust in Artificial Intelligence during the online shopping journey, the more likely it is that we consider AI-powered apps and webshops useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Besides, a higher level of trust forms a more positive attitude towards shopping in such webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Perceived usefulness has a central role in this model as it (PU) significantly impacted attitude (H3 accepted) and behavioural intention (H2 accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The more useful we find the use of artificial intelligence in online shopping believing that it allows us to grab the best deals, the more likely we are to consider it a wise decision to do the shopping in AI-powered webshops and apps more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Not surprisingly, attitude towards AI- powered webshops and apps was found to have a strong, significant, positive direct impact on behavioural intention (H1 accepted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It suggests that forming consumers’ attitude plays a vital role in increasing the traffic of AI-powered webshops and apps (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Although there was no significant direct relationship between perceived ease of use and attitude, the indirect effect of PEU on attitude (PEU->ATT=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='48) was quite strong, similar to its indirect impact on behavioural intention (PEU->BI1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Also, trust was found to indirectly influence behavioural intention (T2->BI1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It suggests that if shopping requires much mental effort and seems to be complicated in AI-powered webshops and apps, consumers tend to form stronger negative attitudes towards them and also tend to trust them less, which will result in weaker consumer intention to visit such webshops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In the nested model perceived usefulness had the highest total effect on behavioural intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Therefore, AI-powered webshops and apps are advised to increase the level of perceived usefulness to succeed by enabling customers to maximise purchase effectiveness to grab the best deals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' the ideal product with the highest utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Conclusions This research extends our knowledge of consumer acceptance of the use of artificial intelligence in online shopping in many aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The widely used technology acceptance model (TAM) was proved to be suitable for investigating consumer acceptance of the use of artificial intelligence in online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' As expected, it was confirmed in the nested model that the key factors influencing consumer’ behavioural intention to use AI-powered webshops and apps are trust, perceived usefulness, perceived ease of use and attitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In contrast to the original TAM (Davis, 1986), the direct relationship between perceived ease of use and attitudes was insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Nevertheless, it does not mean that user-friendliness of a webshop is not crucial as perceived ease of use indirectly affects attitude and the behavioural intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Instead, user- friendliness and flawless operation of an artificial intelligence-powered website are the prerequisites for market success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Building trust has a central role in consumer acceptance of the use of artificial intelligence in online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' If consumers do not trust in an AI-powered webshop/app, they tend to consider it less useful and form a negative attitude towards it, which will result in less online traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Also, AI must provide online consumers with tailor-made offerings to grab the best deals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' products with the highest value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' and it is expected to shorten the product search time to enhance shopping effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Not surprisingly, the favourable attitude Artificial Intelligence in Wholesale and Retail AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 23 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' 56 • February 2021 169 towards AI-powered webshops leads to more frequent online traffic in such electronic stores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Considering the strong positive impact of the recent COVID-19 crisis on e-commerce, the use of artificial intelligence in online shopping is expected to expand further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' According to Bloomberg (2020) the pandemic lockdowns have a dual effect on consumer behaviour on the development of AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=" Nowadays, it is more important than ever to create a personalised customer journey, to meet customers' demand and to provide a greater online shopping experience." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' In these efforts, artificial intelligence can be a very effective tool, which was confirmed by the research findings of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' This study has several practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It is useful for webshop owners and online marketing managers to understand how consumers adapt to the new technology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' the use of artificial intelligence in online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It is also beneficial to academics and researchers who are interested in the adaptation of the Technology Acceptance Model in online shopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Those who are interested in the role of trust in consumer choices in the online environment will also benefit from this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' As far as the future research directions are concerned, it would be advisable to repeat this study in a multi-cultural context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' It might also be useful to test the model of the Technology Readiness Index proposed by Parasuraman (2000) and to compare the results presented here with the new findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' Acknowledgements “The described article/presentation/study was carried out as part of the EFOP-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content='1-16- 2016-00011 “Younger and Renewing University – Innovative Knowledge City – institutional development of the University of Miskolc aiming at intelligent specialisation” project implemented in the framework of the Szechenyi 2020 program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=' The realization of this project is supported by the European Union, co-financed by the European Social Fund.” References Ajzen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INAzT4oBgHgl3EQfU_y9/content/2301.01277v1.pdf'} +page_content=', 1985.' metadata={'source': 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We will +denote the associated universal deformation by GE, usually viewed of as a p-divisible +group. As previously, E ∈ CAlg(SpG) will denote the associated Borel-equivariant +G-spectrum for a finite group G, see Notation 3.21. +We recall the following fundamental description of the E-cohomology of finite +abelian groups, see [HKR00, Corollary 5.10]. +Lemma 6.2. The E-cohomology E∗(BA) of any finite abelian group A is con- +centrated in even degrees, even periodic, and E0(BA) is a finitely generated free +E0-algebra. +Notation 6.3. For a finite abelian p-group A, we write A∗ for the Pontryagin dual +of A. We also view any A as a group scheme via the constant functor. +Recollection 6.4. For a finite abelian p-group A, recall that the π0E-algebra π0EBA +corepresents homomorphisms of group schemes A∗ → GE into the universal defor- +mation GE, see [HKR00, Proposition 5.12]. The functoriality in A of this iden- +tification is as follows: for a map f : A1 → A2 we obtain a map of π0E-algebras +π0EBA2 → π0EBA1 which corepresents the composition of f ∗ : A∗ +2 → A∗ +1 with +A∗ +1 → GE. In particular, for every subgroup A′ ⊆ A we obtain a map of affine +schemes +Spec(π0EBA′) −→ Spec(π0EBA). +The universal property of π0EBA, applied to the identity map π0EBA → π0EBA, +provides a universal homomorphism A∗ → GE(π0EBA). Choosing a coordinate on +GE, we thus obtain a map +(6.5) +α: A∗ → GE(π0EBA) ⊆ π0EBA. +From now on, we will implicitly fix such a coordinate on GE. +Proposition 6.6. The map Spec(π0EBA′) �→ Spec(π0EBA) is a closed immersion, +i.e., the map of π0E-algebras π0EBA → π0EBA′ given by restriction is surjective. + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +39 +Proof. Via the universal homomorphism (6.5), (A/A′)∗ ⊆ A∗ defines a subset T of +π0EBA. We denote by I the ideal generated by T and claim that the π0EBA-algebras +π0EBA′ and π0EBA/I are isomorphic. This will in particular verify the claim of +the proposition. Now, the two algebras are isomorphic because they corepresent +the same functors over the one corepresented by π0EBA: Indeed, a homomorphism +defined on A∗ will factor through (A′)∗ if and only if it annihilates (A/A′)∗. +□ +Notation 6.7. We define XA := Spec(π0EBA). For a subgroup A′ ⊆ A, we view XA′ +as a closed subsscheme in XA via Proposition 6.6, and write +X◦ +A′ := XA′ \ +� +� � +A′′⊊A′ +XA′′ +� +� . +Then X◦ +A′ ⊆ XA′ is an open subscheme. +Mapping A′ to XA′ induces a map of lattices from the lattice of subgroups of A +to the lattice of closed subsets of XA. This map preserves meets: +Proposition 6.8. Let A′, A′′ ⊆ A, then we have XA′∩A′′ = XA′ ∩ XA′′. +Proof. Using Proposition 6.6 we see that the scheme theoretic intersection XA′ ∩XA′′ +is also the fiber product XA′ ×XA XA′′. We claim that the given map +XA′∩A′′ → XA′ ×XA XA′′ +is an isomorphism of affine schemes over XA. We check that the map induced on +points +XA′∩A′′(R) → XA′(R) ×XA(R) XA′′(R) +is bijective for every π0EBA-algebra R. Indeed, an R-valued point on the left hand +side corresponds to a homomorphism (A′ ∩ A′′)∗ → GE(R), one of the right hand +side to a pair of homomorphisms ((A′)∗ → GE(R), (A′′)∗ → GE(R)) which agree +when pulled back to A∗, and the comparison map restricts a given homomorphism +to (A′)∗ and (A′′)∗. Observe that A′ ∩ A′′ is the pull-back in abelian groups of A′ +and A′′ mapping to A. Then the claim follows from the fact that the dual of a +pull-back of finite abelian groups is a push-out in the category of (not necessarily +finite) abelian groups. +□ +Corollary 6.9. For every finite abelian group A, we have a decomposition of XA +into a disjoint union of locally-closed subsets: +(6.10) +XA = +� +A′⊆A +X◦ +A′. +Proof. Since for every x ∈ XA, the set {A′ ⊆ A | x ∈ XA′} is non-empty and stable +under intersection by Proposition 6.8, there is a smallest A′ with x ∈ XA′. This +implies the claim, because the collection (X◦ +A′)A′⊆A covers XA. +□ +Remark 6.11. This disjoint union is not topological if A ̸= 0 because XA is connected, +being the Zariski spectrum of a local ring. + +40 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +6.2. Regularity of geometric fixed point spectra. In this subsection, we study +the schemes X◦ +A′ appearing in the decomposition Corollary 6.9 in more detail; +in particular, we will show that they are regular. We begin with a topological +interpretation of X◦ +A′ in terms of geometric fixed points. +For every A′ ⊆ A, the canonical map E → �EPA′ ⊗ E induces a map of commu- +tative ring spectra EBA → EBA′ → ΦA′E to the A′-geometric fixed points of the +genuine A-spectrum E. Concretely, ΦA′E is given by inverting on EBA′ all Euler +classes of all non-trivial characters of A′ (see [LMS86, Proposition II.9.13]). This +implies that the first of the following maps on Zariski spectra +Spec(π0ΦA′E) → Spec(π0EBA′) = XA′ �→ XA +is an open immersion, and we determine the image of the composition in the next +result. +Lemma 6.12. For every subgroup A′ ⊆ A, we have an equality of open subsets +Spec(π0ΦA′E) = X◦ +A′ ⊆ XA′ (⊆ XA) . +Proof. Indeed this is the special case of [BHN+19, Lemma 3.11] in which the family +consists of all proper subgroups. +□ +Proposition 6.13. For any finite abelian p-group A, there is a (set-theoretic) +decomposition of the Zariski spectrum of its E-cohomology as +Spec(E0(BA)) ≃ +� +A′⊆A +Spec(π0ΦA′E). +Proof. This follows from (6.10) combined with Lemma 6.12. +□ +We next establish a regularity statement which is key to our stratification result +for Lubin–Tate theory. +Theorem 6.14. For any finite group G and any Lubin–Tate theory E, the commu- +tative ring π0(ΦGE) is regular Noetherian. +The proof will be given after some preliminaries. +Remark 6.15. Using Example 3.22, we can reduce the proof of Theorem 6.14 to +the case that G = A is a finite abelian p-group which is generated by at most n +elements; otherwise, π0(ΦGE) ≃ 0 by Proposition 3.25. We will achieve this by +comparing with level A-structures, as introduced by Strickland [Str97], generalizing +work of Drinfel’d [Dri74]. +Recollection 6.16. To review the construction of the quotient π0EBA → D parametriz- +ing level-A-structures on GE, we define two monic polynomials F, G ∈ π0EBA[X]. +Choosing a coordinate on the universal deformation GE as in Recollection 6.4, we +identify the universal homomorphism α defined over π0EBA with a map α: A∗ → +GE(π0EBA) ⊆ π0EBA. This lets us define our first polynomial as +(6.17) +F(X) := +� +a∈A∗ : pa=0 +(X − α(a)) ∈ π0EBA[X]. +Note that F is monic and of degree prkp(A), where rkp(A) := dimFp(A⊗Z Fp) denotes +the p-rank of A. We also define G ∈ π0E[X] ⊆ π0EBA[X] to be the unique monic +polynomial such that G generates the same ideal in π0E[[X]] as does the p-series of + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +41 +GE (with respect to the coordinate X chosen above). Existence and uniqueness of +G follow from the Weierstrass preparation theorem for π0E. Note that G is monic +of degree pn, where n is the height of GE. +Some elementary commutative algebra shows that, given the monic polynomials +F, G ∈ π0EBA[X], there is an initial ring map ρ: π0EBA → D such that the image +of F in D[X] divides the image of G. The ring D is known as the Drinfel’d ring. +For more details on this construction of D, see [Str97, Section 7]. +An important result is that D is a regular, local Noetherian ring, see [Str97, +Theorem 7.3]. This implies that all localizations Dq are regular for all prime ideals +q ⊆ D, not just the maximal one, see [Mat89, Theorem 19.3]. +We will use the following criterion for divisibility. Recall that a polynomial over +a field k is said to be separable if all its roots α ∈ k in an algebraic closure k of k +are simple, i.e. satisfy f ′(α) ̸= 0. +Proposition 6.18. Assume R = (R, m, K) is a local π0EBA-algebra such that over +its residue field K, the image of F is separable and divides the image of G. Then +the image of F in R[X] divides the image of G. +Proof. We denote by FR = �N +i=1(X − ri) ∈ R[X] the image of F in R, i.e., +the elements ri enumerate the roots α(a) from (6.17), and likewise we denote by +GR ∈ R[X] the image of G in R. We now show by induction on l ≥ 1 that the +product �l +i=1(X − ri) divides GR. The case l = N is the desired claim. The base +case l = 1 claims that X − r1 divides GR, i.e., that GR(r1) is zero. In other words, +we have to show that GR(α(a)) = 0 for any is a p-torsion point a of A∗. This holds +because all the α(a) are p-torsion points of GE and G has the same zeroes as the +p-series of GE. In the inductive step, we know that GR = �l−1 +i=1(X − ri) · H in +R[X] for some l ≤ N, and a suitable H. Since GR(rl) = 0, we obtain from this by +evaluating at rl the relation 0 = �l−1 +i=1(rl − ri) · H(rl) in R. Since the image of F +over the residue field K is separable, none of the elements rl − ri ∈ R reduces to +zero in K, hence they do not lie in m, thus they are units in R. By cancelling these +units we obtain H(rl) = 0, i.e., H is divisible by X − rl in R[X], and this completes +the induction step. +□ +Now we obtain a relation between the ring π0ΦAE which we want to show to be +regular, and the ring D which we know to be regular. +Proposition 6.19. The canonical ring homomorphism π0EBA → π0ΦAE factors +(uniquely) through the surjection ρ: π0EBA → D, i.e., there is a commutative +diagram +π0EBA +� +ρ +�� +π0ΦAE +D. +� +This says that the base change of the universal homomorphism A∗ → GE to the +geometric fixed points is a level-structure. We first check that Proposition 6.19 +implies our theorem. +Proof of Theorem 6.14 assuming Proposition 6.19. By Remark 6.15, we may as- +sume that G = A is an abelian p-group which is generated by (at most) n elements. + +42 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +As a localization of the Noetherian ring π0EBA, the ring π0ΦAE is also Noetherian. +To see that it is regular, note that Lemma 6.12 and Proposition 6.19 say that the +open immersion X◦ +A �→ XA factors through the closed immersion Spec(D) ⊆ XA. +The resulting map X◦ +A → Spec(D) is necessarily an open immersion. Then, since +Spec(D) is regular, so is X◦ +A by [Mat89, Theorem 19.3], as claimed. +□ +Proof of Proposition 6.19. Set I := ker(ρ) ⊆ π0EBA; the claim is I · π0ΦAE = 0. +It is enough to show this after replacing π0ΦAE with an arbitrary localization +R := (π0ΦAE)p for a prime ideal p ⊆ π0ΦAE. To this end, we will verify that we +are in the situation of Proposition 6.18, i.e., that the conditions of this proposition +apply to the local ring R = (R, pR, K := R/pR): +(a) We first prove that the image of F is separable over K. Looking at the +definition of F, it will be enough to show that the composite +f : A∗ +α−→ GE(π0EBA) → GE((π0ΦAE)) → GE((π0ΦAE)p) → GE(K) +is injective, where the last map is induced by the reduction R = (π0ΦAE)p → +K. To this end, assume otherwise. Then f factors over some non-zero +quotient of A∗, which implies that its classifying map �f : π0EBA → K factors +over some non-trivial restriction map π0EBA → π0EBA′ with A′ ⊆ A a +proper subgroup. This is a contradiction, because by construction �f factors +over the geometric fixed points and hence inverts all non-zero Euler classes +whereas the restriction map annihilates some non-zero Euler class. This +contradiction shows that f is indeed injective. +(b) Secondly, we observe that the image of F in K divides the image of G +because G annihilates all p-torsion points of GE. +Therefore, Proposition 6.18 implies that the image of F in R divides the image of +G in R. Since D was constructed to be initial among π0EBA-algebras with this +property, this is equivalent to saying that I · R = 0, as claimed. +□ +We make the case of height one of the above results explicit and point out a +possible sharpening of the regularity result Theorem 6.14. +Example 6.20. Assume the height is one, then E = E1 is (a form of) p-complete +complex K-theory. Let A = Cpn be the cyclic group of order pn for some n ≥ 0. +Then E0 = Zp and a familiar computation in complex oriented cohomology theories +gives an isomorphism of E0-algebras +E0(BCpn) ≃ E0[[X]]/ +� +(X + 1)pn − 1 +� +. +Here, X corresponds to the Euler class of the defining representation Cpn ⊆ S1. It +is convenient to set Y := X + 1. We can decompose +Y pn − 1 = +n +� +i=0 +Φpi(Y ) = (Y − 1) · Y p − 1 +Y − 1 · . . . · Φpn(Y ) +into irreducible polynomials over Qp. Then Φpi is the pi-th cyclotomic polyno- +mial, the minimal polynomial of any primitive pi-th root of unity ζpi. Each ring +Zp[Y ]/(Φpi) is a discrete valuation ring, the ring of integers in the local cyclo- +tomic field Qp(ζpi), and has residue field Fp. One can check that the obvious ring + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +43 +homomorphism +E0(BCpn) = Zp[Y ]/(Y pn − 1) → +n +� +i=0 +Zp[Y ]/(Φpi(Y )) +glues the spectra of the rings on the right hand side along their common closed +point and that one has X◦ +Cpi = Spec(Qp(ζpi)) for i ≥ 1 and X◦ +e = Spec(Zp). The +Zariski spectrum of E0(BCpn) is depicted in Figure 1. +•Fp +•Qp +•Qp(ζp) +•Qp(ζp2) +. . . +•Qp(ζpn−1) +•Qp(ζpn) +Figure 1. An image of Spec(E0(BCpn)). Each prime is repre- +sented by its residue field, and the lines indicate specialization +relations with closure going upwards. +Remark 6.21. We now turn to discuss a conceivable sharpening of Theorem 6.14; this +remark can be skipped without loss of continuity. Let again E denote a Lubin–Tate +theory of arbitrary height and A a finite abelian p-group. We have a ring extension +R := E0 → S := ΦA(E). +Fix a prime ideal q ⊆ S, and denote p := q ∩ R ⊆ R. The resulting map +Rp → Sq +is a finite flat extension of local Noetherian rings, Rp is regular, and we showed that +Sq is regular, too. We can consider the condition +(6.22) +q = p · Sq. +Note this condition implies the regularity of Sq and more precisely it shows that +any regular system of parameters in Rp gives a regular system in Sq. The condition +(6.22) is immediate to check in height one and can be checked in height two using +computations about modular equations. We have not been able to decide whether +(6.22) holds in a single case of height larger than or equal to three. Observe that the +condition in (6.22) does not claim that Rp → Sq is ´etale: Indeed, it is known that +the ´etale locus of the extension E0 → E0(BA) is exactly E0[p−1], so if (6.22) holds +at a prime p of positive characteristic, as it does in height two, then necessarily +the residue field extension will be inseparable. We remind the reader of the reason +for the claim about the ´etale locus of the finite flat algebra E0 ⊆ E0(BA): The +height stratification of the corresponding finite flat group scheme is given by a +regular sequence v0 = p, v1, . . . , vn−1 ∈ π0E. In particular, the formal part is zero +(equivalently, the group scheme is ´etale), exactly if p is invertible. +6.3. Stratification for E-modules. In this section we show that for any finite +group G the category ModG(E) is stratified. The key input is the stratification of +Mod(ΦA(E)), which uses the regularity of the geometric fixed points established in +Section 6.2. + +44 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +Lemma 6.23. Let A be a finite abelian p-group, then Mod(ΦAE) is cohomological +stratified and the ungraded comparison map +ρ0 : Spc(Perf(ΦAE)) +∼ += +� Spec(π0(ΦAE)) +is a homeomorphism. +Proof. By Theorem 6.14, π0(ΦAE) is a regular Noetherian ring and the graded +ring π∗(ΦA(E)) is concentrated in even degrees. In fact, this ring is even periodic +(as being obtained by inverting finitely many elements in the even periodic ring +E∗(BA)), and hence π∗(ΦAE) is regular as a graded commutative ring. Now we +can apply [DS16, Theorem 1.1 and Theorem 1.3]6, keeping in mind Remark 2.6. +□ +We now use Lemma 6.23 to prove that ModG(E) is stratified for any finite +group G. In the next section, we will then identify this Balmer spectrum with +the Zariski spectrum of the cohomology ring and deduce that ModG(E) is in fact +cohomologically stratified. +Theorem 6.24. For any group G and any Lubin–Tate theory E, the category +ModG(E) is stratified with Noetherian spectrum Spc(PerfG(E)). +Proof. We recall that E is F-nilpotent for the family of abelian p-groups which are +generated by n elements (Example 3.22). Let A be such a group, then it suffices +by Theorem 3.33 to show that Spc(Perf(ΦAE)) is Noetherian and Mod(ΦAE) is +stratified. This follows from Lemma 6.23. +□ +7. The Balmer spectrum for Borel-equivariant E-theory +In this section we compute the spectrum of PerfG(E) for any finite group G +and any Lubin–Tate theory E of height n and prime p, by showing that the +comparison map ρ0 : Spc(PerfG(E)) → Spec(E0(BG)) is a homeomorphism. Along +with Theorem 6.24, this will show that ModG(E) is cohomologically stratified. +Finally, we deduce that the category of modules over the cochain spectrum EBG is +also cohomologically stratified. +7.1. The spectrum of Borel-equivariant E-theory. In order to compute the +spectrum of PerfG(E), we first show that this category is Noetherian (Definition 2.9). +Lemma 7.1. For any finite group G, the rings E0(BG) and E∗(BG) are Noether- +ian. Moreover for any subgroup H ⊆ G, the ring maps E0(BG) → E0(BH) and +E∗(BG) → E∗(BH) are finite. In particular, the category PerfG(E) is Noetherian. +Proof. The ring E∗(BG) is graded Noetherian and the composite +E∗ → E∗(BG) → E∗(BH), +is finite by [GS99, Cor. 4.4]. This also implies that E∗(BG) → E∗(BH) is finite. +Passing to degree zero elements and using that E∗ is concentrated in even degrees +and even periodic, we deduce that the composite E0 → E0(BG) → E0(BH) is finite. +It then follows that E0(BG) → E0(BH) is finite and that E0(BG) is Noetherian. +The final claim then follows from Lemma 3.4. +□ +6Note that there is a gap in one of the lemmas of that paper, which however does not affect +the main theorems, see [DS22]. + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +45 +Lemma 7.2. For any finite abelian p-group A which can be generated by n elements, +the ungraded comparison map +ρA +0 : Spc(PerfA(E)) +∼ += +� Spec(E0(BA)) +is a homeomorphism. +Proof. Let Φ: PerfA(E) → � +A′⊆A Perf(ΦA′E) denote the symmetric monoidal +functor induced by the product of the geometric fixed point functors ΦA′. By +naturality of the comparison map [Bal10, Corollary 5.6], there is a commutative +diagram +Spc(PerfA(E)) +� +A′⊆A Spc(Perf(ΦA′E)) +Spec(E0(BA)) +� +A′⊆A Spec(π0(ΦA′E)). +ρA +0 +� ρ0 +Spc(Φ) +Spec(Φ0) +We claim that ρA +0 is a bijection. By Lemma 6.23, the right hand vertical arrow is a +homeomorphism. Using Lemma 6.12, we see that the bottom map is equivalent to +the map +� +A′⊆A +X◦ +A′ ∼= +� +A′⊆A +Spec(π0(ΦA′E)) → Spec(E0(BA)) = XA. +By Proposition 6.13 this map is a bijection. It follows from Corollary 3.29 that +Spc(Φ) is surjective, hence the commutativity of the square implies that ρA +0 is a +bijection. Finally, by Lemma 7.1 and Proposition 2.10, this shows that ρA +0 is a +homeomorphism. +□ +Lemma 7.3. For any finite group G, the comparison map +ρG +0 : Spc(PerfG(E)) +∼ += +� Spec(E0(BG)) +is a homeomorphism. +Proof. This follows from Corollary 4.4(a). Indeed, E is F-nilpotent for the family +of abelian p-subgroups of G which are generated by (at most) n elements, see Ex- +ample 3.22. Then the assumptions of Corollary 4.4(a) hold by Lemma 7.2 and +Lemma 7.1. +□ +Theorem 7.4. For any finite group G and any Lubin–Tate theory E, the category +ModG(E) is cohomologically stratified by E0(BG). +Proof. Combine Theorem 6.24 and Lemma 7.3. +□ +Remark 7.5. Theorem 7.4 provides a chromatic counterpart in intermediate charac- +teristic to the celebrated stratification theorem of Benson–Iyengar–Krause [BIK11b], +extending their work from height ∞ to all finite heights. The strategy of proof, +however, is fundamentally different: we first establish stratification relative to +the Balmer spectrum of ModG(E) and then lift our chromatic Quillen stratifica- +tion to an identification of the Balmer spectrum. As mentioned, the fundamental +input to our approach is a regularity result for the geometric fixed points of E +via Drinfel’d level structures, while the proof of [BIK11b] ultimately relies on the +Bernstein–Gelfand–Gelfand(BGG)-correspondence. + +46 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +As a corollary, we obtain a generalization of Proposition 6.13. +Corollary 7.6. Let E be a Lubin–Tate theory at height n and prime p. For any +finite group G, we have a disjoint decomposition into locally closed subsets +Spec(E0(BG)) = +� +A +Spec(π0(ΦAE))/W Q +G (A), +where the coproduct ranges over all G-conjugacy classes of abelian p-subgroups of G +which are generated by (at most) n elements. +Proof. Recall from Example 5.6 that E arises from a global spectrum, and that it +is F-nilpotent for the family of abelian p-subgroups of G which are generated by n +elements by Example 3.22. The result then follows from Theorem 5.18, Lemma 6.23, +and Lemma 7.3. +□ +Remark 7.7. Let R be a commutative ring spectrum and G a finite group. By +[MNN17, Cor. 6.21], Borel-completion is an exact symmetric monoidal and fully +faithful functor +ψ(R): PerfG(R) +� Fun(BG, Perf(R)), +whose essential image is given by the thick subcategory generated by the permutation +modules {R[G/H]}H⊆G. We say that Fun(BG, Perf(R)) is generated by permuta- +tion modules if ψ(R) is an equivalence. For example, for a discrete commutative +ring A, the tt-category Fun(BG, Perf(A)) is equivalent to the bounded derived +category of A[G]-representations whose underlying A-module is perfect. A result of +Rouquier, Mathew [Tre15], and Balmer–Gallauer [BG22a] then implies that ψ(A) +is an equivalence for any regular Noetherian A. It is an open question, first raised +in [Tre15, Question A.2], whether Fun(BG, Perf(E)) is generated by permutation +modules for any Lubin–Tate theory E. This is answered affirmatively by Mathew +in loc. cit. for the case of height 1 and G = Cp. +7.2. Stratification for cochains. In this subsection, we study the category of +modules over the cochain algebra EBG ∈ CAlg(Sp). We will deduce from Theo- +rem 7.4 that Mod(EBG) is cohomologically stratified. We start off with the following +observation. +Lemma 7.8. The endomorphism ring of the unit E in ModG(E) can be identified +with EBG as a commutative ring spectrum. +Proof. By adjunction, we have equivalences of commutative algebras +MapModG(E)(E, E) ≃ MapSpG(S0, E) ≃ (E)G +so we only need to identify the right hand side with the spectrum EBG. To this end +recall that E belongs to the full subcategory (SpG)Borel ⊆ SpG of Borel-equivariant +G-spectra as defined in [MNN17, Definition 6.14]. Moreover by [MNN17, Proposition +6.19], there is a commutative square +(SpG)Borel +SpG +Fun(BG, Sp) +Sp, +∼ +⊗ +(−)G +(−)hG + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +47 +where the left vertical functor takes the associated underlying spectrum with G- +action, and this is a symmetric monoidal equivalence by [MNN17, Proposition +6.17]. Note that all the other functors in the diagram are lax monoidal so they +preserve commutative algebra objects. Chasing E around the diagram, we obtain +the identification (E)G ≃ EhG as commutative algebras. It is only left to note that +EhG ≃ EBG, as E has trivial G-action. +□ +Terminology 7.9. Given a rigidly-compactly generated tt-category T we let +Tcell := LocT ⟨1⟩ ⊆ T +denote the localizing subcategory of T generated by the unit, referred to as the full +subcategory of cellular objects in T. +Remark 7.10. Because the unit is compact in T by assumption, Neeman’s theorem +[Nee96, Theorem 2.1] implies that (Tcell)c = Tcell ∩ Tc. +The next result identifies the category of modules over the cochain algebra as +the subcategory of cellular objects in E-modules. +Lemma 7.11. There is a fully faithful tt-functor +Mod(EBG) �→ ModG(E) +with essential image given by LocModG(E) ⟨E⟩. +Proof. By Morita theory and Lemma 7.8, there is a cocontinuous symmetric monoidal +functor +Mod(EBG) +∼ +−→ LocModG(E) ⟨E⟩ ⊆ ModG(E). +The first functor is an equivalence, and it is symmetric monoidal by [Lur17, Corollary +4.8.1.14]. +□ +Theorem 7.12. The category Mod(EBG) is cohomologically stratified. In particular, +the comparison map +ρG +0 : Spc(Perf(EBG)) +∼ += +� Spec(E0(BG)) +is a homeomorphism. +Proof. By Lemma 7.11 we can identify Mod(EBG) with the subcategory of cellular +objects in ModG(E). Applying [BIK11a, Theorem 1.2] and noting that ModG(E) +is cohomologically stratified (Remark 2.20), we see that stratification for ModG(E) +implies stratification for Mod(EBG). +□ +7.3. Some calculations of Spec(E0(BG)) at height one. Throughout this sec- +tion we will assume that the height is one so that E = E1 is (a form of) p-complete +complex K-theory. We will describe Spec(E0(BG)) explicitly for a large class of +finite groups G. +Consider a finite group G and let C denote the family of cyclic p-subgroups of +G. The Borel-equivariant G-spectrum E is C-nilpotent by Example 3.22 and arises +from a global homotopy type by Example 5.6. It follows from Theorem 5.18 that +we have a homeomorphism +(7.13) +Spec(E0(BG)) ∼= +colim +A∈OQ +C (G) +Spec(E0(BA)). +The Zariski spectrum Spec(E0(BA)) has already been discussed in detail in Exam- +ple 6.20. We make the following additional observations: + +48 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +(1) For n ≥ m ≥ 1, the canonical inclusion Spec(E0(BCpm)) ⊆ Spec(E0(BCpn)) +hits the prime ideals corresponding to Fp and Qp(ζpi) for 0 ≤ i ≤ m. +(2) The Quillen–Weyl group W Q +G (Cpn) acts trivially on Spec(E0(BCpn)). To see +this one checks that any automorphism σ of Cpn preserves the decomposition +(6.10), namely satisfies σ.X◦ +Cpi = X◦ +Cpi for all i ≤ n. Since X◦ +e = Spec(Zp) +and X◦ +Cpi = Spec(Qp(ζpi)) by Example 6.20, one immediately deduces that +σ must act trivially on each X◦ +Cpi and so on all Spec(E0(BCpn)). +Using these observations together with (7.13) we can calculate Spec(E0(BG)) for +many examples of G. +Example 7.14. Let G be a finite group with a p-Sylow subgroup Gp of exponent +p. This means that p is the least common multiple of the orders of all elements of +Gp. For instance, Gp could be cyclic of order p or an elementary abelian p-group. +Our assumptions force any A ∈ C to be cyclic of order p. In this case the Zariski +spectrum of E0(BG) is depicted in Figure 2. +• +• +• +• +. . . +• +• +Figure 2. An image of Spec(E0(BG)) for a finite group G with +p-Sylow subgroup of exponent p. The lines indicate specialization +relations with closure going upwards. The bottom row has n + 1 +prime ideals where n is the maximal number of cyclic subgroups of +G of order p which are pairwise non-conjugate. +Example 7.15. Suppose that p = 2 and G = Dn a dihedral group for some n ≥ 1. +One checks that there is a unique maximal cyclic subgroup A ⊆ G which has order +n. Therefore we immediately get that Spec(E0(BG)) ∼= Spec(E0(BA)). In this case +the Zariski spectrum is depicted in Figure 1. +Remark 7.16. Inspired by Quillen’s work [Qui71], one might wonder if the action +of W Q +G (A) on Spec(π0ΦAE) is free on closed points, see discussion in Remark 5.26. +To see this is not the case take p = 3, G = Σ3 the symmetric group on three letters, +and A = Z/3 the Sylow 3-subgroup generated by the cycle (123). The Quillen–Weyl +group W Q +G (A) is not trivial but acts trivially on all of Spec(E0(BA)) and so on +Spec(π0ΦAE) too, by observation (2) above. +8. Further examples +In this final section, we discuss some further examples for which we can prove +stratification for equivariant module spectra as a consequence of our general tech- +niques, namely for HZ-modules when G = Cpn, equivariant complex K-theory KUG +for all finite groups G, and the real K-theory spectrum KR when G = C2. +8.1. Stratification for modules over the integral constant Mackey functor. +We recall that Theorem 4.3 determines the spectrum of PerfG(R) in terms of the +spectra of the non-equivariant tt-categories Perf(ΦHR), up to the question how the +strata are glued together. In order to settle this question, different techniques are + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +49 +required, such as a study of the Tate squares governing the local-to-global principles. +This ambiguity is illustrated by the next example. +Example 8.1. Let G = Cpn be a cyclic p-group and write HZ ∈ CAlg(SpCpn) for +the Eilenberg–MacLane spectrum associated to the constant Green functor with +value Z. By [HHR16, Proposition 3.18] for p = 2 and [Zen17, p. 34] for p > 2, its +geometric fixed points have coefficients +π∗ΦHHZ ∼= +� +Z +if H = e, +Z/p[t] +non-trivial H ⊆ Cpn, +as graded commutative rings, where t is in degree 2. In particular, all geometric +fixed points of HZ are even and regular. It then follows from [DS16] that the +tt-categories Mod(ΦHHZ) are stratified with Balmer spectrum +Spc(Perf(ΦHHZ)) ∼= +� +Spec(Z) +if H = e, +Spech(Z/p[t]) +non-trivial H ⊆ Cpn. +Therefore, Theorem 3.33 and Theorem 4.3 imply that ModCpn (HZ) itself is stratified, +and we depict its Balmer spectrum in Figure 3. The solid part of this figure is +e +Cp +Cp2 +Cpi +Cpn +Spec(Z) +• +• +. . . +• +• +• +• +• +Figure 3. An image of Spc(PerfCpn(HZ)). The lines indicate +specialization relations, with closure going upwards. Contributions +from the same subgroup are displayed in the same color. +determined by Theorem 4.3 together with the observation that the corresponding +Weyl groups act trivially on the geometric fixed points in this case, since HZ +arises from a global homotopy type. Consequently, we obtain a parametrization of +localizing ideals in this setting: +�Localizing ⊗-ideals +of ModCpn (HZ) +� +∼ +� � +Subsets of Spec(Z) ⊔ �n +i=1 Spech(Z/p[t]) +� +. +The dashed lines in the figure—gluing the strata of the spectrum together—are +known to exist by forthcoming work of Balmer and Gallauer, see [BG20]. In fact, +they further determine the underlying set of Spc(PerfG(HZ)) for all finite groups +G and establish stratification of ModG(HZ) in this generality. +8.2. Stratification for modules over equivariant K-theory. The goal of this +subsection is to use our methods to show that the category ModG(KUG) is cohomo- +logically stratified, where KUG denotes G-equivariant complex K-theory, [Seg68a]. +Our argument is modelled on the case of Borel-equivariant Lubin–Tate theory +treated in previous sections. We will also show cohomological stratification for the +category of modules over the C2-spectrum of real K-theory KR. + +50 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +Remark 8.2. The G-equivariant stable homotopy groups are +πG +∗ (KUG) ∼= R(G)[β±1], +|β| = −2, +where R(G) denotes the complex representation ring of G. In particular, πG +∗ (KUG) +is an even periodic Noetherian ring. Using the fact that these rings are even periodic, +we deduce the following from Segal’s seminal work on πG +0 (KUG). +Lemma 8.3. For any finite group G, the category PerfG(KUG) is Noetherian. +Proof. For any finite group G, the ring πG +∗ (KUG) is Noetherian. Moreover, if H +is a subgroup of G, then πH +∗ (KUG) ∼= πH +∗ (KUH) is a finitely generated πG +∗ (KUG)- +module. Both of these statements were proven in [Seg68b, Proposition 3.2 and +Corollary 3.3]. The result then follows from Lemma 3.4. +□ +We will also need the following result, proved in [MNN19, Proposition 5.6]. +Lemma 8.4. The G-spectrum KUG has derived defect base equal to the family of +cyclic subgroups of G. +Remark 8.5. By equivariant Bott periodicity, KUG is complex stable ([Gre99, Section +4]): for every complex representation V , there are compatible isomorphisms +πG +k+|V |(SV ∧ X) ∼= πG +k (X). +For such complex stable equivariant theories, geometric fixed points are given by +inverting the Euler class of the reduced regular representation, and this is the key +ingredient in the computational part of the following result. +Lemma 8.6. Let G be a finite group. The homotopy of the G-geometric fixed points +of KUG are given by +π∗(ΦGKUG) = +� +π∗(KU)[1/n, ζn] +G ∼= Cn +0 +otherwise, +where ζn denotes a primitive n-th root of unity. In particular, the ring π∗(ΦG(KUG)) +is even periodic, regular and Noetherian. +Proof. The nilpotence result of Lemma 8.4 implies that ΦGKUG = 0 unless G ∼= Cn +(Lemma 3.24). In the case G is a cyclic group, the computation follows from [tD87, +Proposition 7.7.7] (see also [Gre99, Lemma 3.1]). For the regularity claim, we use +the explicit description observing that Z[ζn] ⊆ Q(ζn) is the full ring of integers, in +particular it is a Dedekind ring, and thus regular. +□ +Proposition 8.7. For any finite group G, the category Mod(ΦGKUG) is cohomo- +logically stratified. In particular, the comparison map +ρG +0 : Spc(Perf(ΦGKUG)) +∼ += +� Spec(π0(ΦGKUG)) +is a homeomorphism. +Proof. By Lemma 8.6, the ring π∗(ΦGKUG) is even periodic, regular and Noetherian. +Now apply [DS16, Theorem 1.1 and 1.3] noting that we can use the spectrum of degree +zero elements instead of the graded homogeneous spectrum (see Remark 2.6). +□ +Theorem 8.8. For any finite group G, the tt-category ModG(KUG) is stratified. + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +51 +Proof. Note that the restriction of KUG to an H-spectrum is KUH. By Theo- +rem 3.33, the theorem is then a consequence of Proposition 8.7, since ΦHKUG ≃ +ΦHKUH for any subgroup H in G. +□ +Remark 8.9. It remains to show how the Balmer spectrum in Theorem 8.8 is +determined by the representation ring of G. The next result resolves this in the case +of a cyclic group. +Lemma 8.10. For any cyclic group Cn, the comparison map +ρCn +0 +: Spc(PerfCn(KUCn)) +∼ += +� Spec(R(Cn)) +is a homeomorphism. +Proof. Arguing as in Lemma 7.2, we consider the commutative diagram +Spc(PerfCn(KUCn)) +� +A⊆Cn Spc(Perf(ΦA(KUCn))) +Spec(R(Cn)) +� +A⊆Cn Spec(π0(ΦA(KUCn))). +ρCn +0 +� ρ0 +Spc(Φ) +Spec(Φ0) +We first prove that the bottom map is bijective. This can be deduced from work of +Segal and Bojanowska for general finite groups, see [Seg68b, Proposition 3.7] and +[Boj83, Theorem 4.13]. But the special case at hand of a cyclic group admits the +following direct argument. Our map is induced by the ring homomorphism +π: R(Cn) ∼= Z[X]/(Xn − 1) → +� +1≤d|n +Z[1 +d, ζd] ∼= +� +A⊆Cn +π0(ΦA(KUCn)) +sending X to the tuple (ζd)d|n of d-th primitive roots of unity. To say this induces +a bijection on Zariski spectra is equivalent to saying that every ring homomorphism +ω: Z[X]/(Xn − 1) → Ω into an algebraically closed field Ω factors uniquely through +exactly one of the component maps πd : Z[X]/(Xn − 1) → Z[ 1 +d, ζd] of π. Indeed, +denoting x := ω(X) ∈ Ω∗, the map ω factors uniquely through πd for d the order of +x ∈ Ω∗. +It follows that the bottom map in the commutative diagram is a bijection. +Moreover, the right-hand vertical map is a homeomorphism by Proposition 8.7, +while the top horizontal map is a surjection by Corollary 3.29. It follows that ρCn +0 +is a bijection, and hence a homeomorphism by Proposition 2.10, which is applicable +by Lemma 8.3. +□ +Lemma 8.11. For any finite group G, the comparison map +ρG +0 : Spc(PerfG(KUG)) +∼ += +� Spec(R(G)) +is a homeomorphism. +Proof. Recall from [Seg68b, Proposition 3.2, Corollary 3.3] that πG +0 (KUG) ∼= R(G) +is a Noetherian ring and that for all subgroup H ⊆ G, the R(G)-module πH +0 (KUG) ∼= +R(H) is finitely generated. Combining this with Lemma 8.4 and Lemma 8.10 we +see that the assumptions of Corollary 4.4(a) are satisfied so the result applies to +give the claim. +□ +Putting together Theorem 8.8 and Lemma 8.11 we deduce the desired stratification +result for equivariant K-theory. + +52 +BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL +Theorem 8.12. For any finite group G, the category ModG(KUG) is cohomologi- +cally stratified. +Example 8.13. We give an explicit description for the case G = Cp following [DM21, +Remark 6.4]. In this case R(Cp) ∼= Z[x]/(xp − 1). We have (xp − 1) = (x − 1)Φp(x) +where Φp(x) := 1 + x + · · · xp−1. The two quotient maps +R(Cp) → R(e) ∼= Z +and +R(Cp) → Z[x]/Φp(x) +give rise to jointly surjective embeddings +Spec(Z) +ψe +−→ Spec(R(Cp)) +ψCp +←−−− Spec(Z[x]/Φp(x)). +There is only one prime ideal in the intersection of their images, namely +ψe((p)) = (p, x − 1) = (p, Φp(x)) = ψCp((p)). +Correspondingly, we have a decomposition +(8.14) +Spec(R(Cp)) ∼= Spec(Z) ⊔ Spec(Z[x, p−1]/Φp(x)). +Write Z[θ, p−1] := Z[x, p−1]/Φp(x), where θ is the image of x under the quotient +map (so θ is a p-th root of unity). The proof of Lemma 8.10 also shows that the +following diagram commutes +Spec(Perf(KU)) +Spc(PerfCp(KUCp)) +Spc(Perf(ΦCpKUCp)) +Spec(Z) +Spec(R(Cp)) +Spec(Z[θ, p−1]). +ϕe +ϕCp +ρe +0 ∼ += +ρ +Cp +0 +∼ += +ψe +ψCp +ρ0 ∼ += +When p = 2, we depict Spec(R(C2)) ∼= Spc(PerfC2(KUC2)) in Figure 4; it consists +of a copy of Spec(Z[x]/(x − 1)) and a copy of Spec(Z[x](x + 1)) (both of which are +homeomorphic to Spec(Z)) glued together at the prime 2. +•F3•F5•F7 · · · +��F2 +•F3•F5•F7 · · · +•Q +•Q +Figure 4. An illustration of the spectrum Spc(PerfC2(KUC2)) ∼= +Spec(R(C2)). Here closure goes upwards, and the primes are la- +beled by their residue fields. The blue points denote primes in +Spec(Z[x]/(x − 1)) ∼= Spec(Z), while the red ones denote primes in +Spec(Z[x]/(x + 1)) ∼= Spec(Z). The point denoted F2 shown in blue +and red denotes the gluing of the two copies of Spec(Z) over the +prime ideal (2). +We refer the reader to [Ser78, Section 11.4] for a discussion of Spec(R(G)) for +general finite groups G. For example, there it is shown that Spec(R(G)) is connected. +Example 8.15. Let KR ∈ CAlg(SpC2) denote Atiyah’s K-theory with reality, intro- +duced in [Ati66]. This is a C2-ring spectrum with underlying spectrum given by +KU, whose homotopy groups are regular Noetherian concentrated in even degrees. + +QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY +53 +Its C2-geometric fixed points are trivial by [HS14, Proposition 6.1]. It follows +from Theorem 3.33 that ModC2(KR) is stratified, and by Theorem 4.3 we have an +identification +Spc(PerfC2(KR)) ∼= Spech(KU∗)/C2. +Note that C2 acts trivially on the affine scheme Spech(KU∗) ∼= Spec(Z). It then +follows that Spc(PerfC2(KR)) ∼= Spec(Z), and that Spc(PerfC2(KR)) is actually +cohomologically stratified. 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In preparation. +Tobias Barthel, Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, +Germany +Email address: tbarthel@mpim-bonn.mpg.de +URL: https://sites.google.com/view/tobiasbarthel/ +Nat`alia Castellana, Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, +08193 Bellaterra, Spain, and Centre de Recerca Matem`atica +Email address: natalia@mat.uab.cat +URL: http://mat.uab.cat/∼natalia +Drew Heard, Department of Mathematical Sciences, Norwegian University of Science +and Technology, Trondheim +Email address: drew.k.heard@ntnu.no +URL: https://folk.ntnu.no/drewkh/ +Niko Naumann, Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, Universit¨atsstraße +31, 93053 Regensburg, Germany +Email address: Niko.Naumann@mathematik.uni-regensburg.de +URL: https://homepages.uni-regensburg.de/∼nan25776/ +Luca Pol, Fakult¨at f¨ur Mathematik, Universit¨at Regensburg, Universit¨atsstraße 31, +93053 Regensburg, Germany +Email address: luca.pol@mathematik.uni-regensburg.de +URL: https://sites.google.com/view/lucapol/ + diff --git a/IdE0T4oBgHgl3EQfRwCH/content/tmp_files/load_file.txt b/IdE0T4oBgHgl3EQfRwCH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5dc62207e40108389c666c5e45eae326451ea990 --- /dev/null +++ b/IdE0T4oBgHgl3EQfRwCH/content/tmp_files/load_file.txt @@ -0,0 +1,2507 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf,len=2506 +page_content='QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY TOBIAS BARTHEL, NAT`ALIA CASTELLANA, DREW HEARD, NIKO NAUMANN, AND LUCA POL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We prove a version of Quillen’s stratification theorem in equivariant homotopy theory for a finite group G, generalizing the classical theorem in two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We then apply our methods to the case of Borel-equivariant Lubin–Tate E-theory En, for any finite height n and any finite group G, where we obtain a sharper theorem in the form of cohomological stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing ⊗-ideals of the category of equivariant modules over En, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Tensor-triangular preliminaries 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Stratifications in stable equivariant homotopy theory 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Strong Quillen stratification for equivariant Balmer spectra 27 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Strong Quillen stratification for global equivariant spectra 32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Stratification via a theorem of Drinfel’d–Strickland 38 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The Balmer spectrum for Borel-equivariant E-theory 44 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Further examples 48 References 53 Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 18F99, 55P42, 55P91, 55U35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The image is a visualization of Quillen’s category for the symmetric group S12 at the prime 2, made by Jared Warner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We are grateful to him for allowing us to include it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='02212v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='AT] 5 Jan 2023 2 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Introduction Quillen’s celebrated stratification theorem [Qui71] provides a geometric descrip- tion of the cohomology of any finite group G with coefficients in a field k in terms of a decomposition of its Zariski spectrum into locally closed subsets: (∗) Spec H•(G, k) = � (E)⊆G V+ G,E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Here, the disjoint union is indexed on the conjugacy classes of elementary abelian subgroups E ⊆ G and the strata V+ G,E are given by orbits of the Weyl group action on an open subset of the Zariski spectrum of the cohomology of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since the cohomology of elementary abelian subgroups is well-known, Quillen’s work gives a formula to understand the cohomology ring of any finite group geometrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The first goal of the present paper is to develop and prove a far-reaching general- ization of Quillen’s theorem, in the following two directions: (a) We establish an analogue of Quillen stratification for an arbitrary commu- tative equivariant ring spectrum R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Specializing to the Borel-equivariant theory R = k recovers a version of Quillen’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) We categorify the stratification of the group cohomology to a decomposition of the entire category of modules over R, for appropriate equivariant ring spectra R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The special case of R = k is contained in the seminal work of Benson, Iyengar, and Krause [BIK11b] on the stable module category of k-linear G-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our results take place in the context of equivariant homotopy theory and are formulated in the language of tensor-triangular geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This ‘equivariant tensor- triangular geometry’ was initiated by Strickland (unpublished), Balmer [Bal16b], and Balmer and Sanders in [BS17] and pursued further in [BHN+19], [BGH20], [PSW22], and [BHS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The present paper conceptualizes these developments in the form of a unifying perspective encompassing equivariant tensor-triangular geometry, Quillen’s stratification of group cohomology, and stratifications in modular representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our first main theorem is: Theorem (Informal version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The category of equivariant modules over any com- mutative G-equivariant ring spectrum R is stratified in terms of the geometric fixed points ΦHR equipped with their Weyl group actions for all subgroups H of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Of particular importance to us is the special case of Borel-equivariant Lubin–Tate E-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Their non-equivariant counterparts constitute outstanding examples of commutative rings of mixed chromatic characteristic ([HS98]) and play a pivotal role in chromatic stable homotopy theory (see for example [Mor85, GHMR05]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' While they are known to afford a generalized character theory in the sense of Hopkins, Kuhn, and Ravenel [HKR00], a classification of isomorphism classes of E-linear representation theory of finite groups seems out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In light of this, our second main theorem provides a ‘coarse’ parametrization of all (permutation) representations of a finite group G over any Lubin–Tate E-theory in terms of the Zariski spectrum of the E-cohomology of G: Theorem (Informal version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The category of G-equivariant modules over Borel- equivariant Lubin–Tate E-theory is cohomologically stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 3 In other words, we obtain a chromatic analogue in all mixed intermediate charac- teristics of the aforementioned work of Benson, Iyengar, and Krause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This completes the stratification program for the higher representation theory of finite groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', representations in all characteristics over the sphere spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Overview of cohomological stratification for representations of finite groups Coefficients Chromatic height Spectrum Reference Q 0 Spec(Q) [Mas99] Fp (p, ∞) Spech(H∗(G, Fp)) [BCR97, BIK11b] Zp 0, (p, ∞) Spech(H∗(G, Zp)) [Lau21, Bar21] En 0, (p, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' , (p, n) Spec(E0 n(BG)) [Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4] Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Based on previous ideas by Balmer–Favi [BF11], Benson–Iyengar– Krause [BIK11c], as well as Stevenson [Ste13] and extending their works, a subset of the present authors together with Sanders have developed a general framework of stratification in tensor-triangular geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This theory utilizes a universal support function to parametrize localizing ideals of a suitable compactly generated tensor-triangulated category T in terms of subsets of the Balmer spectrum of Tc: Supp: �Localizing ⊗-ideals of T � � Subsets of Spc(Tc) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We then call the tt-category T stratified if Supp is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This separates the problem of classifying localizing ⊗-ideals into two questions: Is T stratified ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' What is Spc(Tc) as a set?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In [BHS21], we provide various techniques for approaching both questions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in practice, this often turns out to be easier than a full computation of Spc(Tc) which is equivalent to a parametrization of all thick ⊗-ideals of compact objects Tc in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The general paradigm goes as follows: First construct a suitable family of jointly conservative tt-functors fi : T Si with all Si stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Secondly, show that stratification descends along the family of functors fi, using the above tt-methods established in [BHS21] and [Bar21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Assuming for simplicity that the family is finite, it follows that there is a continuous surjection � i Spc(Sc i) Spc(Tc), see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The third step relies on the geometry of the functors fi to determine the spectrum of Tc in terms of Spc(Sc i), potentially using additional structure available in the given context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our first aim is to carry out this strategy in the context of stable equivariant homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consider a finite group G, fixed throughout, and let SpG be the category of genuine G-equivariant spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R be a commutative G-equivariant ring spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', a commutative algebra in the symmetric monoidal category SpG, and let ModG(R) be the tt-category of G-equivariant modules over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Write PerfG(R) for the full subcategory of compact objects in ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Equivariant 4 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL homotopy theory supplies excellent candidates for the family fi, namely the geometric fixed points ΦH : ModG(R) Mod(ΦHR) for H varying over the subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is a natural action of the Weyl group WG(H) = NG(H)/H on Mod(ΦHR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' With this preparation, the next result is then a precise statement of our first main theorem written above: Theorem A (Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R be a commutative equivariant ring spectrum and write ModG(R) for the category of G-equivariant modules over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then: (a) The spectrum of perfect R-modules admits a locally-closed decomposition (†) Spc(PerfG(R)) ≃ � (H)⊆G Spc(Perf(ΦHR))/WG(H), with the set-theoretic disjoint union being indexed on conjugacy classes of subgroups of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) ModG(R) is stratified if the categories Mod(ΦHR) are stratified with Noe- therian spectrum for all subgroups H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In both (a) and (b) it suffices to index on a family F of subgroups H ⊆ G such that R is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To orient intuition and reconnect to Quillen’s work on group cohomology, recall that any field k may be viewed as a commutative equivariant ring spectrum k by passing to the Borel-completion of the associated Eilenberg–MacLane ring spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The corresponding tt-category of equivariant modules identifies1 on compacts with the bounded derived category of finitely generated kG-modules: PerfG(k) ∼= Db(mod(kG)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Coupled with the computation of the spectrum in [BCR97], Theorem A, (a) ap- plied to R = k then essentially recovers Quillen’s stratification theorem2 from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For a more thorough discussion of the comparison, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Part (b) reduces equivariant stratification to a non-equivariant question, one for each subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In the case of R = k, this reduction does not constitute a significant simplification, but it’s surprisingly powerful in many other instances, as we will demonstrate below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Specializing in a different direction, Theorem A is also interesting for equivariant ring spectra R = infl(Re) with trivial G-action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', those that are inflated from a non- equivariant commutative ring spectrum Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The initial example is the equivariant sphere spectrum S0 G, the monoidal unit of SpG: in this case, Theorem A, (a) reduces to the computation of the underlying set of Spc(Spc G) obtained by Strickland and Balmer–Sanders [BS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' More generally, for such R, the tt-category ModG(R) is equivalent to the tt-category of spectral Mackey functors with coefficients in Mod(Re), see [PSW22, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11] (for Re = Z) and [BHS21, Proposition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3] (for the general case), all Weyl group actions on the relevant spectra are trivial by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10, and Theorem A recovers the results of [BHS21, Part 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 1This identification requires a result on generation by permutation modules due to Rouquier, Mathew, and Balmer–Gallauer, see Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2but for homogeneous prime ideals only QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 5 Under additional assumptions on the equivariant ring spectrum R, we can find a more economical model of the decomposition of Theorem A: Theorem B (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If R arises as the restriction of a global equivariant ring spectrum Rgl, then the Weyl group WG(H) in Theorem A may be replaced by the ‘global Weyl group’ W gl G (H) = NG(H)/H · CG(H), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', the centralizer acts trivially in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, if H is abelian, then it suffices to consider the action of the ‘Quillen–Weyl group’ W Q G (H) = NG(H)/CG(H) on Spc(Perf(ΦHR)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We provide an example that demonstrates that the conclusion of this theorem fails for arbitrary equivariant ring spectra (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19) and that, even if it holds, the resulting action is in general neither free nor trivial (Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' While Theorem B looks innocuous, it becomes very useful in practice when we try to compute actions in explicit examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The next theorem collects several new stratification results for prominent equivariant ring spectra from Section 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' each of them follows in a straightforward way from the novel techniques discussed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The category ModG(R) is stratified in each of the following cases: (a) (Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1) The integral constant Green functor R = HZ for any cyclic p-group G, with stratified Balmer spectrum described in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) (Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12) Equivariant K-theory R = KUG for any finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this case, Spc(PerfG(KUG)) ∼= Spec(π0KUG), where π0KUG ∼= R(G) is the complex representation ring of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) (Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15) Atiyah’s K-theory with reality R = KR for G = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this case, Spc(PerfC2(KR)) ∼= Spec(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In summary, our equivariant stratification theorem expresses a substantial part of the equivariant tt-geometry of any equivariant ring spectrum R in terms of the non-equivariant tt-geometry of their geometric fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' As such, the list in Theorem C is not exhaustive, but rather intended as a proof of concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In order to give a complete description of the equivariant tt-geometry of R, it remains to determine how the strata of Spc(PerfG(R)) in Theorem A, (a) are glued together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In general, this is a problem that poses substantial difficulties, as witnessed by the case of SpG for which we can still only resolve this question for abelian groups [BHN+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To illustrate this point further, note that Theorem C, (a) gives a short and independent proof of a recent result of Balmer–Gallauer [BG22b] for cyclic p-groups, the case relevant for the motivic tt-geometry of Artin motives over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' However, it requires additional techniques to determine the remaining specialization relations in Figure 3, and this corresponds precisely to the gluing data between the strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' While for cyclic groups, this could be done ‘by hand’ and will also be part of their forthcoming work [BG20], a more systematic understanding for arbitrary equivariant ring spectra R would be desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Turning attention to the main example of interest to us in this paper, let E = En be a Lubin–Tate E-theory of height n at the prime p and consider its G-Borel completion E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' From the point of view of chromatic homotopy theory, E is a commutative ring spectrum of mixed characteristic3 ((0), (p, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' , (p, n)) and it is arguably the most important example of such a spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It plays the same 3Here, being of characteristic (p, n) is tested against the Morava K-theories K(p, n), which form the prime fields of the stable homotopy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 6 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL fundamental role in higher algebra as the p-adic integers Zp in ordinary algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' note that, viewed chromatically, the latter is of characteristic ((0), (p, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Unsurprisingly, understanding the E-cohomology of finite groups has a long history, starting with the work of Ravenel from the 1970s, but also stimulating much recent activity, such as [Sta13], [MNN19], [Lur19], and [CMNN20] with connections to algebraic K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A version of Quillen’s stratification for Spec(E0(BG)) was previously found by Greenlees and Strickland [GS99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It is therefore natural to wonder if the categorification of Quillen’s classical stratification theorem in the work of Benson–Carlson–Rickard [BCR97] and Benson–Iyengar–Krause [BIK11b] admits a chromatic counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our next theorem provides an affirmative answer: Theorem D (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4 and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let E = En be a G-Borel-equivariant Lubin–Tate E-theory of height n and at the prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The category ModG(E) is cohomologically stratified, and there is a decomposition into locally-closed subsets4 Spc(PerfG(E)) ∼= Spec(E0(BG)) ≃ � A Spec(π0ΦAE)/W Q G (A), where the disjoint union is indexed on abelian p-subgroups A of G generated by at most n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, the generalized telescope conjecture holds for ModG(E) and there are explicit bijections �Thick ⊗-ideals of PerfG(E) � � Specialization closed subsets of Spec(E0(BG)) � ∼ and �Localizing ⊗-ideals of ModG(E) � � Subsets of Spec(E0(BG)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ∼ We include a brief outline of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since E is Borel-complete and En is complex orientable, in a first step we utilize work of Mathew, Naumann, and Noel [MNN19] to reduce to the case of a finite abelian p-group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' From here on, our strategy diverges from the approach of Benson, Iyengar, and Krause at chromatic height ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The next step applies Theorem A and Theorem B to reduce further to the following two tasks: (a) Show that Mod(ΦAE) is stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) Identify Spc(PerfA(E)) with Spec(E0(BA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In order to solve (b), we use Balmer’s comparison map [Bal10] to compare the stratification of Spc(PerfA(E)) to the analogous stratification of Spec(E0(BA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In light of results proven by Dell’Ambrogio and Stanley [DS16], both (a) and (b) are then consequences of the following key technical input to the proof of Theorem D: Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The commutative ring π0ΦAE is regular Noetherian for any finite abelian p-group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This regularity result is an elaboration on a theorem of Drinfel’d [Dri74] and its extension by Strickland [Str97] on moduli of level structures, and its proof follows the analysis undertaken in [BHN+19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' With that, we conclude our outline of the proof of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 4Note that the first map is homeomorphism, while the second map provides only a stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 7 Relation to previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Quillen’s work has found numerous extensions in various contexts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' here, we briefly review the ones closest to the present work and indicate their relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In the classical setting of a finite group G and a field k of characteristic p dividing the order of G, Quillen first established a ‘weak version’ of the stratification theorem, in the form of a homeomorphism (‡) colim G/E∈OE(G) Spec H•(E, k) ∼= Spec H•(G, k), where the colimit is indexed on the orbit category of elementary abelian p-subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' On the one hand, Mathew–Naumann–Noel [MNN19] produce a generalization of this formula for coefficients in an arbitrary commutative equivariant ring spectrum in place of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' On the other hand, (‡) has found a tt-geometric incarnation for the spectrum of a k-linear tt-category K(G) which admits a tt-functor Db(FpG) → K(G) in Balmer’s [Bal16b]: colim G/E∈OE(G) Spc(K(E)) ∼= Spc(K(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The proof of Theorem A uses parts of both of these results as input, and refines them tt-geometrically in three ways: firstly, (†) establishes a ‘strong version’ of Quillen’s theorem as in (∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in particular, the strata in our decomposition are given by non-equivariant data via geometric fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Secondly, we work in the generality of equivariant modules over an arbitrary equivariant ring spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' And thirdly, we also control the tt-geometry of not-necessarily compact objects in terms of stratification, akin to the passage from [BCR97] to [BIK11b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This last part is modelled on the approach taken in [BHS21] for equivariant ring spectra with trivial action, and draws from the general theory of stratification developed therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The approach to probe the tt-geometry of equivariant categories through their geometric fixed points is based on the work of Strickland and Balmer–Sanders [BS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our results have found inspiration and precursors in their work (R = S0 G), [BGH20] (R = S0 G, compact Lie groups), [PSW22] (R = HZG), as well as [BHS21] (R = infl(Re));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' as such, the present paper unifies these different developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A related notion of stratification also appears in the work of Ayala, Mazel-Gee, and Rozenblyum [AMR19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There, the authors construct an adelic decomposition of the category of G-spectra in terms of geometric fixed points together with gluing data between these;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' a generalization to equivariant ring spectra with trivial action was subsequently given in [AMR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' However, in contrast to Theorem A, this does not result in a classification of ⊗-ideals in these categories: from the point of view of tt-stratification as considered here, [AMR19, AMR21] give an explicit manifestation of the local-to-global principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' An analogue of Quillen’s stratification for the En-cohomology of finite group has also been studied by Greenlees and Strickland [GS99], extending earlier work of Hopkins–Kuhn–Ravenel [HKR00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' They prove a decomposition of the Zariski spectrum Spec(E0 n(BG)) into irreducible varieties indexed on abelian p-subgroups of G of rank at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Pulled back to a pure (or monochromatic) stratum, their decomposition is into disjoint subsets, thereby establishing a version of the strong form (∗) of Quillen stratification for these monochromatic strata .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Outline of the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We begin with preliminaries on tensor-triangular geometry in Section 2, in particular reviewing some background material on stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In Section 3 we introduce the equivariant 8 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL stable homotopy category, and prove that stratification can be detected by the geometric fixed point functors (part (b) of Theorem A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Sections 4 and 5 are dedicated to strong Quillen stratification in equivariant homotopy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in particular, they contain the proofs of part (a) of Theorem A and of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In Sections 6 and 7 we establish stratification for Borel-equivariant Lubin–Tate E-theory and compute the Balmer spectrum, proving Theorems D and E, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Finally, Section 8 is dedicated to the proofs of the remaining examples in Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For a finite group G, we let SpG denote the ∞- category of genuine G-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given an algebra object R ∈ Alg(SpG), we write ModG(R) for the ∞-category of left R-modules in G-spectra, and PerfG(R) for its full subcategory of compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will often implicitly view a symmetric monoidal stable ∞-category as a tensor-triangulated category via the homotopy category functor Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For example given an essentially small symmetric monoidal stable ∞-category C, we will write Spc(C) for the Balmer spectrum of the associated tensor-triangulated category Ho(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Similarly, we will use the homotopy category functor to extend common notions in tensor-triangulated geometry (such as that of a finite ´etale functor) to the world of ∞-categories, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given a closed symmetric monoidal (∞-)category C, we denote the symmetric monoidal structure by ⊗, the internal hom by Hom and the unit object by 1, and write D(−) = Hom(−, 1) for the duality functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We are grateful to Scott Balchin, Paul Balmer, David Ben- son, Clover May, Beren Sanders, and Nat Stapleton for useful conversations about the subject matter of this paper, and we thank Akhil Mathew and Nat Stapleton for pointing out that something like Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14 should be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Special thanks to Beren for helpful comments on an earlier version of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' TB is supported by the European Research Council (ERC) under Horizon Eu- rope (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 101042990) and would like to thank the Max Planck Institute for its hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' NC is partially supported by Spanish State Research Agency project PID2020-116481GB-I00, the Severo Ochoa and Mar´ıa de Maeztu Pro- gram for Centers and Units of Excellence in R&D (CEX2020-001084-M), and the CERCA Programme/Generalitat de Catalunya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' DH is supported by grant number TMS2020TMT02 from the Trond Mohn Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' NN and LP are supported by the SFB 1085 Higher Invariants in Regensburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This material is partially based upon work supported by the Swedish Research Council under grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2016-06596 while TB, NC, and LP were in residence at Institut Mittag-Leffler in Djursholm, Sweden during the semester Higher algebraic structures in algebra, topology and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The authors would also like to thank the Hausdorff Research Institute for Mathemat- ics for the hospitality in the context of the Trimester program Spectral Methods in Algebra, Geometry, and Topology, funded by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Tensor-triangular preliminaries In this section we recall the key concepts from tensor-triangular geometry that we will use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' With the exception of our abstract nilpotence theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25), the material presented here is standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' we therefore invite the reader familiar with basic tt-geometry to skip this section and only refer to it for notation and terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollections on basic tt-geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We begin by fixing some relevant terminology in tt-geometry, as developed by Balmer [Bal05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A tensor-triangulated category (tt-category for short) is a triple (T, ⊗, 1) consisting of a triangulated category T with symmetric monoidal product ⊗ which is exact in each variable, and tensor unit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A tt-functor F : T → S is an exact symmetric monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let T be a tt-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A triangulated subcategory J ⊆ T is thick if it is closed under retracts and it is called a ⊗-ideal if T ⊗ J ⊆ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any class of objects E ⊆ T, we will write thick⟨E⟩ for the smallest thick subcategory of T containing E and thick⊗⟨E⟩ for the smallest thick ⊗-ideal containing E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If in addition T admits arbitrary (set-indexed) coproduts, we call a triangulated subcategory L ⊆ T localizing if it is closed under coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any class of objects E ⊆ T, we will write Loc⟨E⟩ for the smallest localizing subcategory containing E, and Loc⊗⟨E⟩ for the smallest localizing ⊗-ideal containing E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Observe that, for any exact functor f : T → S and any collection E of objects in T, there is an inclusion f(thick⟨E⟩) ⊆ thick⟨f(E)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This uses that the collection of t ∈ T with f(t) ∈ thick⟨f(E)⟩ is thick and contains E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If the exact functor f additionally preserves coproducts, then the corresponding statement is true for localizing subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The analogous results hold for tt-functors and ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Warning 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' When the tt-category T admits set-indexed coproducts, one may consider thick ⊗-ideals either in T or in the full subcategory Tc of T spanned by the compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Both concepts will appear in this paper, and we will usually specify the ambient category in case it is not clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let K be an essentially small tt-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Balmer [Bal05] con- structed a topological space Spc(K), called the spectrum of K, defined as follows: the points of Spc(K) are the prime ⊗-ideals of K, that is those thick ⊗-ideals J ⊆ K which are proper and satisfy ∀ a, b ∈ T : (a ⊗ b ∈ J =⇒ a ∈ J or b ∈ J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The support of an object a ∈ K is defined to be supp(a) = � P ∈ Spc(K) �� a ̸∈ P � , and the topology of Spc(K) is the one having {supp(a)}a∈K as a basis of closed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In [Bal05, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2] Balmer proves that the pair (Spc(K), supp) is the final support datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Explicitly, this means that given a pair (X, σ) consisting of a topological space X and an assignment σ which associates to any object a ∈ K a closed subset σ(a) ⊆ X satisfying the conditions given in [Bal05, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1], then there exists a unique map f : X → Spc(K) such that σ(a) = f −1(supp(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, by [Bal05, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10] there is an order-preserving bijection �Radical thick ⊗-ideals of K � supp � supp−1 � �Thomason subsets of Spc(K) � , where supp(J) = � a∈J supp(a), and supp−1(V) = � a ∈ K �� supp(a) ⊆ V � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Here a thick ⊗-ideal J is radical if it satisfies a⊗n ∈ J =⇒ a ∈ J and a subset Y ⊆ Spc(K) is Thomason if it is a union of closed subsets each of which has quasi-compact 10 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that if K is rigid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', every object has a dual (which it will be in all examples that we consider), then every thick ⊗-ideal is automatically radical [Bal07, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, if Spc(K) is Noetherian, then Thomason subsets are exactly specialization closed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In any essentially small tt-category K the graded endomorphism ring R := End∗ K(1) is graded commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In [Bal10, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6] Balmer defines a natural continuous comparison map ρ: Spc(K) → Spech(R) P �→ {f | cone(f) ̸∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R0 denote the commutative ring of degree zero elements in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By combining ρ with the continuous map Spech(R) → Spec(R0) which sends a homogeneous prime ideal p to p0 := p ∩ R0, we also obtain an ungraded comparison map ρ0 : Spc(K) → Spec(R0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We recall that if R is a 2-periodic evenly graded commutative ring, then q �→ q ∩ R0 : Spech(R) � Spec(R0) : p · R ←� p � are inverse homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Suppose the graded endomorphism ring R of the unit 1 in K is 2-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then, if the comparison map ρ: Spc(K) → Spech(R) is a homeomorphism, we deduce from the commutative diagram Spc(K) Spech(R) Spec(R0) ρ ∼ = ρ0 of [Bal10, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6] that the ungraded comparison map ρ0 is also a homeomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Under certain conditions ρ is a homeomorphism if and only if it is bijective, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Stratification in tt-geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This subsection contains a rapid review of the theory of stratification of rigidly-compacty generated tt-categories T relative to the Balmer spectrum Spc(Tc) as developed in [BHS21], as well as a comparison to the notion of cohomological stratification of Benson, Iyengar, and Krause [BIK11c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A tt-category T with arbitrary coproducts and a closed symmetric monoidal structure is rigidly-compactly generated if it is compactly generated, the unit 1 is compact, and every compact object is rigid, or dualizable (under these conditions, the compact and rigid objects of T coincide, see [HPS97, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3(d)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We write K = Tc for the subcategory of rigid and compact objects of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that Tc is an essentially small tt-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Here are two examples, studied in [HPS97, Section 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (a) For any commutative ring R, the derived category T = D(R) of unbounded chain complexes of R-modules is a tt-category, with K the subcategory of complexes quasi-isomorphic to a bounded complex of finitely generated projective R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 11 (b) For any finite group G, the homotopy category of G-spectra (studied in more detail in Section 3) is a rigidly-compactly generated tt-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A rigidly-compactly generated tt-category T is called Noetherian if the endomorphism ring End∗ T(1) is graded Noetherian and the module End∗ T(C) is finitely generated over End∗ T(1) for any compact object C ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10 ([Lau21, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If T is Noetherian, then the comparison map ρ is a homeomorphism if and only if it is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If T is an affine weakly regular tensor-triangulated category in the sense of [DS16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1], then T is Noetherian, see [DS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For R a commutative ring, work of Hopkins and Neeman [Hop87, Nee92] (for R Noetherian) and Thomason [Tho97] (in general) imply that the comparison map ρ: Spc(D(R)c) ∼ = −→ Spec(R) is a homeomorphism, see [Bal10, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, Neeman has shown in [Nee92, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8] that if R is Noetherian, then in fact Spec(R) ∼= Spc(D(R)c) parameterizes all localizing ⊗-ideals of D(R): there is a bijection �localizing ⊗-ideals of D(R) � ∼ � �Subsets of Spc(D(R)c)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Following Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12, one would like to know for which rigidly- compactly generated tt-categories (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7) the Balmer spectrum of compact objects also parameterizes the localizing ⊗-ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The first step towards doing this is to extend the notion of support from compact objects to all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To that end, suppose that T is a rigidly-compactly generated tt-category whose Balmer spectrum of compact objects is Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then, Balmer–Favi [BF11] and Stevenson [Ste13] have extended the universal support function from Tc to all of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We denote this extension by Supp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' we thus obtain maps: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14) �Localizing ⊗-ideals of T � Supp � Supp−1 � �Subsets of Spc(Tc)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Following and extending work of Benson–Iyengar–Krause [BIK11c] and Stevenson [Ste13], Barthel–Heard–Sanders [BHS21] introduced the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let T be a rigidly-compactly generated tt-category with Spc(Tc) Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If the maps (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14) are inverse bijections, then we say that T is stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The theory developed in [BHS21] allows for the more general case where Spc(Tc) is only weakly Noetherian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', every point in Spc(Tc) is the inter- section of a Thomason subset and the complement of a Thomason subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this paper, we will only consider the case where Spc(Tc) is Noetherian, in which case the local-to-global principle of [BHS21, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8] holds automatically [BHS21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let T be a rigidly-compactly generated tt-category with Spc(Tc) Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To each prime P ∈ Spc(Tc) one can associate a tensor idempotent ΓP1 ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5 By [BHS21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='21 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1], stratification for T is equivalent to the following condition: 5Alternatively denoted g(P) in [BHS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 12 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL (∗) For each P ∈ Spc(Tc), ΓPT := Loc⊗⟨ΓP1⟩ is a minimal localizing ⊗-ideal of T, or equivalently, for all t ∈ T Loc⊗⟨t⟩ = Loc⊗⟨ΓP1 | P ∈ Supp(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We say that T has minimality at P if ΓPT is a minimal localizing ⊗-ideal of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If T is a rigidly-compactly generated tt-category which is stratified and has a Noetherian spectrum Spc(Tc), then we can deduce further consequences beyond a classification of the localizing ⊗-ideals of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For example: (a) the telescope conjecture holds [BHS21, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', the kernel of every smashing (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', coproduct-preserving) localization is generated by compact objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) the Balmer–Favi support satisfies the tensor product property, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', there is an equality Supp(t1 ⊗t2) = Supp(t1)∩Supp(t2) for all t1, t2 ∈ T, by [BHS21, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) and the Bousfield lattice is isomorphic to the lattice of subsets of Spc(Tc) [BHS21, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The theory of stratification is based on previous work of Benson– Iyengar–Krause [BIK11c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Rather than working with the Balmer spectrum, they consider an action of a graded commutative Noetherian ring R on T, and parametrize localizing ⊗-ideals of T in terms of subsets of the homogeneous spectrum Spech(R) via a support theory SuppR⟳T that depends on the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For a Noetherian tt-category T the following conditions are equivalent: (a) T is stratified and Balmer’s comparison map ρ is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) T is stratified by the action of R = End∗ T(1) in the sense of Benson, Iyengar, and Krause [BIK11c], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', the analogue of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15 holds for the support theory SuppR⟳T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The implication (b) implies (a) is proven in [BHS21, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14], while the converse has been observed by Changhan Zou [Zou].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Indeed, assume (a) holds, then stratification and the homeomorphism ρ give a bijection �Localizing ⊗-ideals of T � � Subsets of Spech(R) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ∼ By [BIK11c, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2] it suffices to show that this bijection is given by the Benson–Iyengar–Krause theory of support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This follows from [Ste13, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A more thorough discussion will appear in Zou’s forthcoming work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A rigidly-compactly generated Noetherian tt-category T is said to be cohomologically stratified if the equivalent conditions of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20 hold for T equipped with the action of End∗ T(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If R is a commutative Noetherian ring, then Neeman’s theorem (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12) can be reinterpreted as the statement that D(R) is cohomologically stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' More generally, if X is a Noetherian scheme and Dqc(X) denotes the derived category of complexes of OX-modules with quasi-coherent cohomology, then Spc(Dqc(X)c) ∼= |X|, the underlying topological space of X, and Dqc(X) is stratified, see [Ste13, Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13] and [BHS21, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that the comparison map ρ will not be a homeomorphism in general in this case [Bal10, Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2], so that Dqc(X) is not cohomologically stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 13 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The approach of stratification via the Balmer spectrum developed systematically in [BHS21] separates the classification of localizing ideals into two parts: one is to determine the set underlying the spectrum Spc(Tc), and the other is to show that it stratifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is the approach that we will take in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, note that stratification in this sense only relies on an understanding of the underlying set of Spc(Tc), rather than its topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In a second step, one can then try to determine the topology on Spc(Tc), which by Balmer’s work [Bal05] is tantamount to the classification of thick tensor ideals of the compact objects in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is in contrast to the approach of [BIK11c], which solves both classification problems simultaneously via the auxiliary action of a Noetherian commutative ring on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In practice, this separation is very useful, as there are examples where we can show stratification without a full understanding of the topology on Spc(Tc), see [BHS21, Remark 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12] as well as the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A tt-nilpotence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The next result constitutes a tt-theoretic gener- alization of the equivariant nilpotence theorem of [BGH20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19], which in turn was inspired by the abstract nilpotence theorem of [HPS97, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Ultimately, the idea of the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25 is due to Devinatz, Hopkins, and Smith [DHS88, HS98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We say that a collection (Fi : T → Si)i∈I of exact symmetric monoidal functors between tt-categories is jointly conservative if an object X ∈ T is zero if and only if Fi(X) = 0 for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The collection (Fi)i∈I is said to detect ⊗-nilpotence of morphisms with dualizable source if the following condition holds: a morphism α: C → X in T with C dualizable satisfies α⊗m = 0 for some m ≥ 0 if and only if for all i ∈ I there exists mi such that Fi(α)⊗mi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that in the definition of detection of ⊗-nilpotence we do not assume that the mi are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let (Φi : T → Si)i∈I be a collection of coproduct-preserving tt- functors between rigidly-compactly generated tt-categories, indexed on a set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If the functors Φi are jointly conservative on T, then they detect ⊗-nilpotence of morphisms with dualizable source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consider a morphism α: C → X in T with C dualizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let Y = Hom(C, X), and write y: 1 → Y for the adjoint of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The dualizability of C implies that Hom(C⊗k, X⊗k) ≃ D(C)⊗k ⊗ X⊗k ≃ Hom(C, X)⊗k for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Under these identifications, the adjoint of α⊗k is the composition y(k) : 1 y−→ Y y−→ Y ⊗2 y−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' y−→ Y ⊗k, as one easily checks by observing that y(k) identifies with y⊗k (see also [HS98, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, α is ⊗-nilpotent if and only if y is nilpotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', y(m) : 1 y−→ Y y−→ Y ⊗2 y−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' y−→ Y ⊗m is zero for some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Now, denote the homotopy colimit of the infinite composite sequence by T(y) = hocolim(1 y−→ Y y−→ Y ⊗2 y−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since the unit in T is compact, the argument in [HS98, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4] shows that y is nilpotent if and only if T(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' With this preparation, we can now argue as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The map α is ⊗-nilpotent if and only if T(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since the collection (Φi)i∈I is jointly conservative, T(y) = 0 if and only if Φi(T(y)) = 0 for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' As Φi preserves colimits and tensor products, 14 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL Φi(T(y)) = T(Φi(y)) for any i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then, in turn, α is ⊗-nilpotent if and only if Φi(y) is nilpotent for all i ∈ I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', that Φi(α) is ⊗-nilpotent for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The last statement uses that Φi(C) is dualizable, following the argument in the first paragraph of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let Φ: T → S be a coproduct-preserving tt-functor between rigidly- compactly generated tt-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If the functor Φ is conservative, then Φ induces a surjective map Spc(Φc): Spc(Sc) � Spc(Tc) on Balmer spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25, the functor Φ detects ⊗-nilpotence between compact objects in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It then follows from [Bal18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3] that Spc(Φc) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This result is a ‘big’ variant of Balmer’s [Bal18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2], with a stronger hypothesis and a stronger conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Base-change, separable algebras, and ´etale morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this section we collect some facts about module categories internal to a tt-category (or symmetric monoidal stable ∞-category), base-change, and recall the notions of separable algebras and ´etale morphisms for tt-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The main references for this material are [BDS15] and [Bal16b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let C be a rigidly-compactly generated symmetric monoidal stable ∞-category and consider a commutative algebra A ∈ CAlg(C) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is then an associated symmetric monoidal ∞-category of modules over A internal to C, denoted ModC(A), along with an extension of scalars functor FA : C → ModC(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If G is a set of compact generators for C, then ModC(A) is rigidly-compactly generated by the set A ⊗ G := {FA(G) | G ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Indeed, the claim about compact generation follows because the forgetful functor UA : ModC(A) → C is conservative and preserves colimits [Lur17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5], and so its left adjoint FA : C → ModC(A) preserves generating sets and compact objects [Lur09, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Combining [Lur17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1] we see moreover that ModC(A) inherits a symmetric monoidal structure from C and that FA is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It follows that FA preserves dualizable objects, and hence that ModC(A) is rigidly-compactly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let θ: C → D be a lax monoidal functor between presentable symmetric-monoidal ∞-categories whose tensor product commutes with all colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any commutative algebra A ∈ CAlg(C), there is a diagram C D ModC(A) ModD(θ(A)), UA θ θ Uθ(A) FA Fθ(A) where F and U denote extension and restriction of scalars along the units of A and θ(A), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The functor θ is compatible with restriction of scalars, in the sense that θ ◦ UA ≃ Uθ(A) ◦ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If θ is symmetric monoidal and preserves geometric realizations of simplicial objects (for example, if θ preserves colimits), then we additionally have θ ◦ FA ≃ Fθ(A) ◦ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Both of these statements follow from the discussion in [Erg22, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We also note that under these QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 15 stronger assumptions, the induced functor θ: ModC(A) → ModD(θ(A)) is symmetric monoidal [Lur17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2], and that if θ: C → D preserves colimits, then so does θ: ModC(A) → ModD(θ(A)), as it is a left adjoint, see [Lur17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In general, one does not expect the category of modules over a commutative algebra object A in a tt-category to inherit the structure of a tt- category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' However, under suitable conditions on A, it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' One such notion is that of a separable algebra, first introduced in the tt-context by Balmer in [Bal11], which turns out to be especially well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, it allows us to prove an extension result for natural transformations, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='35 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let T be a tt-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A ring object A ∈ T is separable if the multiplication map µ: A ⊗ A → A admits a section as a map of (A, A)-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Explicitly, there exists σ: A → A ⊗ A such that µσ = idA and (1 ⊗ µ) ◦ (σ ⊗ 1) = σµ = (µ ⊗ 1) ◦ (1 ⊗ σ): A ⊗ A → A ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Under some mild conditions on the tt-category, any separable ring object has a notion of tt-degree by work of Balmer [Bal14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The tt-degree can either be a non-negative integer or be equal to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In practice, the additional assumption of having finite tt-degree is harmless, see for instance [Bal14, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let T and S be rigidly-compactly generated tt-categories and let F : T → S be a coproduct-preserving tt-functor with right adjoint G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Following [Bal16b], we say that F is finite ´etale if there is a compact commutative separable algebra A of finite tt-degree in T and an equivalence of tt-categories S ∼= ModT(A) under which the adjunction F : T ⇆ S: G is identified with the extension-of- scalars/restriction adjunction FA : T ⇆ ModT(A): UA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A colimit-preserving sym- metric monoidal functor between rigidly-compactly generated symmetric monoidal stable ∞-categories F : C → D is finite ´etale if Ho(F): Ho(C) → Ho(D) is finite ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let C be a rigidly-compactly generated symmetric monoidal stable ∞-category and consider A ∈ CAlg(C) such that A ∈ Ho(C) is a separable algebra of finite degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Combining the results of [DS18] and [San22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8] (for the symmetric monoidal structure), there is then a tt-equivalence Ho(ModC(A)) ≃ ModHo(C)(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, it follows that the base-change functor C → ModC(A) is finite ´etale in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In other words, this provides the compatibility between enhanced notions of ´etale and their effect on homotopy categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will need the next technical result later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let T and S be tt-categories and let A ∈ CAlg(T) be separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let FA : T ⇆ ModT(A): UA denote the free-forgetful adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Suppose we are given two tt-functors H0, H1 : ModT(A) → S and a natural transformation η: H0 ◦ FA ⇒ H1 ◦ FA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then there exists a unique natural transformation �η: H0 ⇒ H1 which extends η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since A is separable the counit of the adjunction ϵ: FAUA ⇒ 1 admits a section σ: 1 ⇒ FAUA [Bal11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any M ∈ ModT(A), we define 16 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL �ηM to be given by the composite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='36) H0M H0σM −−−−→ H0FAUAM ηUAM −−−−→ H1FAUAM H1ϵM −−−−→ H1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is natural in M since ηUAM, ϵM and σM are so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any X ∈ T, we have a commutative diagram H0FAX H1FAX H0FAX H0FAUAFAX H1FAUAFAX H1FAX ηX 1 1 �ηFAX H0σFAX H0ϵFAX ηUAFAX H1ϵFAX H1ϵFAX showing that �η does extend η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Finally, we note that any natural transformation extending η must be compatible with ϵ and σ and hence must be given by the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Stratifications in stable equivariant homotopy theory In this section we begin our study of the tt-geometry of modules internal to the category of G-equivariant spectra, for G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We establish a nilpotence theorem for any commutative equivariant ring spectrum R (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='28), general- izing the one for the equivariant sphere due to Balmer and Sanders [BS17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15], show that the Balmer spectrum of perfect equivariant R-modules is covered by the spectra of perfect modules over the geometric fixed points (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29), and finally prove that stratification of ModG(R) can be detected by geometric fixed point functors (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollections on equivariant homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Here we introduce some relevant notions from equivariant stable homotopy theory in the context of equivariant module categories over a commutative equivariant G-ring spectrum, for G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular, with a view towards our tt-theoretc applications, we establish the naturality of the various constructions with respect to change of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This material is standard, albeit not readily available in the literature as far as we are aware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' our exposition in this subsection and the next one loosely follows and generalizes the approach taken in [PSW22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We let SpG denote the stable ∞-category of G-spectra as constructed in [GM20, Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1] with the symmetric monoidal structure arising from [GM20, Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By [GM20, Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9] this is symmetric monoidally equivalent to the underlying ∞-category associated to the category of orthogonal G-spectra with the stable model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We denote the symmetric monoidal structure on SpG by ⊗ and the unit object by S0 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The ∞-category SpG admits a set of compact and dualizable generators G/H+ for H ⊆ G (we omit writing the G-suspension spectrum functor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This statement can be checked in the homotopy category (see [Lur17, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3] for the compact generation), where it is shown in, for example, [HPS97, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The homotopy category Ho(SpG) is then a rigidly-compactly generated tt-category in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 17 Let SG ∗ be the symmetric monoidal ∞-category of pointed G-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any group homomorphism α: G′ → G we get an essentially unique symmetric monoidal left adjoint α∗ : SpG → SpG′ (see [PSW22, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13]) making the diagram SG′ ∗ SG ∗ SpG′ SpG α∗ Σ∞ α∗ Σ∞ commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' At the level of orthogonal G-spectra, this functor is constructed in [Sch18, Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15], see also the remark at the top of page 353 of [Sch18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If α: H �→ G is the inclusion of a subgroup, then we call the resulting functor restriction, denoted resH : SpG → SpH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If α: G ↠ G/N is the quotient by a normal subgroup, then the resulting functor will be denoted inflG/N : SpG/N → SpG and will be referred to as inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For a finite group G and R ∈ CAlg(SpG) we let ModG(R) denote the ∞-category of R-modules internal to SpG, and write PerfG(R) for its full subcategory of compact (or perfect) modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We now collect some basic facts about the structure of these equivariant module categories and base-change functors between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If R ∈ CAlg(SpG), then the ∞-category of modules ModG(R) is a rigidly-compactly generated symmetric monoidal stable ∞-category, with a set of compact generators given by � R ⊗ G/H+ �� H ⊆ G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='28 and Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1 for C = SpG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(SpG) with πG ∗ (R) graded Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Suppose that for all subgroups H ⊆ G, πH ∗ (R) is a finitely generated πG ∗ (R)-module via restriction, then ModG(R) is Noetherian (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Firstly, we note that End∗ ModG(R)(R) = πG ∗ (R) which is graded Noetherian by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let X ∈ PerfG(R) and write D(X) for the internal dual of X in ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We are required to show that End∗ ModG(R)(X) is a finitely generated πG ∗ (R)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By adjunction and dualizability of X we have EndModG(R)(X) ≃ HomModG(R)(R, D(X) ⊗R X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since D(X) ⊗R X is still in PerfG(R), we are reduced to showing that Hom∗ ModG(R)(R, X) is a finitely generated πG ∗ (R)-module for each X ∈ PerfG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Using a thick subcat- egory argument and the fact that {G/H+ ⊗ R}H⊆G is a set of compact generators for ModG(R), we then reduce to showing that Hom∗ ModG(R)(R, G/H+ ⊗ R) is a finitely generated πG ∗ (R)-module for all H ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This now follows from our assumptions since Hom∗ ModG(R)(R, G/H+ ⊗ R) ∼= Hom∗ SpG(S0, G/H+ ⊗ R) ∼= πH ∗ (R) is finitely generated over πG ∗ (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ 18 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let α: G′ → G be a group homomorphism and R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By the discussion in Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29, there is a commutative diagram SpG SpG′ ModG(R) ModG′(α∗(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' α∗ FR α∗ Fα∗(R) UR Uα∗(R) If α: H �→ G is the inclusion of a subgroup, then we refer to the induced symmetric monoidal functor resH : ModG(R) → ModH(resH R) as restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' From now on, we will usually omit writing the restriction functor on an object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', we simply write resH : ModG(R) → ModH(R) for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The restriction functors SpG → SpH and ModG(R) → ModH(R) are symmetric monoidal and preserve colimits, so they admit lax monoidal right adjoints, both denoted coindH and called coinduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' These make the following diagram commute (by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5) SpH SpG ModH(R) ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' coindH UR coindH UR The top horizontal functor is conservative by the proof of [MNN17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='32] and the vertical functors are conservative by [Lur17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It then follows that the bottom horizontal functor is also conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any subgroup H ⊆ G and R ∈ CAlg(SpG), the restriction functor resH : ModG(R) � ModH(R) is a finite ´etale tt-functor with corresponding separable commutative algebra given by D(G/H+) ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is a commutative diagram of ∞-categories (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5) SpG SpH ModG(R) ModH(R), resH FR FR resH where the vertical arrows are the extension of scalars functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Passing to homotopy categories, we obtain a commutative diagram of rigidly-compactly generated tt- categories in which the top horizontal arrow is finite ´etale with corresponding separable commutative algebra A = D(G/H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Indeed, A is separable by [BDS15, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We can check that D(G/H+) has finite tt-degree by testing on all geometric fixed points [Bal14, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7(b)] and applying [Bal14, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' These results together with the second part of [BDS15, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1] show that the restriction functor resH : Ho(SpG) → Ho(SpH) is finite ´etale with A = D(G/H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 19 Using Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7 and [MNN17, Equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15] we can apply [San22, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5] to see that the bottom horizontal map in the diagram is finite ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By the compatibility of the adjunctions, the separable algebra associated to resH must be coindH R ≃ D(G/H+) ⊗ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(SpG) and a subgroup H ⊆ G be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is a symmetric monoidal equivalence LH : ModG(D(G/H+) ⊗ R) → ModH(R) which makes the following diagram commute up to isomorphism ModG(R) ModH(R) ModG(D(G/H+) ⊗ R), resH FD(G/H+)⊗R ∼ LH where FD(G/H+)⊗R denotes the extension of scalars functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Furthermore, the equivalence LH is natural with respect to the extension of scalars functors along morphisms in CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The equivalence for R = S0 G is proved in [MNN17, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='32] as an ap- plication of [MNN17, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The commutativity of the diagram follows from the diagram in [MNN17, Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='23] by passing to left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The general case follows from this by combining [BS20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9] and [MNN17, Propo- sition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Implicitly, we are using the fact that D(G/H+) ⊗ R ≃ coindH(resH R), see for instance [MNN17, Equations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Finally, the equivalence LH is explicitly given by the composite ModG(D(G/H+) ⊗ R) resH −−−→ ModH(D(G/H+) ⊗ R) −⊗D(G/H+)⊗RR −−−−−−−−−−−→ ModH(R) see [MNN17, Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='23], and this is clearly natural in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ The following corollary establishes the compatibility between our ∞-categorical constructions and the tensor-triangular context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in particular, it serves as a gateway for importing results from Balmer’s work to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be a finite group, H ⊆ G a subgroup, and R a commutative G-equivariant ring spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then there is a tt-equivalence Ho(ModG(D(G/H+) ⊗ R)) ≃ ModHo(ModG(R))(D(G/H+) ⊗ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is a special case of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='34, but we can also give a direct argu- ment: On the one hand, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8 shows that resH is an ´etale tt-functor, which by definition (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33) means that Ho(resH) is ´etale, with corresponding separable algebra given by R ⊗ D(G/H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It follows that there is a tt-equivalence ModHo(ModG(R))(D(G/H+) ⊗ R) ≃ Ho(ModH(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' On the other hand, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9 provides an equivalence Ho(LH): Ho(ModG(D(G/H+) ⊗ R)) ∼ −→ Ho(ModH(R)) of tt-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Combining these two equivalences yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ 20 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' R-linear geometric fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this subsection, we discuss geometric fixed point functors and their R-linear version for R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Fix a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We recall the construction of the geometric fixed point functors ΦH = ΦH G : SpG → Sp for each subgroup H ⊆ G, with the subscript G usually omitted from notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' First, for H = G, we define ΦG to be the finite localization of SpG away from � G/K+ �� K ⊊ G � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' it is a theorem of Lewis–May–Steinberger–McClure that the target category of this finite localization is indeed Sp, see [LMS86, Corollary II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6] at the level of homotopy categories, or [MNN17, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11] for a lift to the level of ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, ΦG splits the inflation functor infle : Sp → SpG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', ΦG ◦ infle is naturally isomorphic to the identity functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For arbitrary H ⊆ G, we define ΦH as the composite functor ΦH = ΦH G : SpG resH � SpH ΦH � Sp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8 and the definition, this is the composition of a finite ´etale extension and a finite localization, and that ΦH is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Fix a finite group G and let R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The functor ΦG : SpG → Sp induces a symmetric monoidal functor on module categories ΦG : ModG(R) → Mod(ΦGR) which we refer to as the (R-linear) G-geometric fixed point functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For an arbitrary subgroup H ⊆ G we refer to the composite ΦH = ΦH G : ModG(R) resH � ModH(R) ΦH � Mod(ΦHR) as the (R-linear) H-geometric fixed point functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consider an element g ∈ G, a subgroup H ⊆ G and R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then for all M ∈ ModG(R) we have ΦHM ≃ 0 if and only if ΦHgM ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We observe that the claim can be checked in the homotopy category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The case R = S0 G follows for example from [BS17, Section 2, (L)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The general case follows from this one by considering the commutative diagrams SpG Sp SpG Sp ModG(R) Mod(ΦHR) ModG(R) Mod(ΦHgR) ΦH ΦHg UR ΦH UΦH R UR ΦHg UΦHg R from Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5 and using the fact that the forgetful functors are conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any R ∈ CAlg(SpG) the geometric fixed point functor ΦG : ModG(R) → Mod(ΦGR) is the finite localization with respect to � R ⊗ G/K+ �� K ⊊ G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' More generally, for every subgroup H ⊆ G, ΦH : ModG(R) → Mod(ΦHR) is the composite of a finite ´etale extension and a finite localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Using Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11 the first claim follows from [PSW22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The second claim follows from the first and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We recall that a family of subgroups of a finite group G is a non- empty collection of subgroups of G that is closed under conjugation and passage to subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A pivotal example arises from a given subgroup H ⊆ G by taking [⊂ H] to be the family of subgroups of G which are G-conjugate to a proper subgroup of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We note that [⊂ H] ∩ H = PH is the family of proper subgroups of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let F be a family of subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is a pointed G-CW-complex �EF which is characterized, up to homotopy equivalence, by the property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='16) ( �EF)K ≃ � S0 if K ̸∈ F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ∗ if K ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By abuse of notation, we also write �EF ∈ SpG for the resulting suspension spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Specializing Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15 to F = [⊂ H] for a subgroup H in G, we see that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14 implies that ΦHR ≃ (R⊗ �E[⊂ H])H as objects in CAlg(Sp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, there is an equivalence ΦHM ≃ (M ⊗ �E[⊂ H])H ∈ Mod(ΦHR) for all M ∈ ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' When H = G, we recover the formula ΦGM ≃ (M ⊗ �EPG)G (see [MNN17, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We observe that �E[⊂ H] ∈ SpG is naturally an idempotent commutative algebra object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We now prove a variant of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(SpG) and a subgroup H ⊆ G be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is a symmetric monoidal equivalence ΨH : ModG(D(G/H+) ⊗ R ⊗ �E[⊂ H]) ∼ −→ Mod(ΦHR) which makes the following diagram commute up to isomorphism ModG(R) Mod(ΦHR) ModG(D(G/H+) ⊗ R ⊗ �E[⊂ H]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ΦH FD(G/H+)⊗R⊗ � E[⊂H] ΨH ∼ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consider the following diagram ModG(R) ModH(R) ModH(R ⊗ �EPH) Mod(ΦHR) ModG(D(G/H+) ⊗ R) ModG(D(G/H+ ⊗ R ⊗ �E[⊂ H])), resH FD(G/H+)⊗R F � EPH (−)H ∼ LH ∼ F � E[⊂H] �LH ∼ where the vertical symmetric monoidal equivalences are obtained by applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9 to R and R ⊗ �E[⊂ H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The same result also shows that the above diagram commutes up to isomorphism: the left triangle commutes by definition of LH and the middle square commutes by naturality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Finally we note that the top horizontal composite is equivalent to ΦH by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' One checks via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 that ModH(R ⊗ �EPH) is compactly generated by R ⊗ �EPH, so Morita theory ([Lur17, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1]) implies that ModH(R ⊗ �EPH) ≃ Mod(ΦHR) as symmetric monoidal ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The functor inducing the equivalence can be identified with the H-fixed points functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Now set ΨH := (−)H ◦ �LH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ 22 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Nilpotence and surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We recall the concept of nilpotent algebra objects in SpG from [MNN17] and explain how this interacts with the geometric fixed point functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' As a consequence, we show that, for any R ∈ CAlg(SpG), the geometric fixed point functors provide a cover of the Balmer spectrum of PerfG(R) by non-equivariant spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The following definition was given in [MNN17, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='36], up to identifying D(G/H+) with G/H+, which follows because G is finite: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given a finite group G and a family F of subgroups of G, we consider the commutative algebra object AF := � H∈F D(G/H+) ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We say that M ∈ SpG is F-nilpotent if M is in the thick ⊗-ideal of SpG generated by AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that here we allow tensoring with arbitrary objects of SpG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' As noted in [MNN19, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4] there is always a minimal family such that M is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It is called the derived defect base of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given R ∈ CAlg(Sp), we let RG denote the Borel completion of R in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By definition, RG is the image of R under the composite of functors Sp infle −−−→ SpG B −→ (SpG)Borel ⊆ SpG, where: infle is the inflation functor from Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' B is the Borel completion functor constructed as the Bousfield localization of SpG with respect to G+, given by F(EG+, −);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (SpG)Borel is the full subcategory of SpG spanned by the Borel-equivariant G-spectra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', the essential image of B, see [MNN17, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The functor B is symmetric monoidal by general facts about localizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' see the discussion around [MNN17, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20], and the composite with the right adjoint inclusion (SpG)Borel ⊆ SpG is then lax symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, the composite functor Sp → SpG is lax symmetric monoidal and hence preserves commutative algebra objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It follows that RG ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will also refer to RG as Borel-equivariant R-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Oftentimes, when the group is clear from context, we will omit the subscript and simply write R for RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We note that resH RG ≃ RH as an easy consequence of [MNN17, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will use this as an identification throughout the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If M = S0 G is the G-sphere spectrum, then M has derived defect base the family of all subgroups [MNN19, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If M = E = EG is Borel-equivariant Lubin–Tate E-theory of height n at a prime p, then M has derived defect base the family of abelian p-subgroups which can be generated by n elements [MNN19, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The commutative algebra R ⊗ AF ∈ CAlg(ModG(R)) is separable of finite degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that this holds more generally if F is just a collection of subgroups of G rather than a family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The following result, which is due to Mathew, Naumann, and Noel, relates the concept of F-nilpotence applied to ring spectra with the vanishing of geometric fixed points: QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 23 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(SpG), then R is F-nilpotent if and only if ΦHR = 0 for all H ̸∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, if R is F-nilpotent and M ∈ ModG(R), then ΦHM = 0 if H ̸∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The statement about R is [MNN17, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The second part follows because ΦHM is a module over ΦHR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be a finite group, R ∈ CAlg(SpG) and F a family of subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then the following are equivalent: (a) ModG(R) is compactly generated by R ⊗ AF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) R is F-nilpotent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) the geometric fixed point functor ΦF = (ΦH)H∈F : ModG(R) → � H∈F Mod(ΦHR) is conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (d) the fixed point functor (−)F = ((−)H)H∈F : ModG(R) → � H∈F Mod(RH) is conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (e) the restriction functor resF = (resH)H∈F : ModG(R) → � H∈F ModH(R) is conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To avoid confusion, throughout this proof we will write ΦH R and (−)H R for the R-linear version of these functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We now prove all implications in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (a) =⇒ (b): Assuming part (a) we know that R ∈ Loc⟨R ⊗ AF⟩ ⊆ ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since R and R⊗AF are compact objects of ModG(R), we can apply [Nee96, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1] and deduce that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26) R ∈ Loc⟨R ⊗ AF⟩ ∩ PerfG(R) = thick⟨R ⊗ AF⟩ ⊆ ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Applying the forgetful functor ModG(R) → SpG to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26) and keeping in mind Recollection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2, we see that there are inclusions (computed in SpG) R ∈ thick⟨R ⊗ AF⟩ ⊆ thick⊗⟨AF⟩, so (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) =⇒ (c): Suppose now that (b) holds and let us prove (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' When R = S0 G, then F is the family of all subgroups, and the claim is classical, see, for example, [MNN17, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The general case follows from this case and the observation that in the commutative square (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='27) SpG Sp ModG(R) Mod(ΦHR) ΦH UR ΦH R UΦH R the forgetful functors are conservative [Lur17, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Indeed, suppose M ∈ ModG(R) has ΦF(M) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We want to show that M ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By the conservativity of UR and the previous case, it suffices to show that ΦH(URM) = 0 for all H ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 24 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL By commutativity of the diagram above, we have ΦH(URM) = UΦHRΦH R (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If H ∈ F, this is zero by assumption, and if H ̸∈ F then this is zero by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) =⇒ (d): Suppose that (c) holds and consider M ∈ ModG(R) such that 0 ≃ M H ∈ Mod(RH) for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5 there is a commutative square SpG Sp ModG(R) Mod(RH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (−)H UR (−)H R URH It follows that 0 ≃ URH((M)H R ) ≃ (URM)H for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since F is closed under subgroups, we also have (URM)K ≃ 0 for all subgroups K ⊆ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore resH(URM) ≃ 0 for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We deduce that ΦHURM ≃ 0 for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The commutativity of the diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='27) tells us that 0 ≃ ΦHURM ≃ UΦHRΦH R M for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since (c) holds by assumption, conservativity of UΦHR then implies that M ≃ 0, which proves (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (d) =⇒ (e): Similarly (d) implies (e) since resF(M) ≃ 0 in particular implies that 0 ≃ M H ∈ Mod(RH) for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (e) =⇒ (a): Consider M ∈ ModG(R) such that HomModG(R)(R ⊗ AF, M) ≃ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' we are required to show that M ≃ 0, compare with [SS03, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By adjunction, there are equivalences 0 ≃ HomModG(R)(R ⊗ AF, M) ≃ HomSpG(AF, URM) ≃ � H∈F (URM)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Thus (URM)H ≃ 0 for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since F is closed under subgroups, then resH URM ≃ 0 for all H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, the commutativity of the following diagram (Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5) ModG(R) ModH(R) SpG SpH resH UR UR resH and conservativity of the forgetful functor UR, implies that 0 ≃ resH M ∈ ModH(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Hence by part (e), we conclude M ≃ 0 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be a finite group, R ∈ CAlg(SpG), and F a family of subgroups of G such that R is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then the geometric fixed point functors (ΦH)H∈F detect ⊗-nilpotence of morphisms with dualizable source in ModG(R), in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In fact, it suffices to choose one representative (H) for each G-conjugacy class of subgroup in F and use the collection (ΦH)(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The functor ΦF = (ΦH)H∈F is a coproduct-preserving tt-functor as it is a finite product of such functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It is also conservative by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25(c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in fact, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13 implies that it is enough to take one representative from each G-conjugacy class of subgroups in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, the claim follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 25 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' With notation as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='28, the geometric fixed point functor ΦF : ModG(R) � � (H)∈F Mod(ΦHR) induces a surjective map of topological spaces: Spc(ΦF R): � (H)∈F Spc(Perf(ΦHR)) � � Spc(PerfG(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Here, the product is taken over a set of representatives of conjugacy classes of subgroups contained in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The result is a direct consequence of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26, which applies by the same argument as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For a general R ∈ CAlg(SpG), the map Spc(ΦF R) will not be bijective, see Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19 or Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='27 together with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='23 for some counterexam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' However, in some favourable situations the map is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is the case if R is an equivariant ring spectrum with trivial G-action by [BHS21, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11] or by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will prove later in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 that the map is a homeomorphism for R = EG, the Borel-completion of a Lubin–Tate E-theory of any height at the prime p and any finite abelian p-group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be a finite group and let R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let F be a family of subgroups of G for which R is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Suppose that for each H ∈ F the spectrum Spc(Perf(ΦHR)) is a Noetherian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then so is Spc(PerfG(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29 implies that Spc(PerfG(R)) is covered by the images of finitely many continuous maps with Noetherian source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, Spc(PerfG(R)) is itself Noetherian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Stratification for equivariant ring spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this section we show how to descend stratification for equivariant module categories along geometric fixed point functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This relies on a version of ´etale descent which we now recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let f ∗ : S → T be a finite ´etale functor between rigidly-compactly generated tt-categories with Noetherian spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If T is stratified, then the localizing ideals ΓPS are minimal for each P ∈ Im(Spc(f ∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This follows from the discussion of [Bar21, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2], in particular the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We continue to use the notation of the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be a finite group and let R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let F be a family of subgroups G for which R is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Suppose that the following conditions hold for all subgroups H ∈ F: (a) Spc(Perf(ΦHR)) is a Noetherian space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) Mod(ΦHR) is stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then ModG(R) is stratified with Noetherian spectrum Spc(PerfG(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For a given finite group G, first observe that it suffices to prove the statement for the family of all subgroups of G: Indeed, if R is F-nilpotent for some family F of subgroups of G, then both (a) and (b) hold trivially for all subgroups of G not in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, we might as well take F to be the family of all subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 26 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL The statement that Spc(PerfG(R)) is Noetherian is Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Re- mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17 it therefore remains to verify the minimality of ΓPModG(R) at all prime ideals P ∈ Spc(PerfG(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will argue by induction on the order of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The base of the induction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', the case that G = e, holds by Hypothesis (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For the induction step, we introduce some auxiliary notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let ϕ = Spc(� H⊆G ΦH R ) and write ϕH for its H-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, we assemble the restriction maps for all proper subgroups of G into a tt-functor res⊊G : ModG(R) � � H⊊G ModH(R), and denote the induced map on spectra by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consider a prime ideal P ∈ Spc(PerfG(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We need to show that ΓPModG(R) is a minimal localizing ideal in ModG(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29, P is in the image of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We distinguish two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' First, suppose that P is in the image of ϕG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this case, Zariski descent in the form of [BHS21, Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4] implies our minimality claim at P, because ΦG is a finite localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Now, suppose that P is not in the image of ϕG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in other words, assume that P is in the image of ϕH for some proper subgroup H ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since this map factors through ψ, we have P ∈ im(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8, the functor res⊊G is finite ´etale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='32 then applies to show that our minimality claim holds at P if � H⊊G ModH(R) is stratified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We can thus deduce the required minimality from our induction hypothesis, thereby finishing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The prototypical example of an equivariant ring spectrum R for which ModG(R) is stratified is the Borel-completion of a field k of characteristic p dividing the order of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In this case, there is a tt-equivalence PerfG(k) ≃ Db(kG), the bounded derived category of finitely-generated kG-modules equipped with the symmtric monoidal structure coming from the coproduct of kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This equivalence extends to a symmetric monoidal equivalence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='35) ModG(k) ≃ K(Inj kG), where the right hand side denotes the homotopy category of unbounded chain complexes of injective kG-modules [BK08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This category admits the stable mod- ule category StMod(kG) of kG as a finite localization away from the localizing subcategory generated by kG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, the equivalence of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='35) passes to the quotients: ModG(k)/ Loc⟨k ⊗ G+⟩ ≃ StMod(kG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Both of these equivalences rely on generation by permutation modules, see also Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The corresponding stratification theorem is then due to Benson, Iyengar, and Krause [BIK11b];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' in fact, their main theorem gives the cohomological stratification of ModG(k) in the sense of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19, from which they deduce the stratification of StMod(kG) over Spc(StMod(kG)c) ∼= Proj H•(G, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' More generally, the analogous results hold with coefficients in any regular commu- tative ring in place of k, see [Bar21, Bar22, BIKP22], so in particular over mixed characteristic discrete valuation rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4, we establish a chromatic analogue of their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Strong Quillen stratification for equivariant Balmer spectra The goal of this section is to obtain a decomposition of the Balmer spectrum of PerfG(R) for R ∈ CAlg(SpG) in terms of the Balmer spectra of the non-equivariant categories Perf(ΦH(R)) for H ∈ F together with their Weyl-group actions, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The decomposition result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The main result of this section will compare to the stratification theorem of Quillen [Qui71, Stratification Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2] for the spectrum of the mod p cohomology of a finite group, so we follow his lead in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let H ⊆ G be a subgroup and R ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (a) We set V(R, H) := Spc(PerfG(D(G/H+) ⊗ R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By the functoriality of this construction, the map of G-sets G/H → G/G induces a map ψH : V(R, H) → V(R, G) on Balmer spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9 identifies V(R, H) with Spc(PerfH(R)) and ψH with Spc(resH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) The map S0 G → �E[⊂ H] is the unit map of the finite localization −⊗ �E[⊂ H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consequently, we can identify Spc(PerfG(D(G/H+) ⊗ R ⊗ �E[⊂ H])) with an open subset of V(R, H), which we denote as V+(R, H) ⊆ V(R, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that V+(R, H) identifies with Spc(Perf(ΦHR)) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) We denote by VG(R, H) the image of V(R, H) under the map ψH : V(R, H) → V(R, G), and we endow VG(R, H) ⊆ V(R, G) with the subspace topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We observe that by [Bal16b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4(b)] the map ψH is closed (but not in general a closed immersion), and in particular that VG(R, H) ⊆ V(R, G) is closed (and equal to the support of R ⊗ G/H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (d) We write V+ G(R, H) for the image of V+(R, H) in V(R, G) under ψH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We observe that V+ G(R, H) ⊆ VG(R, H) is open as its complement can be identified with the support of R ⊗ A[⊂H], for A[⊂H] as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given a subgroup H ⊆ G, we let WG(H) = NG(H)/H denote the Weyl group of H ⊆ G, that is, the automorphism group of G/H in FinSetop G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(SpG) as before and consider R⊗ �E[⊂ H] ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is an action of WG(H) on D(G/H+)⊗R by functoriality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If we replace R with R⊗ �E[⊂ H] in the previous construction, we obtain a WG(H)-action on D(G/H+) ⊗ R ⊗ �E[⊂ H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We emphasize that in both cases the Weyl group acts through D(G/H+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The canonical map S0 G → �E[⊂ H] induces a ring map D(G/H+) ⊗ R → D(G/H+) ⊗ R ⊗ �E[⊂ H] which is WG(H)-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By functoriality we then get induced WG(H)-actions on the corresponding Balmer spectra which make V+(R, H) ⊆ V(R, H) into a WG(H)-equivariant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Throughout this section we will always consider the Balmer spectra V(R, H) and V+(R, H) together with this Weyl group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our main decomposition result for the Balmer spectrum is stated in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It generalizes Balmer’s tt-theoretic Quillen stratification theorem [Bal16a, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6] from the Fp-linear setting to arbitrary equivariant tt-categories and extends the ‘weak’ decomposition (in the form of part (a) below) to its ‘strong’ form (parts (b) and (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We remark that it’s the latter that will be crucial for the proof of our stratification result (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4) for Lubin–Tate theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 28 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Assume G is a finite group, R ∈ CAlg(SpG), and F is a family of subgroups of G such that R is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then: (a) The restriction maps induce a homeomorphism colim G/H∈OF(G) V(R, H) ∼ = � V(R, G) , where OF(G) denotes the category of non-empty, transitive G-sets with isotropy in the family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) There is a decomposition into locally closed disjoint subsets V(R, G) = � (H)∈F V+ G(R, H), where the index runs over a set of representatives of conjugacy classes of subgroups in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) For every H ∈ F, restriction induces a homeomorphism V+(R, H)/WG(H) ∼ = � V+ G(R, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We first deduce a corollary which shows that the condition of the comparison map of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6 being a homeomorphism descends in the present situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Under the hypothesis of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3, assume in addition that (1) for every H ∈ F, the comparison map ρH 0 : V(R, H) ∼ = � Spec(πG 0 (D(G/H+) ⊗ R)) ∼= Spec(πH 0 (R)) is a homeomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (2) the ring πG 0 (R) is Noetherian, and the restriction morphism πG 0 (R) → πH 0 (R) makes πH 0 (R) into a finitely generated πG 0 (R)-module for every H ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then: (a) There is a commutative diagram of homeomorphisms colim G/H∈OF(G) V(R, H) V(R, G) colim G/H∈OF(G) Spec(πH 0 (R)) Spec(πG 0 (R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ρG 0 ∼ = ∼ = colim ρH 0 ∼ = ∼ = (b) There is a decomposition into a disjoint union of locally closed subsets Spec(πG 0 (R)) = � (H)∈F VH/WG(H), where VH ⊆ Spec(πH 0 (R)) denotes the open subset complementary to the van- ishing locus of the ideal � x ∈ πH 0 (R) | ResH H′(x) = 0 ∀ H′ ⊊ H � ⊆ πH 0 (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (a) The diagram commutes by the naturality of ρ(−) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The upper horizontal map is a homeomorphism by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The left vertical map is a homeomorphism by our assumption and the lower horizontal map is a homeomorphism by [MNN17, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The final reference requires the Noetherian and finiteness assumptions in the hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 29 (b) The statement follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 (b) and (c) after observing that the homeomorphism ρH 0 identifies the open subset V+(R, H) ⊆ V(R, H) with the open subset VH ⊆ Spec(πH 0 (R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is a direct argument using support, which we leave to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In favorable situations VH can be identified with Spec(π0(ΦHR)), see for instance (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In general, we do not see a reason why VH should even be an affine scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The rest of this subsection is devoted to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' An important ingredient is Balmer’s descent for separable algebras, which we now review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For subgroups H, K ⊆ G, there is a decomposition of the product of transitive G-sets into transitive G-sets as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7) β := � βg : � [g]∈H\\G/K G/(Hg ∩ K) ∼ = −→ G/H × G/K, where the coproduct on the left side runs through a set of double coset representatives and βg sends [x] to ([xg−1], [x]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let S be a set of subgroups of G, and set AS := � H∈S D(G/H+) ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is a coequalizer diagram of topological spaces � H,K∈S [g]∈H\\G/K V(R, Hg ∩ K) � H∈S V(R, H) supp(R ⊗ AS) ⊆ V(R, G), ψ1 ψ2 ψS where the map ψS is induced by the extension of scalars functor, and the maps ψ1 and ψ2 are induced by the maps of G-sets αg : G/(Hg ∩ K) βg −→ G/H × G/K π1 −→ G/H and α1 : G/(Hg ∩ K) βg −→ G/H × G/K π2 −→ G/K, see notation from Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is an adaptation of the proof of [Bal16b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Set C = Ho(PerfG(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We first note that the commutative algebra R ⊗ AS is separable of finite degree by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore by [Bal16b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14], there is a coequalizer diagram of topological spaces Spc(ModC(R ⊗ A⊗2 S )) Spc(ModC(R ⊗ AS)) supp(R ⊗ AS), φ1 φ2 φS where the maps φ1 and φ2 are induced by left and right unit maps, and φS is induced by the extension of scalars functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that D(G/H)⊗D(G/K) ≃ D(G/H ×G/K) and that the maps φ1 and φ2 are induced by the projection maps of G-sets G/H × G/K π1 −→ G/H and G/H × G/K π2 −→ G/K for H, K ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We note that for any subgroup H ⊆ G, there is a homeomorphism (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9) Spc(ModC(R ⊗ D(G/H+))) ∼= V(R, H) by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Using this and the definition of AS, we get a decomposition Spc(ModC(R ⊗ AS)) ∼= � H∈S V(R, H), 30 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL and under this identification φS corresponds to the map ψS induced by the individual extension of scalars functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Similarly, using the Mackey decomposition formula (Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9), we get a decomposition Spc(ModC(R ⊗ A⊗2 S )) ∼= � H,K∈S [g]∈H\\G/K V(R, H ∩ Kg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, we can rewrite the above coequalizer as follows � H,K∈S [g]∈H\\G/K V(R, H ∩ Kg) � H∈S V(R, H) supp(R ⊗ AS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ψ1 ψ2 ψS The maps ψ1 and ψ2 are obtained by composing the left/right unit maps R ⊗ AS ⇒ R ⊗ A⊗2 S , with the maps R⊗D(G/H+)⊗D(G/K+) ≃ R⊗D((G/H×G/K)+) R⊗D((βg)+) −−−−−−−−→ R⊗D(G/Hg∩K+) on components, and then passing to spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Thus we can calculate these composi- tions directly in the category of finite G-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It follows that ψ1 and ψ2 are induced by the maps of G-sets αg : G/(Hg ∩ K) βg −→ G/H × G/K π1 −→ G/H and α1 : G/(Hg ∩ K) βg −→ G/H × G/K π2 −→ G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The argument is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (a) The colimit description follows by considering S = F in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since R is F-nilpotent, we have supp(R ⊗ AF) = V(R, G), for instance by com- bining [Bal16b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15(iii)] with [Mat18, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8 gives a coequalizer diagram of topological spaces � H,K∈F [g]∈H\\G/K V(R, Hg ∩ K) � H∈F V(R, H) V(R, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' ψ1 ψ2 ψF It remains then to identify this coequalizer description with the colimit over the orbit category in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The map ψF factors through the canonical quotient map π: � H∈F V(H, R) → colim H∈OF(G)V(R, H) from the coproduct to the colimit, inducing a continuous map ϕ: colim H∈OF(G)V(R, H) → V(R, G), since V(R, −) is a functor on the orbit category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' On the other hand, the maps ψ1 and ψ2 are equalized by the canonical map π to the colimit, because they are induced by maps of G-sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', morphisms in the orbit category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The universal property of the coequalizer then provides us with a continuous map π: V(R, G) → colim H∈OF(G)V(R, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 31 Since both maps to V(R, G) are topological quotient maps, and both π and ϕ factor the identity, π and ϕ are inverse homeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) We will show that V(R, G) is a disjoint union � (H)∈F V+ G(R, H) of locally closed subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recall that V+ G(R, H) ⊆ VG(R, H) is open, and that VG(R, H) ⊆ V(G, R) is closed (Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Hence every V+ G(R, H) ⊆ V(G, R) is locally closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, it follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29 that there is a union V(R, G) = � (H)∈F V+ G(R, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It remains to argue that this union is disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To this end, consider the commutative diagram of symmetric monoidal left-adjoint functors ModG(R) � (H) ModG(D(G/H+) ⊗ R ⊗ �E[⊂ H]) � (H) Mod(ΦHR) SpG � (H) Sp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' F (ΦH)H ∼ ΨH FR (ΦH)H (FΦH R)H Here (H) runs through a set of representatives for subgroups of G up to conjugacy, the upper part of the diagram comes from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18 and so it commutes, and the outer diagram commutes by Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' On Balmer spectra, the previous diagram induces a commutative diagram V(R, G) � � (H) V+(R, H) � � Spc(Spc G) � (H) Spc(Spc), ∼ � where the bottom map is a bijection by a result of Balmer and Sanders, see [BS17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The components of � (H)∈F V+(R, H) have pairwise disjoint images in Spc(Spc G), as seen by considering the composition through the lower right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It follows that their images in V(R, G), which by definition are V+ G(R, H), must be disjoint as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) The final claim is that for every subgroup H ∈ F, the map V+(R, H) → V+ G(R, H) induced by restriction gives a homeomorphism V+(R, H)/WG(H) → V+ G(R, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' To see this, we set S = {H} in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8, and obtain the following coequalizer � g∈H\\G/H V(R, H ∩ Hg) ψ2 � ψ1 � V(R, H) ψH � VG(R, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The key observation now is that the open subset V+(R, H) ⊆ V(H, R) is saturated for the equivalence relation generated by the maps ψ1 and ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Two elements x, y ∈ V(R, H) are equivalent exactly if there is a chain x = x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' , xn = y such that for each i there is some g ∈ G and some z ∈ V(R, H ∩ Hg) such that xi = ψ1(z) and xi+1 = ψ2(z), or the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For our claim about being saturated, we can assume by induction that n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since at least one of x1, x2 lies in V+(R, H), the subgroup H ∩ Hg ⊆ H cannot be proper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=', g ∈ NG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' So we are reduced to seeing that V+(R, H) ⊆ V(R, H) is WG(H)-stable, which is 32 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Observe that this argument also shows that the equivalence relation induced on V+(R, H) is the one afforded by the action of the Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Saturation and a general fact from topology [Mun00, Theorem 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1] now imply that the composition V+(R, H) ⊆ V(R, H) → V(R, G) restricted to its image is a topological quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Since this image is by definition V+ G(H, R) and we identified the resulting equivalence relation with the Weyl group action, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ We now apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 to give an example where the map in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='29 is in fact bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This gives an alternative proof of [BHS21, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(Sp) and let RG := infle(R) ∈ CAlg(SpG) be the image of R under the inflation functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then the canonical map � (H)⊆G V+(RG, H) ∼ = � V(RG, G) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We observe first that ΦHRG ≃ R for all H ⊆ G (for example, [BHS21, Equation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This implies that if R is non-trivial then the derived defect base of RG is the family of all subgroups, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover, the composite Ho(Mod(R)) infle −−−→ Ho(ModG(RG)) ΦH −−→ Ho(Mod(ΦHRG)) ≃ Ho(Mod(R)) is naturally isomorphic to the identity functor [BHS21, Remark 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4 and Equation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Passing to Balmer spectra, we see that the induced map V+(RG, H) → V(RG, G) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Comparing this with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3(c) shows that the Weyl group WG(H) must act trivially on V+(RG, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We can then apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3(b) to deduce that the required map is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Strong Quillen stratification for global equivariant spectra The goal of this section is to prove a refined version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 under the additional assumption that the commutative algebra R ∈ CAlg(SpG) arises from a global homotopy type in the sense of [Sch18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Before proving our result, let us recall some background on global stable homotopy theory following [LNP22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recollections on global homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let Glo denote the global category indexed on the family of finite groups, see [LNP22, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This is the ∞-category whose objects are all finite groups G, and whose morphism spaces are given by Hom(H, G)hG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' the homotopy orbits of the conjugation G-action on the set of group homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The global category contains a wide subcategory Orb ⊆ Glo where we take the path connected components spanned by the injective group homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Finally, we let Rep denote the homotopy category of Glo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' this is the category with the same objects of Glo but with morphisms given by conjugacy classes of group homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recall that two group homomorphisms f, g: A → B are conjugate if there exists b ∈ B such that cb ◦ f = g where cb(x) = bxb−1 for all x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 33 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By [LNP22, Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6] there is a functor into the ∞-category of symmetric monoidal ∞-categories (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3) Sp• : Gloop → Cat⊗ ∞ which sends a finite group G to SpG, and a group homomorphism α: H → G to the restriction-inflation functor α∗ : SpG → SpH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that α∗ is restriction if α is injective and inflation if α is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In general, it will be the composite of two such functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We define the ∞-category of global (equivariant) spectra Spgl as the partially lax limit of Sp• marked at Orb, see [LNP22, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Informally, an object X ∈ Spgl is determined by the following data: a G-spectrum XG for all finite group G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' compatible maps fα : α∗XG → XH in SpH for all group homomorphisms α: H → G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' subject to the condition that fα is an equivalence whenever α is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The ∞-category Spgl is symmetric monoidally equivalent to Schwede’s model of global spectra, see [LNP22, Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By the discussion in [Sch18, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19], any Borel-equivariant G-spectrum arises from a global homotopy type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Also, topological G-equivariant K-theory admits a global refinement by [Sch18, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given a global spectrum X ∈ Spgl, we will denote by XG its image under the canonical symmetric monoidal functor Spgl → SpG, and call this the underlying G-spectrum of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Quillen stratification for commutative global ring spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Our goal in this section is to prove a refined version of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 under the additional assumption that R arises from a global homotopy type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The first step is to construct a more refined version of the functor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8) ΘR 0 : OF(G)op → Cat, G/H �→ Ho(PerfG(RG ⊗ D(G/H+))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let R ∈ CAlg(Spgl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then there is a functor Mod•(R•): Gloop → Cat⊗ ∞, G �→ ModG(RG) extending (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3) to module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Moreover there is also an induced functor with values G �→ PerfG(RG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The existence of the first functor follows from [LNP22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The existence of the second functor follows from the first one by observing that by construction, the functor α∗ is symmetric monoidal for all group homomorphism α, and that dualizable and perfect objects agree in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any g ∈ G, the conjugation isomorphism cg : G → G lies in the same connected path component of MapGlo(G, G) as the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The existence of the functor in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9, then provides a natural isomorphism c∗ g ⇒ 1 between endofunctors of PerfG(RG) or ModG(RG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We now introduce a variation of Quillen’s category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 34 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let F be a family of subgroups of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The Quillen category AF(G) is the category whose objects are subgroups of G in F, and mor- phisms from H to K in AF(G) are group homomorphims of the form cg : H → K, h �→ ghg−1 for some g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that the automorphism group of H ∈ AF(G) is NG(H)/CG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let E be the family of elementary abelian p-subgroups of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then AE(G) is the category A(G) defined by Quillen in [Qui71, page 567].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We now introduce the orbit category associated to the Quillen category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let F be a family of subgroups of a finite group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The Quillen orbit category OQ F(G) is the category whose objects are subgroups of G in F, and morphisms from H to K in OQ F(G) are given by K-conjugacy classes of morphisms in HomAF(G)(H, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The automorphisms of H ∈ OQ F(G) are then NG(H)/H · CG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let H ⊆ G be a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We set W gl G (H) := NG(H)/H · CG(H) and refer to it as the global Weyl group of H in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If H is abelian so that H ⊆ CG(H), we write W Q G (H) instead of W gl G (H) and refer to it as the Quillen–Weyl group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Assume G is a finite group, R ∈ CAlg(Spgl), and F is a family of subgroups of G such that RG is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Consider the functor Gloop → Cat, H �→ Ho(PerfH(RH)), which factors through Rep by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' There is also a functor J : OF(G) → Rep which sends a coset G/H to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Given a G-equivariant map f : G/H → G/K satisfying f(eH) = gK we set J(f) = [cg−1] ∈ Rep(H, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' One easily verifies that J is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We note that the image of J is precisely OQ F(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Therefore, by precomposing, we obtain a functor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='16) ΘR 1 : OF(G)op → Cat, G/H �→ Ho(PerfH(RH)), which by construction factors through OQ F(G)op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The functors ΘR 0 and ΘR 1 from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='8) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='16) are naturally pseudo- isomorphic and they factor through OQ F(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any subgroup H ⊆ G, we let LH : Ho(PerfG(RG ⊗ D(G/H+))) ∼ → Ho(PerfH(RH)) denote the equivalence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will show that the equivalences LH assemble to give a pseudonatural equivalence between ΘR 0 and ΘR 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' As the latter factors through OQ F(G), so does ΘR 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let f : G/H → G/K with f(eH) = gK be given so that J(f) = cg−1 : H → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We will show that the following diagram commutes up to natural isomorphism Ho(PerfK(RK)) Ho(PerfH(RH)) Ho(PerfG(RG ⊗ D(G/K+))) Ho(PerfG(RG ⊗ D(G/H+))), c∗ g−1 ∼ LK Ff ∼ LH where Ff is a shorthand for the extension of scalars functor along R ⊗ D(f+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' QUILLEN STRATIFICATION IN EQUIVARIANT HOMOTOPY THEORY 35 To this end, consider the following diagram Ho(PerfK(RK)) Ho(PerfH(RH)) Ho(PerfG(RG ⊗ D(G/K+))) Ho(PerfG(RG ⊗ D(G/H+))) Ho(PerfG(RG)) Ho(PerfG(RG)) Ff ∼ LK LH ∼ c∗ g−1 resH FD(G/H+)⊗RG FD(G/K+)⊗RG c∗ g−1 resK in which the right and left triangles commute by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10, there is a natural isomorphism c∗ g−1 ⇒ 1 between endofunctors of Ho(PerfG(RG)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This natural isomorphism makes the bottom square commute as one readily verifies that Ff ◦ FD(G/K+)⊗RG ≃ FD(G/H+)⊗RG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that the outer diagram commutes as well, using that iK ◦ cg−1 = cg−1 ◦ iH, where iK : K ⊆ G and iH : H ⊆ G denote the inclusion of subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This shows that there is a natural isomorphism η: LH ◦ Ff ◦ FD(G/K+) ⇒ c∗ g−1 ◦ LK ◦ FD(G/K+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We then apply Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='35 (keeping in mind Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='10) to obtain a natural transformation �η: LH ◦ Ff ⇒ c∗ g−1 ◦ LK extending η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' A thick subcategory argument then shows that �η is also an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' This uses that Ho(PerfG(RG ⊗ D(G/K+))) admits a set of generators in the image of FD(G/K+)⊗RG, namely the orbits, see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ We are finally ready to state the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recall the notation introduced in Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Assume G is a finite group, R ∈ CAlg(Spgl), and F is a family of subgroups of G such that RG is F-nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Then: (a) The restriction maps induce a homeomorphism colim H∈OQ F (G) V(RG, H) ∼ = � V(RG, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (b) There is a decomposition into locally closed disjoint subsets V(RG, G) = � (H)∈F V+ G(RG, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The index runs over conjugacy classes of subgroups in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' (c) For every H ∈ F, restriction induces a homeomorphism V+(RG, H)/W gl G (H) ∼ = � V+ G(RG, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' With the work done so far, all the claims follow from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3 as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Recall from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='17 that the functor G/H �→ V(RG, H) factors through J : OF(G) → OQ F(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Note that J is cofinal since it is bijective on objects and surjective on morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Combining this with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3(a) gives part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Part (b) is precisely Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Finally, part (c) follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3(c) and the observation that the centralizer CG(H) must act trivially on V+(RG, H) as G/H �→ V(RG, H) factors through J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' □ The globality assumption in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18 implies that the action of CG(H) on Spc(Perf ΦHR) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The next example shows that it can be non-trivial in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 36 BARTHEL, CASTELLANA, HEARD, NAUMANN, AND POL Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let G be the cyclic group of order 2 and let R = D(G+) ∈ CAlg(SpG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We note that R does not come from a global spectrum as the restriction map resG e : πG 0 (R) → π0(R) is not surjective, compare with [Sch18, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' We also note that ΦGR ≃ 0 and that ΦeR ≃ S0 ⊕ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20) Spc(PerfG(R)) = Spc(Perf ΦeR)/G = (Spc(Spc) ⊔ Spc(Spc))/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' On the other hand, Spc(PerfG(R)) = Spc(Spc) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' It follows that the Weyl group WG(e) = G does not act trivially in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' In particular we cannot replace WG(e) with W gl G (e) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Quillen stratification, revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The goal of this subsection is to show that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='18 contains a weak version of Quillen’s stratification theorem as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' The precise formulation of this stratification result appears in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Notation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' Let k be a field of characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' For any finite group G, we set Vh G = Spech(H∗(G, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdE0T4oBgHgl3EQfRwCH/content/2301.02212v1.pdf'} +page_content=' If E is an elementary abelian p-group, we also set Vh,+ E = Vh E \\ � E′ 0.3, the magnitude of |χzzz +2 (ω,ω)| starts decreasing +and it becomes nearly overdamped for αz +LRC > 0.6. The over- +all shape of the spectrum is maintained in the whole range of +αz +LRC considered. By applying the same procedure to the zxx +and xxz components we have determined the remaining coef- +ficient αx=y +LRC = 1.8. +In Ref. [6], two additional measurements were conducted at +≃ 1.55 eV for varying angle θ of linearly-polarized light, with +the field oriented along the [1,1,-1] (parallel setup, ∥) and [1,- +1,0] (perpendicular setup, ⊥) directions. Making use of the +appropriate combination of the SHG tensor components (see +Eqs. 3 and 4 of the Supplementary Information in Ref. [6]), we +have calculated the angular dependence of the SHG intensity +and compared it to the experimental polar plot, as shown in +Fig. 6(b). For the parallel configuration, the response shows +an elongated shape along the θ = 0 axis that is remarkably +well captured by our MB result. For the perpendicular config- +uration, the response shows a four-fold structure with maxima +at π/4+n·π/2 and minima at n·π/2 for any integer n. While +the KS calculation fails in both magnitude and shape, our MB +result nicely agrees with the experimental measurement, thus +capturing the main characteristics of the photoresponse at this + +11 +particular energy. +As the last step, we proceed to study the low-energy +region accessed in Ref. [87], +where a narrow reso- +nance was measured at ≃ 0.75 eV. In Fig. 6(c) we +compare the experimentally measured |σzxx +M,2|, |σxxz +M,2| and +|σeff +M,2| ≡ |σzzz +M,2 +4σxxz +M,2 +2σzxx +M,2| with our calculations using +the optimal values of αa +LRC quoted previously. Our results un- +derestimate the main exciton-like peak by an order of magni- +tude, and we have been unable to strike a substantial improve- +ment by further varying αa +LRC. The description of this low- +energy peak appears therefore to be beyond the scope of the +linear tensorial LRC xc kernel considered here. It is tempting +to speculate that it might be induced by MB corrections not +included in our calculations, e.g., a frequency dependence in +the LRC xc coefficients αa +LRC(ω) [94, 95], or the quadratic +tensorial xc kernel of Eq. 5. +V. +SUMMARY +In summary, we have described a general scheme for cal- +culating the quadratic optical response to light tensor of +crystals taking into account many-body interactions. +We +have formally included excitonic effects by means of a ten- +sorial long-range exchange-correlation kernel whose coeffi- +cients have been calculated self-consistently using the boot- +strap method. +We have also generalized previous expres- +sions [12–14, 27, 28] for the transition matrix elements to +account for all metallic contributions, allowing an exhaustive +study of materials like Weyl semimetals. +Linking the formalism with the Wannier interpolation of +the transition matrix elements [44, 45, 96], we have per- +formed calculations of the second-harmonic generation pho- +toresponse tensor in a range of materials. +Besides bench- +marking our approach, we have shown that the electron- +hole attraction can give rise to strong and sharply localized +one- and two-photon resonances that are absent in the Kohn- +Sham photoresponse. In the graphitic-layered crystal BC2N, +an space-averaged isotropic approach overestimates the elec- +tronic renormalization by orders of magnitude, highlighting +the need of accounting for the space-resolved anisotropic na- +ture of many-body interactions in tensorial form. +Finally, +with the use of a highly dense k-space mesh, our calcula- +tions have reproduced the magnitude and angular dependence +of the photoresponse for the Weyl semimetal TaAs measured +recently [6]. +We hope that the presented scheme together with its imple- +mentation in the Wannier90 and WannierBerri code pack- +ages will facilitate an efficient and accurate calculation of the +quadratic optical photoresponse of materials beyond the SHG +process analyzed here. We note that the procedure adopted for +including many-body excitonic effects requires only a fraction +of the computational time as compared to the calculation of +the Kohn-Sham photoresponse. +The proposed method can be improved in several fronts. +Adopting a Wannier-based strategy for the calculation of the +linear xc kernel in metals and semimetals (see e.g., Ref. [97]) +would allow a fully parameter-free analysis in these type of +materials. An improved description of many-body effects can +be achieved by extending the LRC xc coefficients to frequency +domain [94, 95] or by working out an approximation for the +second-order xc kernel, which would open the way to study +potentially new excitonic effects that have been barely de- +scribed in the literature up to now. The method can also model +crystal local-field corrections, although their effect has been +found to be minor when employing a localized Wannier ba- +sis [98, 99]. Finally, accounting for quasiparticle self-energy +corrections due to electron-electron or electron-phonon inter- +actions would allow modelling extrinsic quadratic contribu- +tions such as the ballistic current [2, 100–102]. We expect to +address these subjects in future works. +VI. +ACKNOWLEDGEMENTS +We are very grateful to Ivo Souza and Fernando de Juan +for helpful discussions. +This project has received funding +from the European Union’s Horizon 2020 research and in- +novation programme under the European Research Council +(ERC) grant agreement No 946629. +Appendix A: DERIVATION OF THE QUADRATIC +DYSON-LIKE RESPONSE TENSOR EQUATION +Here we outline the steps involved in the derivation of the +Dyson-like equation relating the MB and KS conductivity ten- +sors at second order in Eq. 15 of the main text. We start by +applying the chain rule twice in the definition of the quadratic +MB conductivity tensor in Eq. 14a, +σabc +1 +(1,2,3) = +δ +δEb +ext(2) +� +∑ +d +� +δJa(1) +δEd +tot(4) +δEd +tot(4) +δEc +ext(3)d4 +� +=∑ +d +� � +δ 2Ja(1) +δEb +ext(2)δEd +tot(4) +δEd +tot(4) +δEc +ext(3) + δJa(1) +δEd +tot(4) +δ +δEb +ext(2) +� δEd +tot(4) +δEc +ext(3) +�� +d4 +=∑ +de +�� +δ 2Ja(1) +δEe +tot(5)δEd +tot(4) +δEe +tot(5) +δEb +ext(2) +δEd +tot(4) +δEc +ext(3)d4d5+∑ +d +� +δJa(1) +δEd +tot(4) +δ 2Ed +tot(4) +δEb +ext(2)δEc +ext(3)d4. +(A1) + +12 +The first term on the right-hand side (r.h.s.) of the last line in Eq. A1 can be expressed in terms of σKS +2 +and ε using Eqs. 14b +and 10, respectively. As for the second term, the piece δJa(1)/δEd +tot(4) can be written in terms of σKS +1 +using Eq. 8b, while the +calculation of the remaining piece requires applying the chain rule again, +δ 2Ed +tot(4) +δEb +ext(2)δEc +ext(3) = +δ +δEb +ext(2) +� +δ(4,3)δdc + +� +∑ +e +δEd +Hxc(4) +δJe(5) +δJe(5) +δEc +ext(3)d5 +� += +� +∑ +e +� +δ 2Ed +Hxc(4) +δEb +ext(2)δJe(5) +δJe(5) +δEc +ext(3) + δEd +Hxc(4) +δJe(5) +δ 2Je(5) +δEb +ext(2)δEc +ext(3) +� +d5 += +�� +∑ +e f +δ 2Ed +Hxc(4) +δJ f (6)δJe(5) +δJ f (6) +δEb +ext(2) +δJe(5) +δEc +ext(3)d5d6+ +� +∑ +e +δEd +Hxc(4) +δJe(5) +δ 2Je(5) +δEb +ext(2)δEc +ext(3)d5, +(A2) +where we used ε−1,ab(1,2) = δEatot(1) +δEbext(2) and Eq. 14b. The first term on the r.h.s. of the last line in Eq. A2 can be expressed in terms +of Kabc +xc,2(1,2,3) and σab +1 (1,2) using +δ 2Ea +Hxc(1) +δJb(2)δJc(3) = Kabc +xc,2(1,2,3) and Eq. 8a, respectively. As for the second piece, it can be recast +in terms of Kab +Hxc,1(1,2) and σ2 using Kab +Hxc,1(1,2) = δEa +Hxc(1) +δJb(2) and Eqs. 14a, respectively. +Taking into account all the previous observations, we can rewrite Eq. A1 as +σabc +2 +(1,2,3) = +�� +∑ +de +σKS,aed +2 +(1,5,4)ε−1,eb(5,2)ε−1,dc(4,3)d4d5 ++ +��� +∑ +de f +σKS,ad +1 +(1,4)Kd fe +xc,2(4,6,5)σ fb +1 (6,2)σec +1 (5,3)d4d5d6 ++ +�� +∑ +de +σKS,ad +1 +(1,4)Kde +Hxc,1(4,5)σebc +2 +(5,2,3)d4d5. +(A3) +Moving now the last term on the r.h.s. of Eq. A3 to the left-hand side (l.h.s.), we can rewrite this side with the quadratic MB +conductivity tensor as a common factor. Taking advantage from the definition of the dielectric tensor in Eq. 12, we obtain that +� +∑ +e +� +δ(1,5)δae − +� +∑ +d +σKS,ad +1 +(1,4)Kde +Hxc,1(4,5)d4 +� +σebc +2 +(5,2,3)d5 ≡ +� +∑ +d +εda(1,4)σdbc +2 +(4,2,3)d4 = +�� +∑ +de +σKS,aed +2 +(1,5,4)ε−1,eb(5,2)ε−1,dc(4,3)d4d5+ +��� +∑ +def +σKS,ad +1 +(1,4)Kd fe +xc,2(4,6,5)σ fb +1 (6,2)σec +1 (5,3)d4d5d6. +(A4) +Finally, inverting the transpose of the dielectric tensor from the r.h.s. to the l.h.s., we arrive at the Dyson-like equation 15 quoted +in the main text: +σabc +2 +(1,2,3) = +��� +∑ +de f +ε−1,da(1,4)σKS,def +2 +(4,5,6)ε−1,eb(5,2)ε−1, fc(6,3)d4d5d6 ++ +��� +∑ +de f +σad +1 (1,4)Kdef +xc,2(4,5,6)σeb +1 (5,2)σ fc +1 (6,3)d4d5d6. +(A5) +Appendix B: KS OPTICAL RESPONSE TENSOR +EXPRESSIONS UP TO SECOND ORDER +In this appendix we provide the expressions of all optical +KS response tensors up to second order within the formalism +of Sipe and co-workers [12–14] (see Sec. II B). These expres- +sions are valid for any combination of ω1 and ω2 and include +metallic terms proportional to k-space derivatives of the occu- +pation factors. Here we merely quote the final expressions; for +details on the derivation steps, we refer the reader to Sec. IV +in Ref. [14] or to Appendix A in the Supplemental Material of +Ref. [27]. +At first order, the optical KS interband polarizability and +intraband conductivity tensors are respectively expressed as +αKS,ab +ter,1 (ω) = e2 +¯hV ∑ +kmn +fnm +ra +nmrb +mn +ωmn − ˜ω , +(B1a) +σKS,ab +tra,1 (ω) = e2 +¯hV ∑ +kn +fn +� +iωn;ab +˜ω +−εcabΩc +n +� +, +(B1b) +while at second order they are respectively expressed as + +13 +αKS,abc +ter,2 +(ω1,ω2) = +e3 +2¯h2V +� +∑ +kmnl +ra +nm +ωmn −ω12 +� +fnl +� rb +lnrc +ml +ωln −ω1 ++ +rc +lnrb +ml +ωln −ω2 +� +− flm +� +rb +mlrc +ln +ωml −ω1 ++ +rc +mlrb +ln +ωml −ω2 +�� ++i ∑ +kmn +� fnmrb;c +mn + fnm;crb +mn +ωmn −ω1 +− fnmrb +mnΛc +mn +(ωmn −ω1)2 − fnm;brc +mn +ω1 ++ fnmrc;b +mn + fnm;brc +mn +ωmn −ω2 +− fnmrc +mnΛb +mn +(ωmn −ω2)2 − fnm;crb +mn +ω2 +�� +. +(B2a) +σKS,abc +tra,2 +(ω1,ω2) = +e3 +2¯h2V +� +− ∑ +kmn +� fnmΛa +nm +ω12 +� rc +nmrb +mn +ωmn −ω1 ++ +rb +nmrc +mn +ωmn −ω2 +� ++ fnm +� rc;a +nmrb +mn +ωmn −ω1 ++ rb;a +nmrc +mn +ωmn −ω2 +�� ++∑ +kn +� +i +� fn;bεdac +ω1 ++ fn;cεdab +ω2 +� +Ωd +n − va +n fn;bc +ω1ω2 +�� +. +(B2b) +All quantities appearing in the expressions above have been +introduced in Sec. II B except for ωn;ab in Eq. B1b, which +stands for the inverse effective mass tensor. As a remark, the +metallic terms of the quadratic optical KS intraband polariz- +ability tensor in Eq. B2a are shown here for the first time to +the best of our knowledge. +Appendix C: TENSORIAL KERNELS +Here we describe the calculation of the tensorial kernels in +the optical limit. Let us start by reviewing the Hartree contri- +bution. The Hartree potential is defined by +VH(1) = +� +vc(1,2)ρ(2)d2, +(C1) +where vc(1,2) = δ(t1 −t2)/|r1 −r2| is the static Coulomb +scalar potential and ρ(1) is the charge density. With the aid +of Maxwell’s equation, E(r,t) = −∇∇∇V(r,t) and the continu- +ity equation, ∇∇∇·J(r,t) = −∂tρ(r,t), the Hartree electric field +in wavevector and frequency space is expressed as +EH(q1,ω) = ∑ +q2 +KH(q1,q2,ω)·J(q2,ω), +(C2) +with the kernel given by +Kab +H (q1,q2,ω) = qa +1 +vc(q1,q2) +iω +qb +2 = qa +1 +4πδq1,q2 +iω|q1||q2|qb +2. +(C3) +Above, q1 and q2 represent momenta and the Fourier trans- +form of the Coulomb potential was used. +Applying the +q1,q2 → 0 optical limit in Eq. C3, the tensorial Hartree kernel +takes the usual form +Kab +H (ω) = δab +4π +iω , +(C4) +which is a diagonal and isotropic tensor owing to the longitu- +dinal and radial nature of the Coulomb force. +Coming now to the xc piece, its electric field up to linear +order is written as +Exc,1(q1,ω) = ∑ +q2 +Kxc,1(q1,q2,ω)·J(q2,ω). +(C5) +Assuming that the nonlocal long-range behaviour of excitonic +effects completely dominates over all other terms in the op- +tical limit [68], the xc contribution can be modelled by a +Coulomb-like attractive interaction with LRC xc coefficients +αa +LRC. In the wavevector and frequency domain, the corre- +sponding tensorial xc kernel reads +Kab +xc,1(q1,q2,ω) = −qa +1 +αa +LRCδab +iω|q1||q2|qb +2, +(C6) +which is a diagonal tensor owing to the longitudinal nature of +Coulomb-like forces. At variance with the Hartree contribu- +tion in Eq. C3, the tensorial coefficients αa +LRC in Eq. C6 allows +a space-resolved anisotropic response of the xc electric field +along the crystal axes. By taking the optical limit in Eq. C6, +we arrive at the simplified expression used in our calculations, +Kab +xc,1(ω) = −αa +LRCδab +iω +. +(C7) +The bootstrap method is a parameter-free approxima- +tion that was originally proposed for self-consistenty cal- +culating the space-averaged isotropic scalar α-coefficient in +TDDFT [70]. We have adopted this method to compute αa +LRC +by means of the expression +αa +LRC = ε−1,aa +M +(0) +� +αKS +1 (0) +�−1,aa , +(C8) +which requires calculating the LRC xc coefficients indepen- +dently for each of the three Cartesian directions. As in the +original bootstrap kernel, the calculation of the coefficients is +done iteratively; firstly, the microscopic optical MB conduc- +tivity is calculated by means of Eqs. 24 and 25; secondly, the +macroscopic optical dielectric tensor by means of Eq. 27; and +finally, the coefficients αa +LRC by means of Eq. C8. The iter- +ative loop starts with the initial guess αa +LRC = 0 and finishes +when self-consistency is reached. + +14 +Appendix D: THE OPTICAL +MACROSCOPIC-MICROSCOPIC CONNECTION +In this appendix we derive the relations that connect calcu- +lable response tensors at the microscopic scale with their mea- +surable macroscopic counterparts in the optical limit. This is +largely based on the work of Del Sole and Fiorino for the first +order [52], and on the work of Luppi and co-workers for the +second order [35]. +1. +General definitions and useful relations +The response of a material to an applied external electric +field can be mainly described in two ways. On the one hand, +the ability of a material to conduct an electric current is de- +scribed by the electric conductivity, which relates the electric +current-density vector to the electric field. On the other hand, +the ability of a material to electrically polarize is described by +the electric susceptibility or polarizability, which relates the +electric polarization-density vector to the electric field. +At the macroscopic scale (M), these relations are expressed +in terms of the macroscopic total electric field EM +tot(r,t), in +such a way that the j-th order power series expansion of the +macroscopic electric current- and polarization-density vectors +are respectively defined as +JM,j(1) = +� +... +� 1 +0 σM, j(1,..., j +1)∏ +j +Etot,M( j +1)d j +1 +(D1a) +PM, j(1) = ε0 +� +... +� 1 +0 χ j(1,..., j +1)∏ +j +Etot,M( j +1)d j +1 +(D1b) +where JM, j(r,t) and PM, j(r,t) are the j-th order macro- +scopic electric current- and polarization-density vectors, re- +spectively, and σM,j(1,..., j +1) and χ j(1,..., j +1) are +the +j-th order macroscopic conductivity and suscepti- +bility tensors, respectively. +The complete macroscopic +current- and polarization-density vectors are given by +JM(r,t) = ∑ j JM,j(r,t) and PM(r,t) = ∑j PM, j(r,t), respec- +tively. +In turn, at the microscopic scale the relations are expressed +in terms of the microscopic external electric field Eext(r,t), in +such a way that the j-th order power series expansion of the +microscopic electric current- and polarization-density vectors +are respectively defined as +Jj(1) = +� +... +� 1 +0 σ j(1,..., j +1)∏ +j +Eext( j +1)d j +1, (D2a) +P j(1) = +� +... +� 1 +0 α j(1,..., j +1)∏ +j +Eext( j +1)d j +1 (D2b) +where Jj(r,t) and P j(r,t) are the j-th order microscopic +electric current- and polarization-density vectors, respec- +tively, and σ j(1,..., j +1) and α j(1,..., j +1) are the j- +th order microscopic conductivity and polarizability ten- +sors, respectively. +The complete microscopic current- and +polarization-density vectors are given by J(r,t) = ∑j Jj(r,t) +and P(r,t) = ∑j Pj(r,t), respectively. +In the absence of magnetization, and free charge and cur- +rent densities, the current- and polarization-density vectors are +related by J(M),( j)(r,t) = ∂tP(M),(j)(r,t), both at the macro- +scopic and microscopic levels, as well as at any order of the +power series expansion. Using the latter relation and com- +paring Eq. D1a and Eq. D1b, we can derive the connections +between the macroscopic conductivity and susceptibility up to +second order. In the reciprocal space and frequency domain, +the connection at first order in the optical limit is given by +σM,1(ω) = −iωε0χ1(ω), +(D3) +and at second order by +σM,2(ω1,ω2) = −i(ω1 +ω2)ε0χ2(ω1,ω2). +(D4) +In an analogous way, we can derive the connection between +microscopic conductivity and polarizability up to second or- +der, but this time comparing Eq. D2a and Eq. D2b. At first +order it is given by +σ1(ω) = −iωα1(ω), +(D5) +and at second order by +σ2(ω1,ω2) = −i(ω1 +ω2)α2(ω1,ω2). +(D6) +2. +Macroscopic optical susceptibility +Our main goal is to express macroscopic response tensors +as a function of their respective microscopic counterpart. To +this end, the simplest option is to switch to the KS electronic +system, where the observables in Eqs. D2a and D2b are de- +fined in terms of the microscopic total electric field Etot(r,t) +as in Eq. 6 for the current, and then take a macroscopic spatial +average of the microscopic quantities. In the so-called long- +wavelength limit, where the real-space variation of the total +electric field over distances of the order of the lattice param- +eter is neglected and therefore the total electric field is per +se of macroscopic character, the macroscopic spatial average +of microscopic quantities is straightforward; it is sufficient to +retain the G = 0 reciprocal lattice vector [49]. Furthermore, +the averaging is even more direct in the optical limit, since +microscopic quantities are calculated assuming ideally a non- +variational character in space. Therefore, under this point of +view, one can state that the macroscopic optical conductiv- +ity is equal to its microscopic KS counterpart at any order, +i.e. σM,j(1,..., j +1) = σKS +j (1,..., j +1). +Nevertheless, the previous approach does not account for +many-body effects in the response, since those are assumed to +be already included in the total electric field. In order to over- +come this limitation, one can obtain an expression of the exter- +nal electric field as a function of the total electric field at the +microscopic level by using Maxwell’s equations and related +constitutive relations. Then, the resulting expression is used +to define microscopic observables in Eqs. D2a and D2b in + +15 +terms of the total electric field, whose macroscopic spatial av- +erages give access to the formulation of macroscopic response +tensors including many-body effects. Following Ref. [35], in +the reciprocal space and frequency domain, the longitudinal- +longitudinal (LL) component of the linear macroscopic sus- +ceptibility tensor is given by [52] +χLL +1 (q,ω) = 4παLL +1 (q,ω)εLL +M (q,ω), +(D7) +where +αLL +1 (q,ω) ≡ α1LL +GG′(q,ω)δG,0δG′,0 +is +the +LL +component +of +the +macroscopic +spatial +averaged +mi- +croscopic MB polarizability tensor at first order, +and +εLL +M (q,ω) = [1−4παLL +1 (q,ω)]−1 is the LL component of +the macroscopic dielectric tensor. In an analogous way, the +longitudinal-longitudinal-longitudinal (LLL) component of +the quadratic macroscopic susceptibility tensor is expressed +as +χL12L1L2 +2 +(q1,q2,ω1,ω2) = 4πεL12L12 +M +(q12,ω12)αL12L1L2 +2 +(q1,q2,ω1,ω2)εL1L1 +M +(q1,ω1)εL2L2 +M +(q2,ω2), +(D8) +where L1, L2 and L12 stand for the longitudinal component along the directions q1, q2 and q12 ≡ q1 + q2, respectively, and +αL12L1L2 +2 +(q1,q2,ω1,ω2) ≡ α2 +L12L1L2 +G12G1G2(q1,q2,ω1,ω2)δG12,0δG1,0δG2,0 is the LLL component of the spatially averaged micro- +scopic MB polarizability tensor at second order. +The adopted framework is valid for any q and describes lon- +gitudinal responses to longitudinal perturbations [52]. In the +optical limit (q → 0), one can always find three principal axes +for any crystal symmetry in which the macroscopic dielectric +tensor is diagonal [103]. In this reference frame a longitudinal +perturbation induces a longitudinal response, hence any opti- +cal property of the crystal can be deduced from a longitudinal +calculation [104]. 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Gatti, The Microscopic Description of a +Macroscopic Experiment (Springer Berlin Heidelberg, Berlin, +Heidelberg, 2012), pp. 29–50, ISBN 978-3-642-23518-4, +URL https://doi.org/10.1007/978-3-642-23518-4_ +3. + diff --git a/R9AyT4oBgHgl3EQft_n6/content/tmp_files/load_file.txt b/R9AyT4oBgHgl3EQft_n6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a1b0a312cc68fffc42a2cc80141e530b3f2ddf6 --- /dev/null +++ b/R9AyT4oBgHgl3EQft_n6/content/tmp_files/load_file.txt @@ -0,0 +1,2109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf,len=2108 +page_content='Including many-body effects into the Wannier-interpolated quadratic photoresponse tensor Peio Garcia-Goiricelaya,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ∗ Jyoti Krishna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='1 and Julen Ibañez-Azpiroz1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2 1Centro de Física de Materiales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Universidad del País Vasco UPV/EHU,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 20018 San Sebastián,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Spain 2Ikerbasque Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 48013 Bilbao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Spain (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2023) We present a first-principles scheme for incorporating many-body interactions into the unified description of the quadratic optical response to light of noncentrosymmetric crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The proposed method is based on time- dependent current-density response theory and includes the electron-hole attraction via a tensorial long-range exchange-correlation kernel, which we calculate self-consistently using the bootstrap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' By bridging with the Wannier-interpolation of the independent-particle transition matrix elements, the resulting numerical scheme is very general and allows resolving narrow many-body spectral features at low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We show- case its potential by inspecting the second-harmonic generation in the benchmark zinc-blende semiconductor GaAs, the layered graphitic semiconductor BC2N and the Weyl semimetal TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Our results show that ex- citonic effects can give rise to large and sharply localized one- and two-photon resonances that are absent in the independent-particle approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We find overall good agreement with available experimental measure- ments, capturing the magnitude and peak-structure of the spectrum as well as the angular dependence at fixed photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The implementation of the method in Wannier-based code packages can serve as a basis for performing accurate theoretical predictions of quadratic optical properties in a vast pool of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' INTRODUCTION The field of nonlinear optics [1, 2] has received a consider- able push in recent years, thanks in part to advances of con- temporary techniques in designing novel structures such as layered materials and thin films [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Breakthroughs have come in various fronts like topology [5], with an accute en- hancement of the nonlinear light absorption in Weyl semimet- als [6–8] or the prediction of a quantized photoresponse [9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' but also in more applied aspects like the increasing of power- conversion efficiency in ferroelectric insulators [10] or the en- gineering of new effects for boosting the performance of stan- dard solar cells [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The unified microscopic description of nonlinear optical phenomena is due to Sipe and co-workers [12–14], who de- veloped a general formalism within the independent-particle approximation for calculating the intrinsic contribution to the second-order optical photoresponse tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This approach accounts for the various quadratic optical processes taking place in semiconductors, including injection and shift cur- rents [2, 15, 16] that originate from physical divergences of the response coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Building on this scheme, several studies based on density functional theory (DFT) have re- ported material-specific calculations for various second-order processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [17–26] for a small survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In addition, recent works have extended the formalism to include metallic terms [27, 28] and third-order contributions [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Alter- native approaches have also been proposed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', based on the reduced density matrix formalism [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' While the theory of nonlinear optical photoresponses in the independent-particle approximation has an ample track record, much fewer studies have considered many-body in- teractions beyond this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Among those, a series of works by Luppi, Hübener and Veniard [33–36] casted the ∗Electronic address: peio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='garcia@ehu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='eus second-order susceptibility within the time-dependent DFT (TDDFT), and provided explicit calculations of excitonic ef- fects on the second-harmonic generation (SHG) spectrum for various materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' DFT-based SHG spectra influenced by the electron-hole attraction within a Bethe-Salpeter scheme were also reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' An alternative real-time ap- proach based on the Berry-phase formulation of the dynami- cal polarization was set forth in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' More recently, quasiparticle and excitonic effects on the shift current have been analyzed using the GW plus Bethe-Salpeter equation method [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The relative scarcity of practical implementations is in part a consequence of the technical difficulties involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' An important bottleneck concerns the calculation of the independent-particle quadratic response, due to the intricate form of the transition matrix elements that involve derivatives with respect to the crystal momentum k of Bloch states [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Their calculation requires a careful treatment in order to en- sure gauge invariance and properly handle band degeneracies, and brute-force approaches quickly become time-demanding from the computational point of view [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Recently, it has been shown that the so-called “Wannier interpolation” proce- dure can solve the above difficulties [21, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In this approach, the quadratic matrix elements are reformulated in terms of lo- calized Wannier functions, in the same spirit as the Wannier interpolation of the Berry curvature and anomalous Hall con- ductivity [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The method offers a general and efficient way of calculating second-order optical response tensors, and can serve as the basis for further developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In this work, we incorporate many-body interactions into the Wannier-based scheme by working out an expression for the quadratic optical photoresponse tensor beyond the independent-particle approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Our derivation is based on the time-dependent current-density response theory and formally includes excitonic effects through a tensorial long- range exchange-correlation (xc) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Explicitly adopting the tensorial character of the response is of central importance, as this allows a natural connection with the formalism of the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='00607v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='mes-hall] 2 Jan 2023 2 independent-particle picture in the optical limit and, by ex- tension, with the Wannier-interpolation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To illustrate the generality and accuracy of our method, we analyze the SHG process in three bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In first place, we con- sider GaAs as a benchmark test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Secondly, we study BC2N, a highly anisotropic graphitic-layered semiconductor that show- cases the advantages of the adopted tensorial framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Fi- nally, we apply our scheme to the Weyl semimetal TaAs and discuss the results in the context of recent optical measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' II we present the main theoretical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We first express the microscopic quadratic conductivity tensor renormalized by many-body in- teractions, and compare our main tensorial expression with the scalar counterpart of TDDFT [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We then consider the optical limit and specialize to the SHG process, for which we derive new metallic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In order to establish the link to experimental observables, we analyze the connection be- tween the microscopic and macroscopic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Technical de- tails concerning the electronic-structure ab initio calculations based on maximally localized Wannier functions and the in- clusion of excitonic effects are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The com- puted SHG spectra of GaAs, BC2N and TaAs are presented and discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We provide concluding remarks in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' V, while several technical subjects are kept for the Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' THEORETICAL FRAMEWORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Microscopic response tensors and many-body effects Our starting point considers the microscopic response of a many-body (MB) system of electrons interacting via the Coulomb potential in a crystal that relates the electric current- density vector J(r,t) to the powers of an externally ap- plied time-dependent electric field Eext(r,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In practice, this amounts to expanding the current-density vector in a power series J(r,t) = ∑ j J j(r,t), (1) with the jth-order contribution defined as Jj(1) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' � 1 0 σ j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1)∏ j Eext( j +1)d j +1, (2) where we adopted the notation (rj,t j) ≡ ( j) with j a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The quantity σ j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j+1) denotes the jth-order MB conductivity tensor, and our main goal consists in finding an expression for the second order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' the j = 2 contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To do so, let us adopt the standpoint of an electron in an auxiliary Kohn-Sham (KS) system of independent particles, where the total electric field that it feels can be written as Etot(r,t) = Eext(r,t)+EH(r,t)+Exc(r,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (3) The Hartree (H) electric field as a function of the current- density vector is given by EH(1) = � KH(1,2)J(2)d2, (4) where KH(1,2) is the tensorial Hartree kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In turn, the xc electric field up to second order can be written as Exc(1) = � Kxc,1(1,2)J(2)d2 + � Kxc,2(1,2,3)J(2)J(3)d2d3, (5) with Kxc,1(1,2) and Kxc,2(1,2,3) the first-order and second- order tensorial xc kernels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Within the KS system, the response properties are governed by the so-called KS conductivity tensor, which describes the current-density vector in terms of powers of the total electric field, in such a way that Jj(1) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' � 1 0 σKS j (1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1)∏ j Etot( j +1)d j +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (6) Thus, the task is to express the MB response tensors up to sec- ond order in terms of the necessary KS response coefficients as well as the tensorial Hartree and xc kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Linear response As a warmup, we first review the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Within the time-dependent current-density response theory [46, 47], the first-order current-density vector is given by Ja 1(1) = � 1 0 ∑ b σab 1 (1,2)Eb ext(2)d2, (7a) Ja 1(1) = � 1 0 ∑ b σKS,ab 1 (1,2)Eb tot(2)d2 (7b) where σ1(1,2) and σKS 1 (1,2) are the first-order conductivity tensors of the MB and the KS system, respectively, defined as σab 1 (1,2) = δJa(1) δEb ext(2), (8a) σKS,ab 1 (1,2) = δJa(1) δEb tot(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (8b) Henceforth, superscripts refer to Cartesian components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ap- plying the chain rule in the definition of the MB conductivity tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 8a and taking into account the definition of the KS conductivity tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 8b, the first-order Dyson-like equation relating the MB and KS responses reads σab 1 (1,2) = � ∑ c σKS,ac 1 (1,3)ε−1,cb(3,2)d3, (9) 3 where we have introduced the dielectric tensor ε(1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This quantity accounts for the MB electronic screening effects within the crystal and its inverse links the total and external electric fields as Ea tot(1) = � 1 0 ε−1,ab(1,2)Eb ext(2)d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (10) Considering the implicit definition of the inverse dielectric tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 10 together with the relation between the total and external electric fields in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 3, we apply again the chain rule to obtain ε−1,ab(1,2) = δ(1,2)δab + � ∑ c Kac Hxc,1(1,3)σcb 1 (3,2)d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (11) Above, KHxc,1(1,2) = KH(1,2)+Kxc,1(1,2) is the grouping of the first-order tensorial Hartree and xc kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For practical purposes, it is useful to express the dielectric tensor in terms of the KS conductivity tensor instead of the MB one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Such expression is obtained by reproducing the pre- vious chain rule procedure, but this time starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 8b, and reads εab(1,2) = δ(1,2)δab − � ∑ c Kac Hxc,1(1,3)σKS,cb 1 (3,2)d3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (12) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Quadratic response In analogy with the treatment of the first-order response,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' the second-order current-density vector can be written as Ja 2(1) = �� 1 0 ∑ bc σabc 2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3)Eb ext(2)Ec ext(3)d2d3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (13a) Ja 2(1) = �� 1 0 ∑ bc σKS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='abc 2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3)Eb tot(2)Ec tot(3)d2d3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (13b) where σ2(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3) and σKS 2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3) are the second-order con- ductivity tensors of the MB and the KS systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' defined as σabc 2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3) = δ 2Ja(1) δEb ext(2)δEc ext(3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (14a) σKS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='abc 2 (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3) = δ 2Ja(1) δEb tot(2)δEc tot(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (14b) By sistematically applying the chain rule in the definition of the MB conductivity tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 14a, a procedure that is outlined in Appendix A, one derives the desired second-order Dyson-like equation relating the MB and KS responses σabc 2 (1,2,3) =∑ de f ��� ε−1,da(1,4)σKS,def 2 (4,5,6)ε−1,eb(5,2)ε−1, fc(6,3)d4d5d6 +∑ de f ��� σad 1 (1,4)Kdef xc,2(4,5,6)σeb 1 (5,2)σ fc 1 (6,3)d4d5d6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (15) The above equation can be regarded as the tensorial gener- alization of the expression for the second-order scalar den- sity response function obtained in TDDFT (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 180 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [48] or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 13 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Dealing with the response in the form of a tensorial quantity allows a natural connection with the description of the optical KS response, as we show below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Optical limit To proceed further, one needs explicit expressions for the KS response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This task can be greatly simplified by consider- ing the optical and long-wavelength limit, which assumes that the external electric field remains constant in the length-scale of the crystal’s unit cell [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Within this approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' it is conve- nient to adopt the formalism of Sipe and co-workers [12–14],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' where the current-density vector operator is split into its inter- band (ter) and intraband (tra) parts at any jth order as Jj(t) = dPter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' j(t) dt +Jtra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' j(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (16) 4 where the interband polarization- and intraband current- density vectors are respectively expressed in terms of the charge-density matrix elements ρj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='mn(t) as Pa ter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' j(t) = e V ∑ kmn ra nmρ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='mn(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (17a) Ja tra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='j(t) = e V ∑ kmn � va nmδnm −∑ b � ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='b nm +δnmεcabΩc n �� ρ j,mn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (17b) Above, V denotes the volume of the crystal, while n and m are band indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The transition matrix ele- ments involve several quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ra nm = (1−δnm)ξ a nm and ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='b nm = ∂ra nm/∂kb −i(ξ b nn −ξ b mm)ra nm are the inter- band dipole and its generalized derivative, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ξ a nm = i⟨un|∂/∂ka|um⟩ and εcabΩc n = ∂ξ b nn/∂ka −∂ξ a nn/∂kb stand for the Berry connection and curvature, respectively, with |un⟩ the crystal-periodic part of the Bloch function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' finally, va nm = ⟨un|∂ ˆH/∂ka|um⟩ denotes the velocity matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We kept the dependence on the crystal wave vector k implicit for all these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Based on the dynamical equation of the charge-density operator within the Schrödinger picture, one can solve for ρj,mn(t) employing an iterative scheme at the desired order in the electric field and compute the associated response ten- sors [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In frequency domain, the first-order interband polarization- and intraband current-density vectors can be re- spectively expressed as Pa ter,1(ω) = ∑ b αKS,ab ter,1 (ω)Eb tot(ω), (18a) Ja tra,1(ω) = ∑ b σKS,ab tra,1 (ω)Eb tot(ω), (18b) and similarly for second order Pa ter,2(ω12) = ∑ bc αKS,abc ter,2 (ω1,ω2)Ec tot(ω1)Eb tot(ω2), (19a) Ja tra,2(ω12) = ∑ bc σKS,abc tra,2 (ω1,ω2)Eb tot(ω1)Ec tot(ω2), (19b) with ω12 = ω1 + ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 18a and 18b, αKS ter,1(ω) and σKS tra,1(ω) are the first-order optical KS interband polarizabil- ity and intraband conductivity tensors, respectively, while in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 19a and 19b, αKS ter,2(ω1,ω2) and σKS tra,2(ω1,ω2) are their second-order counterparts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 16, the full optical KS conductivity tensors at first and second order are respectively given by σKS 1 (ω) = −iωαKS ter,1(ω)+σKS tra,1(ω), (20) and σKS 2 (ω1,ω2) = −iω12αKS ter,2(ω1,ω2)+σKS tra,2(ω1,ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (21) The expressions for the optical KS interband polarizabil- ity and intraband conductivity tensors are well established at first order [50, 51], as well as at second order in the case of semiconductors [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the case of metals and semimet- als extra terms appear due to the presence of a Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Recent works [27, 28] have derived and thoroughly discussed the metallic terms of σKS tra,2(ω1,ω2), paying special attention to the direct-current contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As for αKS ter,2(ω1,ω2), its metallic terms have not been previously derived to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Appendix B we provide the general ex- pressions of all optical KS response tensors up to second order valid for any kind of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Second harmonic generation In the remaining of this work, for conciseness we specialize in the calculation of a particular quadratic optical response, namely the second harmonic generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The SHG process considers two initial photons with same frequency which are combined to generate a final photon with twice the initial fre- quency, maintaining the coherence of the excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' By set- ting ω1 = ω2 ≡ ω in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B2a and B2b of Appendix B, the SHG KS interband polarizability intraband conductivity ten- sors are respectively given by αKS,abc ter,2 (ω,ω) = e3 2¯h2V � ∑ kmnl ra nm � rb mlrc ln +rc mlrb ln � ωln −ωml � 2fnm ωmn −2 ˜ω − fnl ωln − ˜ω − fml ωml − ˜ω � +i ∑ kmn � fnm � 2ra nm � rb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='c mn +rc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='b mn � ωmn (ωmn −2 ˜ω) + ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='b nmrc mn +ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='c nmrb mn ωmn (ωmn − ˜ω) + ra nm � rb mnΛc mn +rc mnΛb mn � ω2mn � 1 ωmn − ˜ω − 4 ωmn −2 ˜ω �� − ra nm � fnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='brc mn + fnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='crb mn � ωmn ˜ω �� , (22a) 5 σKS,abc tra,2 (ω,ω) = e3 2¯h2V � − ∑ kmn fnm � rc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='a nmrb mn +rb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='a nmrc mn ωmn − ˜ω + Λa nm � rb mnrc nm +rc mnrb nm � 2ω (ωmn − ˜ω) � +∑ kn � i( fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='bεdac + fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='cεdab)Ωd n ˜ω − va n fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='bc ˜ω2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (22b) Above, ωmn = ωm −ωn and fnm = fn − fm, with ¯hωn and fn = f(¯hωn) the eigenvalue and occupation factor of the eigenstate |kn⟩, respectively, while Λa nm = va n −va m and ˜ω ≡ ω +iη/¯h, with η a positive real infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The terms including the derivatives fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='a = ∂ fn/∂ka and fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab = ∂ fn/∂ka∂kb correspond to the metallic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' With explicit expressions for the SHG KS response coeffi- cients at hand, we can now calculate the SHG MB conduc- tivity tensor from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 15 as σabc 2 (ω,ω) = ∑ de f ε−1,da(2ω)σKS,de f 2 (ω,ω)ε−1,eb(ω)ε−1, fc(ω)+∑ def σad 1 (2ω)Kdef xc,2(ω,ω)σeb 1 (ω)σ fc 1 (ω), (23) where the optical dielectric and MB conductivity tensors sat- isfy respectively (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 12 and 9) εab(ω) = δab −∑ c Kac Hxc,1(ω)σKS,cb 1 (ω), (24) and σab 1 (ω) = ∑ c σKS,ac 1 (ω)ε−1,cb(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (25) Let us inspect Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 23 in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' MB interactions come in two different ways;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' on the one hand, through the inverse dielectric tensor that includes screening effects, and on the other hand, through the second-order tensorial xc ker- nel Kxc,2(ω,ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Due to hierarchy arguments we expect the former to be dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Focusing on the first piece on the right-hand side (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=') of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 23, the response at frequency ω is affected by the screening at that and twice that frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This can lead to double-frequency many-body resonances in the SHG spectrum, as we will show in more detail when ana- lyzing our numerical results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' From the microscopic to the macroscopic response As the final step, we consider the connection between the previous microscopic coefficients and their macroscopic counterparts, which are ultimately the quantities measured in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The macroscopic response to light is described by Maxwell’s equations and can be accessed by performing a macroscopic average of the microscopic response tensors over regions in space that are large in comparison with the crystal unit cell, but small compared to the wavelength of the external perturbation [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In this work we adopt the formula- tion of Del Sole and Fiorino [52] for relating the macroscopic and microscopic scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' the detailed derivation is outlined in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Here we focus on the SHG process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' by setting ω1 = ω2 ≡ ω in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D12, the macroscopic SHG photocon- ductivity tensor is calculated from its microscopic counterpart as σabc M,2(ω,ω) = εaa M (2ω)σabc 2 (ω,ω)εbb M (ω)εcc M(ω), (26) where the macroscopic optical dielectric tensor is given in terms of the microscopic optical conductivity by εaa M (ω) = � 1−i4π ω σ1(ω) �−1,aa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (27) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' TECHNICAL DETAILS In this section we describe in detail the steps followed in the calculations for three bulk materials: the semiconductor GaAs, the semiconductor BC2N and the semimetal TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' DFT calculations In a first step, we performed DFT self-consistent calcula- tions using the QUANTUM ESPRESSO code package [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The interaction between valence electrons and atomic cores was modeled by means of projector-augmented-wave pseu- dopotentials [55] with scalar relativistic corrections for GaAs and BC2N and fully relativistic corrections for TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The pseudopotentials were taken from the QUANTUM ESPRESSO website and generated using the Perdew-Burke-Ernzerhof generalized gradient approximation for thexc energy func- tional [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For GaAs, we considered the zinc-blende crystal structure together with the experimental value of the lattice parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' a = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='68 a0 [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We performed DFT calcula- tions using a 8×8×8 k-point mesh in combination with fixed occupation values and a plane-wave basis set with a cut-off energy of 60 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For BC2N, we considered the graphitic- layered A2 crystal structure, which is the most stable noncen- trosymmetric bulk structure, with orthorhombic space group 6 Γ E S Z N Γ Z X Γ −6 −4 −2 0 2 4 6 Energy [eV] Ab-initio Wannier int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 1: DFT and Wannier-interpolated energy bands of TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The horizontal dashed line at 3 eV denotes the upper limit of the inner energy window used in the disentanglement step of the Wannier con- struction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Pmm2 (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 25) following the theoretical structural parame- ters of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We performed DFT calculations using a 10 × 10 × 10 k-point mesh in combination with fixed occu- pation values and a plane-wave basis set with a cut-off en- ergy of 70 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Finally, for TaAs, we considered its ground- state body-centered-tetragonal crystal structure with nonsym- morphic space group I41md (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 109) following the experi- mental structural parameters of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We performed non- collinear spin-DFT calculations using a 8×8×8 k-point mesh in combination with occupation values calculated by means of the optimized tetrahedron method [60] and a plane-wave basis set with a cut-off energy of 60 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Wannier interpolation In a postprocessing step, we constructed maximally lo- calized Wannier functions (MLWF) using the WANNIER90 code package [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For GaAs, starting from a set of 15 spin-degenerate bands, we constructed 11 disentangled ML- WFs spanning the 4 high-energy valence bands and the 7 low-energy conduction bands using two s and one p trial or- bitals centered on all atoms, as well as one s trial orbital halfway between the two atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For BC2N, starting from a set of 38 spin-degenerate bands, we constructed 8 disentan- gled MLWF spanning the 4 high-energy valence bands and the 4 low-energy conduction bands using pz trial orbitals cen- tered on all atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Finally, for TaAs, starting from a set of 48 spin-polarized bands, we constructed 32 disentangled ML- WFs spanning the 16 high-energy valence bands and the 16 low-energy conduction bands using p and d trial orbitals cen- tered on all As and Ta atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In all cases, the agreement between DFT and Wannier-interpolated bands is in excellent agreement inside the chosen inner energy win- dow [62], as we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 1 for the case of TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Having converged the Wannier basis, we then computed the linear and quadratic optical KS response tensors (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B1a-B1b and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 22a-22b, respectively) using Wannier interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To that end, we used the schemes described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [45] for the calculation of interband dipole matrix ele- ments and Berry curvatures, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [44] for the calculation of generalized derivatives of the dipole matrix and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [63] for the calculation of velocity matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Following the procedure of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [44, 64], in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 22a-22b we regularized the energy denominators of the three-band term and ra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='b mn in- volving intermediate states by means of an auxiliary param- eter ηr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Alongside, the derivatives of the occupation factors were computed by replacing fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='a → d f dωn va n, (28) for the first-order derivative and fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab → d2 f dω2n va nvb n + d f dωn ωn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab, (29) for the second-order derivative, where ωn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab denotes the in- verse effective mass tensor [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We considered Gaussian dis- tributions for the derivatives of the occupation factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In order to obtain well-converged optical spectra, we used dense k-point interpolation meshes of 250 × 250 × 250 for GaAs, 200 × 200 × 200 for BC2N, and 300 × 300 × 300 for TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' With respect to the imaginary part of the complex en- ergy ¯h ˜ω (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 22a and 22b), we set η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='1 eV in the case of GaAs and TaAs, consistent with carrier scattering lifetimes (∼ 10 fs) near the Fermi level observed in both GaAs [65] and TaAs [66], while for BC2N we employed an adaptative scheme [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Regarding the auxiliary parameter for regular- izing energy denominators, we chose ηr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='04 eV for both GaAs and BC2N, following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [44] and [67], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the case of TaAs, we set ηr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='1 meV, in order to prop- erly capture the contriburion of Weyl points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The occupation factors and their derivatives are evaluated at zero temperature (T = 0 K) for the semiconductors GaAs and BC2N and at room temperature (T = 300 K) for TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Bootstrap method for the tensorial xc kernel Within the long-wavelength and optical limit, the general expressions for the tensorial Hartree and xc kernels (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 4 and 5) simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The Hartree contribution reduces to a diago- nal form Kab H (ω) = −i4πδab ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (30) In addition, in this work we assume a tensorial long-range corrected (LRC) xc kernel based on an attractive Coulomb- type potential [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' With these assumptions, the first-order xc contribution simplifies to (see Appendix C) Kab xc,1(ω) = iαa LRCδab ω , (31) 7 which is a diagonal but generally anisotropic 3 × 3 ma- trix composed of three independent, positive-definite and frequency-independent coefficients αa LRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We note that this expression is the tensorial generalization of the scalar α- parameter of LRC xc kernels used in TDDFT [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We calculated the coefficients αa LRC self-consistently by means of the so-called bootstrap method [70] along each principal axis of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Finally, in our calculations we discarded the effect of the second-order tensorial xc kernel Kxc,2 entering Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5, given that its approximate expression is generally un- known and its effects are expected to be minor in comparison to the first-order contribution [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To sum up, in practice, we calculate the microscopic SHG MB conductivity tensor by means of σabc 2 (ω,ω) = ∑ def ε−1,da(2ω)σKS,de f 2 (ω,ω)ε−1,eb(ω)ε−1, fc(ω), (32) where the microscopic optial dielectric tensor is given by εab(ω) = δab +i4π −αa LRC ω σKS,ab 1 (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (33) The last step involves calculating the macroscopic SHG pho- toconductivity tensor from its microscopic counterpart by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' RESULTS In this section we present our numerical results of the macroscopic SHG photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To facilitate comparison with existing literature, we will partly describe our results in terms of the photosusceptibility, whose connection to the pho- toconductivity used in our derivations of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' II is provided in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the materials analyzed in this work, the optical dielectric tensor is diagonal due to symmetry arguments [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' It then follows that the relation between the macroscopic SHG MB and KS photosusceptibilities simplifies to χabc 2 (ω,ω) = β abc(ω)χKS,abc 2 (ω,ω), (34) with β abc(ω) =εaa M (2ω)ε−1,aa(2ω)× εbb M (ω)ε−1,bb(ω)εcc M(ω)ε−1cc(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (35) the enhancement factor, a quantity that will be useful when discussing the impact of MB corrections in our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' GaAs The first SHG measurements in GaAs date back to the 1960’s [72, 73], and it has become the standard material for benchmarking theoretical SHG calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Initial works were based on empirical pseudopotentials [74] and tight- binding models [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' More recently, several first princi- ples studies have been reported [17–19, 21, 76];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' most have −6 −4 −2 0 2 4 6 8 Imχxyz 2 (ω, ω) [102pm/V] (a) KS MB −6 −4 −2 0 2 4 Reχxyz 2 (ω, ω) [102pm/V] (b) KS MB 0 1 2 3 4 5 ω [eV] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='4 Enhancement factor (c) Ebg 2 Ebg FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2: (a) Imaginary and (b) real parts of the macroscopic SHG photosusceptibility for bulk GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Dashed grey and solid black lines represent the KS and MB spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Vertical dotted lines represent the band-gap energy (Ebg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='42 eV) and half its value (Ebg/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (c) Absolute value of the many-body enhancement factor β xyz(ω) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' been performed within the independent-particle approxima- tion plus scissors corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Beyond this approach, only few studies have reported the impact of MB interactions [34, 35, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Since GaAs is a cubic crystal, χabc 2 (ω,ω) = χxyz 2 (ω,ω) for any permutation abc of xyz, while all other components of the tensor vanish by symmetry [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Figures 2(a) and 2(b) show the spectra of the imaginary and real parts, respectively, of the calculated macroscopic SHG photosusceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' KS calcula- tions have been performed by applying a scissors operator that rigidly shifts the conduction bands by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='91 eV in order to re- cover the experimental value of the band-gap energy at room temperature, Ebg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='42 eV [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the MB picture, exci- tonic effects have been included through the self-consistently calculated LRC xc coefficient αa LRC ≡ αLRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='11, which is isotropic in cubic crystals and consistent with that of previous ab initio studies [78–80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We begin by describing the KS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The spectrum of the imaginary part [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2(a)] is finite for energies above Ebg/2 [14] and contains a strong peak near the band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As 8 0 1 2 3 4 5 ω [eV] 0 2 4 6 8 10 |χxyz 2 (ω, ω)| [102pm/V] Ebg Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [81] KS Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [37] MB Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [37] KS Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [38] MB Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [38] KS Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [35] MB Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [35] KS MB FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 3: Absolute value of the macroscopic SHG photosusceptibility for bulk GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Dashed grey and solid black lines represent our cal- culated KS and MB spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Yellow circles represent the experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Light red left triangles, light green up triangles and light blue squares represent theoretical spec- tra within the independent-particle plus scissors approximation from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [37], [38] and [35], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Dark red right triangles, dark green down triangles and dark blue diamonds represent theoretical spectra including excitonic effects by means of BSE from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [37] and [38] and TDDFT from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [35], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' for the real part [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2(b)], it is finite at all energies owing to photons absorbed or emitted in virtual excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The spec- trum grows progressively at low energies and exhibits maxima at Ebg/2 and Ebg due to two- and one-photon resonances, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' At higher energies double resonant transitions take place [14] and the spectrum shows several strong peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The net effect of MB corrections is to increase the magni- tude of both the imaginary and real parts of the SHG spec- trum, as is clearly visible in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2(a) and 2(b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The enhancement factor displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 2(c) shows that the difference ranges between 0 and 50 %, with the largest renor- malization taking place right at the band-edge energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' No new spectral feature is formed as a consequence of excitonic ef- fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 3 we show the absolute value of the macroscopic SHG photosusceptibility and compare our calculations with experimental measurements as well as previous theoretical works including different approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The experimen- tal spectrum is dominated by a peak at the band-edge en- ergy and contains a “V”-shaped form between 2 and 3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' These two spectral features are well described by both our KS and MB calculations, which show similar shape but dif- ferent size as discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Our KS result is in quali- tative agreement with previous KS calculations, specially that of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Our MB calculation strikes the best balance in describing the magnitude and width of the two spectral fea- tures of the experiment, although the height of the “V”-shaped form is somewhat overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Here too we note a qualita- tive agreement with the TDDFT result of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In quantitative terms, our results show sharper peaks than those of previous theoretical works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This can be a conse- quence of the small smearing factors achieveable thanks to Wannier interpolation, which makes it possible to consider on the order of 106 k-points for converging the SHG inte- grals over the BZ (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 22a and 22b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For comparison, the calculations of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [38] and [35] employed on the order of 103 and 104 k-points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This fine sampling has allowed us to model the lifetime of hot carriers (∼ 10 fs) in bulk GaAs [65], which therefore renders more realistic spec- tral widths as compared to experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' BC2N The graphitic-layered semiconductor BC2N has attracted interest in the last years as a potential nonlinear optical ma- terial [82, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Its layered geometry composed of alternat- ing zigzag of C−C and B−N chains makes it a malleable and strongly anisotropic crystal [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Among its several poly- types, the A2 configuration (BC2N-A2) is the most stable non- centrosymmetric structure [58] that allows a finite quadratic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' First-principles calculations in the independent- particle approximation have recently predicted a large SHG for BC2N-A2 in monolayer and nanotube form [85] that is an order of magnitude larger than in bulk GaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A large shift cur- rent has also been calculated recently in bulk [86] and mono- layer [67] form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To our knowledge, no systematic study of MB effects on the SHG has been carried out for bulk BC2N- A2 up to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Owing to its mm2 point group, the symmetry-allowed com- ponents of the SHG photosusceptibility tensor for BC2N-A2 are xxy = xyx, yxx, yyy yzz and zzy = zyz [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Their absolute values in the KS picture are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In order to facilitate the discussion of the spectral features, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 4(b) we show the joint density of states (JDOS) per crystal unit cell [86] for the one- and two-photon signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In these and following figures, Ebg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='18 eV denotes the direct band-gap energy, while EX = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='33 eV refers to the band-gap energy at high symmetry point X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The latter was found to mark the peak absorption of the shift current at low energies [86] and will also play an important role in the SHG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The tensor components xxy and yxx dominate the SHG pho- toresponse with values of the order of the SHG for the mono- layer and nanotube forms [85], and coincide with the domi- nant components of the shift current for bulk BC2N-A2 [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The maximum value of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='8×103 pm/V takes place for the yxx component at EX/2 owing to a two-photon absorption pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' These and further specral features like the peak at ≃ 1 eV can be associated to contributions in the one- and two-photon JDOS [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We focus next on the MB interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Unlike the case of GaAs studied previously, BC2N-A2 is anisotropic and so is the tensorial LRC xc kernel, with self-consistently calculated coefficients are {αx LRC,αy LRC,αz LRC} = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='13,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='29,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='11}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Note that the z component is an order of magnitude larger than the x and y components, as well as the coefficient computed for GaAs (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Therefore, BC2N-A2 represents a clear example where an isotropic treatment of the excitonic 9 0 1 2 3 4 5 6 |χKS,abc 2 (ω, ω)| [103pm/V] Ebg 2 EX 2 Ebg EX (a) xxy yxx yyy yzz zzy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 ω [eV] 0 1 JDOS [eV−1] (b) JDOS(ω) JDOS(2ω) 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 4: (a) Absolute value of the SHG KS photosusceptibility tensor for bulk BC2N-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Solid black, dashed grey, dotted green, dashdot- dashed blue and dashdotdotted red lines represent the spectra of the xxy = xyx, yxx, yyy, yzz and zzy = zyz non-vanishing components, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The inset zooms in the yyy, yzz and zzy components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (b) Joint density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Solid magenta and dashed orange lines rep- resent the one- and two-photon signals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Vertical dotted lines represent the band-edge energy range boundaries (Ebg ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='18 and EX ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='33) and half their values (Ebg/2 and EX/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' effects constitutes a poor choice, given that the value of the space-averaged scalar LRC xc coefficient, αiso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='42, is close to none of the actual space-resolved tensorial compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the following, we illustrate the profound errors that this procedure can induce in the absorption spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5(a) we display the renormalization of the macro- scopic SHG photosusceptibility tensor component xxy by electron-hole corrections at two levels: using the anisotropic and isotropic tensorial LRC xc kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Comparison to the KS response shows that the anisotropic kernel induces a max- imum increase of nearly a factor two [see enhancement factor in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5(b)], but does not alter the overall shape of the spec- trum, in line with what we found for GaAs (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' On the other hand, the isotropic kernel produces a large peak at half the band-edge energy that completely dominates the MB spectrum, with an enhancement of more than one order of magnitude as compared to the anisotropic kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A sec- ondary peak is also visible at the band-edge energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The origin of these two sharp peaks can be determined by inspecting the inverse of the macroscopic optical dielectric tensor along x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' this quantity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5(c) separately for the real and imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' While Imε−1,xx M (ω) is barely 0 20 40 60 80 100 |χxxy 2 (ω, ω)| [103pm/V] Ebg 2 EX 2 Ebg EX (a) 0 1 2 0 5 10 KS MB - iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' MB - aniso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 0 10 20 30 40 Enhancement factor (b) 0 1 2 3 1 2 iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' aniso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 ω (eV) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='10 ε−1,xx M (ω) (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='01 Im - iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Im - aniso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Re - iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Re - aniso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5: (a) Absolute value of the xxy component of the macroscopic SHG photosusceptibility tensor for bulk BC2N-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The solid black line represent the KS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The dashed orange and dashdotted magenta lines represent the MB spectrum using the isotropic (iso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=') and anisotropic (aniso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=') tensorial LRC xc kernel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (b) Absolute value of the enhancement factor β xxy(ω) in the isotropic (dashed orange) and anisotropic (dashdotted magenta) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (c) xx component of the inverse of the macroscopic optical dielectric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The solid red and dotted blue lines represent the real (Re) and imag- inary (Im) parts in the isotropic case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The dashed green and dashdotted grey lines represent the real and imaginary parts in the anisotropic case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' affected by the type of tensorial LRC xc kernel, Reε−1,xx M (ω) shows a strong shift that is nearly frequency-independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' both these features can be qualitatively understood by work- ing out explicit expressions (use Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 33 and 25 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 27) and noting that the Hartree contribution is much stronger than any of the LRC xc components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', 4π ≫ αa LRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the case of the isotropic kernel, Reε−1,xx M (ω) crosses the zero axis very close to the band-edge energy, where Imε−1,xx M (ω) ≃ 0 too, lead- ing to a sharp peak in εxx M(ω) at that energy [see enhancement factor in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This peak is then replicated at half the band-edge energy in the SHG spectrum through the εxx M(2ω) factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 26, and enhanced by transition matrix-elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 ω [eV] 0 5 10 15 20 25 30 35 |χzzz 2 (ω, ω)| [103pm/V] (a) αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='1 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='4 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='6 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='7 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='8 αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='9 αz LRC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 KS Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6] 0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 ISHG(θ) [normalized] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 (b) Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ∥ Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6] Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ⊥ Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6](×8) ∥ MB ⊥ MB(×8) ∥ KS(×6) ⊥ KS(×24) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='4 ω [eV] 0 5 10 15 20 25 |σabc M,2(ω, ω)| [(e2/¯h)V−1] (c) xxz - Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [87] zxx - Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [87] eff - Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [87] xxz - MB(×102) zxx - MB(×102) eff - MB(×10) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 6: (a) Absolute value of the zzz component of the macroscopic SHG photosusceptibility tensor for bulk TaAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Dashed black and solid colored lines represent the KS and MB spectra as a function of αz LRC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Black errorbar corresponds to the experimental datapoint from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6] (b) SHG intensity polar plot in both parallel (∥) and perpendicular (⊥) generator/analyser configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For better visualization, results in the ⊥ configuration are multiplied by a factor 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' KS calculations are multiplied by a factor 6 and 24 for ∥ and ⊥ configurations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Open red and blue circles represent ∥ and ⊥ experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Solid (dashed) red (magenta) and blue (cyan) lines represent our MB (KS) calculations in the ∥ and ⊥ configurations, respectively, for {αx=y LRC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='8,αz LRC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (c) Absolute value of the macroscopic SHG photoconductivity tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Open red, dark blue and black circles represent experimental data from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [87] for the xxz, zxx and effective components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Orange, cyan and grey solid lines represent our calculated MB spectra of the xxz, zxx (multiplied by 100) and effective (multiplied by 10) components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We have verified that a similar effect takes place for the SHG tensor component yxx too (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In this case, the isotropic kernel gives rise to a even larger peak right at the band-edge energy reaching ≃ 600 × 103 pm/V (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5(a) for comparison), while the anisotropic kernel induces only moderate changes to the KS response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' These examples show that sharp, exciton-like peaks in the SHG spectrum can be in- duced by MB effects provided the appropriate conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' These conditions are very sensitive to numerics, which stresses the importance of accounting for the space-resolved anisotropy of the material in the tensorial xc kernel, and there- fore, its advantage over a space-averaged scalar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' TaAs Theoretically predicted [88, 89] and experimentally con- firmed in 2015 [90–92], TaAs is a type I Weyl semimetal [93] without an inversion center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Following its discovery, several experiments have reported remarkable nonlinear optical prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6] measured a “giant” SHG photosusceptibility at ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='55 eV that is an order of magnitude larger than in most other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Shortly after, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [87] extended the mea- surements to lower energies and found a narrow resonance at ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='75 eV with an even larger photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In addition to the SHG, other quadratic optical responses such as the shift current have also been measured to be exceptionally large [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Due to its 4mm point group, the symmetry-allowed compo- nents of the SHG tensor in TaAs are zzz, zxx = zxz = zyy = zyz and xxz = xzx = yyz = yzy, where x and y are equivalent direc- tions of the tetragonal unit cell and the direction perpendicular to the xy plane is the polar axis z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Unlike GaAs and BC2N studied previously, TaAs is a semimetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In this case, the bootstrap method for the self- consistent calculation of the tensorial LRC xc kernel cannot be applied directly since Im[ε(ω = 0)] ̸= 0 [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In conse- quence, we have chosen to renormalize the SHG KS spectrum for a reasonable range of LRC xc coefficients {αx=y LRC,αz LRC} and determine their most appropriate values by comparing to the experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 6(a) we show our calculated |χzzz 2 (ω,ω)| as a func- tion of αz LRC together with the available experimental data- point at ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='55 eV from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6], equal to 7±1×103 pm/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The KS response peaks around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='85 eV, and captures the magnitude of the experimental value but underestimates it by roughly a factor two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The SHG MB spectrum grows with the value of αz LRC until it equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3, where it basically matches the experiment and therefore represents the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For αz LRC > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3, the magnitude of |χzzz 2 (ω,ω)| starts decreasing and it becomes nearly overdamped for αz LRC > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The over- all shape of the spectrum is maintained in the whole range of αz LRC considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' By applying the same procedure to the zxx and xxz components we have determined the remaining coef- ficient αx=y LRC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6], two additional measurements were conducted at ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='55 eV for varying angle θ of linearly-polarized light, with the field oriented along the [1,1,-1] (parallel setup, ∥) and [1,- 1,0] (perpendicular setup, ⊥) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Making use of the appropriate combination of the SHG tensor components (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 3 and 4 of the Supplementary Information in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [6]), we have calculated the angular dependence of the SHG intensity and compared it to the experimental polar plot, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For the parallel configuration, the response shows an elongated shape along the θ = 0 axis that is remarkably well captured by our MB result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' For the perpendicular config- uration, the response shows a four-fold structure with maxima at π/4+n·π/2 and minima at n·π/2 for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' While the KS calculation fails in both magnitude and shape, our MB result nicely agrees with the experimental measurement, thus capturing the main characteristics of the photoresponse at this 11 particular energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As the last step, we proceed to study the low-energy region accessed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [87], where a narrow reso- nance was measured at ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='75 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 6(c) we compare the experimentally measured |σzxx M,2|, |σxxz M,2| and |σeff M,2| ≡ |σzzz M,2 +4σxxz M,2 +2σzxx M,2| with our calculations using the optimal values of αa LRC quoted previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Our results un- derestimate the main exciton-like peak by an order of magni- tude, and we have been unable to strike a substantial improve- ment by further varying αa LRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The description of this low- energy peak appears therefore to be beyond the scope of the linear tensorial LRC xc kernel considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' It is tempting to speculate that it might be induced by MB corrections not included in our calculations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', a frequency dependence in the LRC xc coefficients αa LRC(ω) [94, 95], or the quadratic tensorial xc kernel of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' SUMMARY In summary, we have described a general scheme for cal- culating the quadratic optical response to light tensor of crystals taking into account many-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We have formally included excitonic effects by means of a ten- sorial long-range exchange-correlation kernel whose coeffi- cients have been calculated self-consistently using the boot- strap method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We have also generalized previous expres- sions [12–14, 27, 28] for the transition matrix elements to account for all metallic contributions, allowing an exhaustive study of materials like Weyl semimetals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Linking the formalism with the Wannier interpolation of the transition matrix elements [44, 45, 96], we have per- formed calculations of the second-harmonic generation pho- toresponse tensor in a range of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Besides bench- marking our approach, we have shown that the electron- hole attraction can give rise to strong and sharply localized one- and two-photon resonances that are absent in the Kohn- Sham photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the graphitic-layered crystal BC2N, an space-averaged isotropic approach overestimates the elec- tronic renormalization by orders of magnitude, highlighting the need of accounting for the space-resolved anisotropic na- ture of many-body interactions in tensorial form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Finally, with the use of a highly dense k-space mesh, our calcula- tions have reproduced the magnitude and angular dependence of the photoresponse for the Weyl semimetal TaAs measured recently [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We hope that the presented scheme together with its imple- mentation in the Wannier90 and WannierBerri code pack- ages will facilitate an efficient and accurate calculation of the quadratic optical photoresponse of materials beyond the SHG process analyzed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We note that the procedure adopted for including many-body excitonic effects requires only a fraction of the computational time as compared to the calculation of the Kohn-Sham photoresponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The proposed method can be improved in several fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Adopting a Wannier-based strategy for the calculation of the linear xc kernel in metals and semimetals (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [97]) would allow a fully parameter-free analysis in these type of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' An improved description of many-body effects can be achieved by extending the LRC xc coefficients to frequency domain [94, 95] or by working out an approximation for the second-order xc kernel, which would open the way to study potentially new excitonic effects that have been barely de- scribed in the literature up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The method can also model crystal local-field corrections, although their effect has been found to be minor when employing a localized Wannier ba- sis [98, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Finally, accounting for quasiparticle self-energy corrections due to electron-electron or electron-phonon inter- actions would allow modelling extrinsic quadratic contribu- tions such as the ballistic current [2, 100–102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We expect to address these subjects in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We are very grateful to Ivo Souza and Fernando de Juan for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This project has received funding from the European Union’s Horizon 2020 research and in- novation programme under the European Research Council (ERC) grant agreement No 946629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Appendix A: DERIVATION OF THE QUADRATIC DYSON-LIKE RESPONSE TENSOR EQUATION Here we outline the steps involved in the derivation of the Dyson-like equation relating the MB and KS conductivity ten- sors at second order in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 15 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We start by applying the chain rule twice in the definition of the quadratic MB conductivity tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 14a, σabc 1 (1,2,3) = δ δEb ext(2) � ∑ d � δJa(1) δEd tot(4) δEd tot(4) δEc ext(3)d4 � =∑ d � � δ 2Ja(1) δEb ext(2)δEd tot(4) δEd tot(4) δEc ext(3) + δJa(1) δEd tot(4) δ δEb ext(2) � δEd tot(4) δEc ext(3) �� d4 =∑ de �� δ 2Ja(1) δEe tot(5)δEd tot(4) δEe tot(5) δEb ext(2) δEd tot(4) δEc ext(3)d4d5+∑ d � δJa(1) δEd tot(4) δ 2Ed tot(4) δEb ext(2)δEc ext(3)d4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (A1) 12 The first term on the right-hand side (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=') of the last line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A1 can be expressed in terms of σKS 2 and ε using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 14b and 10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As for the second term, the piece δJa(1)/δEd tot(4) can be written in terms of σKS 1 using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 8b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' while the calculation of the remaining piece requires applying the chain rule again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' δ 2Ed tot(4) δEb ext(2)δEc ext(3) = δ δEb ext(2) � δ(4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='3)δdc + � ∑ e δEd Hxc(4) δJe(5) δJe(5) δEc ext(3)d5 � = � ∑ e � δ 2Ed Hxc(4) δEb ext(2)δJe(5) δJe(5) δEc ext(3) + δEd Hxc(4) δJe(5) δ 2Je(5) δEb ext(2)δEc ext(3) � d5 = �� ∑ e f δ 2Ed Hxc(4) δJ f (6)δJe(5) δJ f (6) δEb ext(2) δJe(5) δEc ext(3)d5d6+ � ∑ e δEd Hxc(4) δJe(5) δ 2Je(5) δEb ext(2)δEc ext(3)d5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (A2) where we used ε−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='2) = δEatot(1) δEbext(2) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 14b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The first term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' of the last line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A2 can be expressed in terms of Kabc xc,2(1,2,3) and σab 1 (1,2) using δ 2Ea Hxc(1) δJb(2)δJc(3) = Kabc xc,2(1,2,3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 8a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As for the second piece, it can be recast in terms of Kab Hxc,1(1,2) and σ2 using Kab Hxc,1(1,2) = δEa Hxc(1) δJb(2) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 14a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Taking into account all the previous observations, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A1 as σabc 2 (1,2,3) = �� ∑ de σKS,aed 2 (1,5,4)ε−1,eb(5,2)ε−1,dc(4,3)d4d5 + ��� ∑ de f σKS,ad 1 (1,4)Kd fe xc,2(4,6,5)σ fb 1 (6,2)σec 1 (5,3)d4d5d6 + �� ∑ de σKS,ad 1 (1,4)Kde Hxc,1(4,5)σebc 2 (5,2,3)d4d5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (A3) Moving now the last term on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' A3 to the left-hand side (l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' ), we can rewrite this side with the quadratic MB conductivity tensor as a common factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Taking advantage from the definition of the dielectric tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 12, we obtain that � ∑ e � δ(1,5)δae − � ∑ d σKS,ad 1 (1,4)Kde Hxc,1(4,5)d4 � σebc 2 (5,2,3)d5 ≡ � ∑ d εda(1,4)σdbc 2 (4,2,3)d4 = �� ∑ de σKS,aed 2 (1,5,4)ε−1,eb(5,2)ε−1,dc(4,3)d4d5+ ��� ∑ def σKS,ad 1 (1,4)Kd fe xc,2(4,6,5)σ fb 1 (6,2)σec 1 (5,3)d4d5d6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (A4) Finally, inverting the transpose of the dielectric tensor from the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' to the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', we arrive at the Dyson-like equation 15 quoted in the main text: σabc 2 (1,2,3) = ��� ∑ de f ε−1,da(1,4)σKS,def 2 (4,5,6)ε−1,eb(5,2)ε−1, fc(6,3)d4d5d6 + ��� ∑ de f σad 1 (1,4)Kdef xc,2(4,5,6)σeb 1 (5,2)σ fc 1 (6,3)d4d5d6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (A5) Appendix B: KS OPTICAL RESPONSE TENSOR EXPRESSIONS UP TO SECOND ORDER In this appendix we provide the expressions of all optical KS response tensors up to second order within the formalism of Sipe and co-workers [12–14] (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' These expres- sions are valid for any combination of ω1 and ω2 and include metallic terms proportional to k-space derivatives of the occu- pation factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Here we merely quote the final expressions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' for details on the derivation steps, we refer the reader to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' IV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [14] or to Appendix A in the Supplemental Material of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' At first order, the optical KS interband polarizability and intraband conductivity tensors are respectively expressed as αKS,ab ter,1 (ω) = e2 ¯hV ∑ kmn fnm ra nmrb mn ωmn − ˜ω , (B1a) σKS,ab tra,1 (ω) = e2 ¯hV ∑ kn fn � iωn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab ˜ω −εcabΩc n � , (B1b) while at second order they are respectively expressed as 13 αKS,abc ter,2 (ω1,ω2) = e3 2¯h2V � ∑ kmnl ra nm ωmn −ω12 � fnl � rb lnrc ml ωln −ω1 + rc lnrb ml ωln −ω2 � − flm � rb mlrc ln ωml −ω1 + rc mlrb ln ωml −ω2 �� +i ∑ kmn � fnmrb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='c mn + fnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='crb mn ωmn −ω1 − fnmrb mnΛc mn (ωmn −ω1)2 − fnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='brc mn ω1 + fnmrc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='b mn + fnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='brc mn ωmn −ω2 − fnmrc mnΛb mn (ωmn −ω2)2 − fnm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='crb mn ω2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (B2a) σKS,abc tra,2 (ω1,ω2) = e3 2¯h2V � − ∑ kmn � fnmΛa nm ω12 � rc nmrb mn ωmn −ω1 + rb nmrc mn ωmn −ω2 � + fnm � rc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='a nmrb mn ωmn −ω1 + rb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='a nmrc mn ωmn −ω2 �� +∑ kn � i � fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='bεdac ω1 + fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='cεdab ω2 � Ωd n − va n fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='bc ω1ω2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (B2b) All quantities appearing in the expressions above have been introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' II B except for ωn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ab in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B1b, which stands for the inverse effective mass tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As a remark, the metallic terms of the quadratic optical KS intraband polariz- ability tensor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' B2a are shown here for the first time to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Appendix C: TENSORIAL KERNELS Here we describe the calculation of the tensorial kernels in the optical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Let us start by reviewing the Hartree contri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The Hartree potential is defined by VH(1) = � vc(1,2)ρ(2)d2, (C1) where vc(1,2) = δ(t1 −t2)/|r1 −r2| is the static Coulomb scalar potential and ρ(1) is the charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' With the aid of Maxwell’s equation, E(r,t) = −∇∇∇V(r,t) and the continu- ity equation, ∇∇∇·J(r,t) = −∂tρ(r,t), the Hartree electric field in wavevector and frequency space is expressed as EH(q1,ω) = ∑ q2 KH(q1,q2,ω)·J(q2,ω), (C2) with the kernel given by Kab H (q1,q2,ω) = qa 1 vc(q1,q2) iω qb 2 = qa 1 4πδq1,q2 iω|q1||q2|qb 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (C3) Above, q1 and q2 represent momenta and the Fourier trans- form of the Coulomb potential was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Applying the q1,q2 → 0 optical limit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C3, the tensorial Hartree kernel takes the usual form Kab H (ω) = δab 4π iω , (C4) which is a diagonal and isotropic tensor owing to the longitu- dinal and radial nature of the Coulomb force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Coming now to the xc piece, its electric field up to linear order is written as Exc,1(q1,ω) = ∑ q2 Kxc,1(q1,q2,ω)·J(q2,ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (C5) Assuming that the nonlocal long-range behaviour of excitonic effects completely dominates over all other terms in the op- tical limit [68], the xc contribution can be modelled by a Coulomb-like attractive interaction with LRC xc coefficients αa LRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the wavevector and frequency domain, the corre- sponding tensorial xc kernel reads Kab xc,1(q1,q2,ω) = −qa 1 αa LRCδab iω|q1||q2|qb 2, (C6) which is a diagonal tensor owing to the longitudinal nature of Coulomb-like forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' At variance with the Hartree contribu- tion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C3, the tensorial coefficients αa LRC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C6 allows a space-resolved anisotropic response of the xc electric field along the crystal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' By taking the optical limit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C6, we arrive at the simplified expression used in our calculations, Kab xc,1(ω) = −αa LRCδab iω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (C7) The bootstrap method is a parameter-free approxima- tion that was originally proposed for self-consistenty cal- culating the space-averaged isotropic scalar α-coefficient in TDDFT [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' We have adopted this method to compute αa LRC by means of the expression αa LRC = ε−1,aa M (0) � αKS 1 (0) �−1,aa , (C8) which requires calculating the LRC xc coefficients indepen- dently for each of the three Cartesian directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' As in the original bootstrap kernel, the calculation of the coefficients is done iteratively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' firstly, the microscopic optical MB conduc- tivity is calculated by means of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 24 and 25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' secondly, the macroscopic optical dielectric tensor by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' and finally, the coefficients αa LRC by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The iter- ative loop starts with the initial guess αa LRC = 0 and finishes when self-consistency is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 14 Appendix D: THE OPTICAL MACROSCOPIC-MICROSCOPIC CONNECTION In this appendix we derive the relations that connect calcu- lable response tensors at the microscopic scale with their mea- surable macroscopic counterparts in the optical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' This is largely based on the work of Del Sole and Fiorino for the first order [52], and on the work of Luppi and co-workers for the second order [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' General definitions and useful relations The response of a material to an applied external electric field can be mainly described in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' On the one hand, the ability of a material to conduct an electric current is de- scribed by the electric conductivity, which relates the electric current-density vector to the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' On the other hand, the ability of a material to electrically polarize is described by the electric susceptibility or polarizability, which relates the electric polarization-density vector to the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' At the macroscopic scale (M), these relations are expressed in terms of the macroscopic total electric field EM tot(r,t), in such a way that the j-th order power series expansion of the macroscopic electric current- and polarization-density vectors are respectively defined as JM,j(1) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' � 1 0 σM, j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1)∏ j Etot,M( j +1)d j +1 (D1a) PM, j(1) = ε0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' � 1 0 χ j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1)∏ j Etot,M( j +1)d j +1 (D1b) where JM, j(r,t) and PM, j(r,t) are the j-th order macro- scopic electric current- and polarization-density vectors, re- spectively, and σM,j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1) and χ j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1) are the j-th order macroscopic conductivity and suscepti- bility tensors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The complete macroscopic current- and polarization-density vectors are given by JM(r,t) = ∑ j JM,j(r,t) and PM(r,t) = ∑j PM, j(r,t), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In turn, at the microscopic scale the relations are expressed in terms of the microscopic external electric field Eext(r,t), in such a way that the j-th order power series expansion of the microscopic electric current- and polarization-density vectors are respectively defined as Jj(1) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' � 1 0 σ j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1)∏ j Eext( j +1)d j +1, (D2a) P j(1) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' � 1 0 α j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1)∏ j Eext( j +1)d j +1 (D2b) where Jj(r,t) and P j(r,t) are the j-th order microscopic electric current- and polarization-density vectors, respec- tively, and σ j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1) and α j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1) are the j- th order microscopic conductivity and polarizability ten- sors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The complete microscopic current- and polarization-density vectors are given by J(r,t) = ∑j Jj(r,t) and P(r,t) = ∑j Pj(r,t), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the absence of magnetization, and free charge and cur- rent densities, the current- and polarization-density vectors are related by J(M),( j)(r,t) = ∂tP(M),(j)(r,t), both at the macro- scopic and microscopic levels, as well as at any order of the power series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Using the latter relation and com- paring Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D1a and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D1b, we can derive the connections between the macroscopic conductivity and susceptibility up to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the reciprocal space and frequency domain, the connection at first order in the optical limit is given by σM,1(ω) = −iωε0χ1(ω), (D3) and at second order by σM,2(ω1,ω2) = −i(ω1 +ω2)ε0χ2(ω1,ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (D4) In an analogous way, we can derive the connection between microscopic conductivity and polarizability up to second or- der, but this time comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D2a and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' At first order it is given by σ1(ω) = −iωα1(ω), (D5) and at second order by σ2(ω1,ω2) = −i(ω1 +ω2)α2(ω1,ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (D6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Macroscopic optical susceptibility Our main goal is to express macroscopic response tensors as a function of their respective microscopic counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' To this end, the simplest option is to switch to the KS electronic system, where the observables in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D2a and D2b are de- fined in terms of the microscopic total electric field Etot(r,t) as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 6 for the current, and then take a macroscopic spatial average of the microscopic quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the so-called long- wavelength limit, where the real-space variation of the total electric field over distances of the order of the lattice param- eter is neglected and therefore the total electric field is per se of macroscopic character, the macroscopic spatial average of microscopic quantities is straightforward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' it is sufficient to retain the G = 0 reciprocal lattice vector [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Furthermore, the averaging is even more direct in the optical limit, since microscopic quantities are calculated assuming ideally a non- variational character in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Therefore, under this point of view, one can state that the macroscopic optical conductiv- ity is equal to its microscopic KS counterpart at any order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' σM,j(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1) = σKS j (1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=', j +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Nevertheless, the previous approach does not account for many-body effects in the response, since those are assumed to be already included in the total electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In order to over- come this limitation, one can obtain an expression of the exter- nal electric field as a function of the total electric field at the microscopic level by using Maxwell’s equations and related constitutive relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Then, the resulting expression is used to define microscopic observables in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D2a and D2b in 15 terms of the total electric field, whose macroscopic spatial av- erages give access to the formulation of macroscopic response tensors including many-body effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' [35], in the reciprocal space and frequency domain, the longitudinal- longitudinal (LL) component of the linear macroscopic sus- ceptibility tensor is given by [52] χLL 1 (q,ω) = 4παLL 1 (q,ω)εLL M (q,ω), (D7) where αLL 1 (q,ω) ≡ α1LL GG′(q,ω)δG,0δG′,0 is the LL component of the macroscopic spatial averaged mi- croscopic MB polarizability tensor at first order, and εLL M (q,ω) = [1−4παLL 1 (q,ω)]−1 is the LL component of the macroscopic dielectric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In an analogous way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' the longitudinal-longitudinal-longitudinal (LLL) component of the quadratic macroscopic susceptibility tensor is expressed as χL12L1L2 2 (q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω2) = 4πεL12L12 M (q12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω12)αL12L1L2 2 (q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω2)εL1L1 M (q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω1)εL2L2 M (q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (D8) where L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' L2 and L12 stand for the longitudinal component along the directions q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' q2 and q12 ≡ q1 + q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' and αL12L1L2 2 (q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω2) ≡ α2 L12L1L2 G12G1G2(q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='ω2)δG12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0δG1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0δG2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content='0 is the LLL component of the spatially averaged micro- scopic MB polarizability tensor at second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' The adopted framework is valid for any q and describes lon- gitudinal responses to longitudinal perturbations [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In the optical limit (q → 0), one can always find three principal axes for any crystal symmetry in which the macroscopic dielectric tensor is diagonal [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In this reference frame a longitudinal perturbation induces a longitudinal response, hence any opti- cal property of the crystal can be deduced from a longitudinal calculation [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Therefore, in the principal frame the linear macroscopic optical susceptibility tensor is expressed as χaa 1 (ω) = 4παaa 1 (ω)εaa M (ω), (D9) and the quadratic macroscopic optical susceptibility tensor as χabc 2 (ω1,ω2) = 4πεaa M (ω12)αabc 2 (ω1,ω2)εbb M (ω1)εcc M(ω2), (D10) where a, b and c are principal axis components of the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Note that for any crystal with a symmetry greater or equal to the orthorhombic symmetry, a, b and c coincide with the Cartesian coordinates [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Macroscopic optical conductivity The derivation of the optical macroscopic-microscopic con- nection in the previous section has been given in terms of the macroscopic susceptibility and the microscopic polarizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Nevertheless, one can also express this connection in terms of the conductivity by means of the identities provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' In particular, inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D3 and D5 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D9 one obtains the linear macroscopic optical conductivity, σaa M,1(ω) = σaa 1 (ω)εaa M (ω), (D11) while inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D4 and D6 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' D10 yields the ex- pression for the quadratic macroscopic optical conductivity, σabc M,2(ω1,ω2) = εaa M (ω12)σabc 2 (ω1,ω2)εbb M (ω1)εcc M(ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' (D12) [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Belinicher, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9AyT4oBgHgl3EQft_n6/content/2301.00607v1.pdf'} +page_content=' Ivchenko, and B.' metadata={'source': 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demonstrate the recently proposed nucleon energy-energy correlator (nucleon EEC) fEEC(x, θ) +can unveil the gluon saturation in the small-x regime in eA collisions. The novelty of this probe is +that it is fully inclusive just like the deep-inelastic scattering (DIS), with no requirements of jets +or hadrons, but still provides an evident portal to the small-x dynamics through the shape of the +θ-distribution. We find that the saturation prediction is significantly different from the expectation +of the collinear factorization. +Introduction. +Small-x gluon saturation [1–6] has +been one of the central focuses in nuclear physics com- +munity in recent years and will be a major research area +in the future Electron Ion Collider (EIC) [7–9]. An effec- +tive field theory called color-glass-condensate (CGC) [4– +6] has been established to compute the hadronic and +nuclear structure functions in deep inelastic scattering +(DIS) at small values of Bjorken-xB [10, 11]. The CGC +predicts the gluon saturation with a characteristic scale +Qs, as a consequence of the small-x nonlinear dynamics +governed by the BK-JIMWLK equation [12–17]. The sat- +uration scale Qs represents the typical size of the gluon +transverse momentum inside the nucleus and grows as +the momentum fraction x → 0. For large nucleus and +small-x, typically Qs > ΛQCD. +Previous experiments from DIS in ep collisions at +HERA and hadron productions in pA collisions at RHIC +and LHC have shown some evidence of gluon saturation +at small-x. With the planned EIC in the horizon, this +physics will be explored in a systematic manner with un- +precedented precision [7–9]. Extensive studies have been +carried out for the EIC experiments, including the in- +clusive DIS structure functions at small-xB [18–20] and +the azimuthal correlations of di-jet/di-hadron/photon- +jet/lepton-jet in the inclusive or diffractive processes [21– +56]. These processes are considered as promising chan- +nels to look for the gluon saturation in eA collisions. +In this manuscript, we present a novel approach to +probe the gluon saturation in eA collisions in terms of +the nucleon energy-energy correlator (nucleon EEC) re- +cently proposed in Ref. [57], which is an extension of +the EEC [58, 59] to the nucleon case. The EEC is the +vacuum expectation of a set of final state correlators to +reformulate jet substructures [60–84], while the nucleon +EEC is the nucleon expectation of the initial-final state +correlator. The latter encodes the partonic angular dis- +tribution induced by the intrinsic transverse momentum +within the nucleon [57]. +Therefore we expect the fea- +tures of the gluon saturation, especially the saturation +scale Qs that measures the size of the intrinsic trans- +verse momentum, should be naturally imprinted in the +nucleon EEC. Our numeric results in Figs. 5 and 6 will +show that the saturation predictions have distinguished +behaviors as compared to those from the collinear fac- +torization. From this comparison, we can further deduce +the saturation scales in ep and eA collisions, respectively. +The quark contribution to the nucleon EEC in the mo- +mentum space is defined as +fq,EEC(x, θ) = +� +dy− +4πEA +e−ixP y−γ+⟨A|¯χ(y−) E(θ)χ(0)|A⟩ , +(1) +where x is the momentum fraction that initiates a scat- +tering process, meanwhile we measure the energy de- +posit in a detector at a given angle θ from the ini- +tial state radiation and the remnants through the en- +ergy operator E(θ) = limr→∞ +� ∞ +0 +dtT0⃗n(t,⃗nr)r2 [90–93], +E(θ)|X⟩ = � +i∈X Eiδ(θ2 +i −θ2)|X⟩ . The measured energy +deposit is normalized to the energy EA carried by the +nucleus A. Here, χ is the gauge invariant collinear quark +field [85–89]. The gluon EEC can be defined similarly. +When θEA ∼ ΛQCD, the fEEC probes the intrinsic trans- +verse dynamics of the nucleus A through the operator +E(θ). +In the collinear factorization, it has been shown that +when θEA ≫ ΛQCD, the fq,EEC(x, θ) can be further fac- +torized as [57] +fi,EEC(x, θ) = +� dξ +ξ Iij +�x +ξ , θ +� � +ξfj/A (ξ) +� +, +(2) +where fj/A(ξ) is the collinear PDF, and Iij is the match- +ing coefficient found to be solely determined by the vac- +uum collinear splitting functions [57]. +As the values of x decreases, the fq,EEC receives dra- +matically enhanced contributions from the low x gluon. +In this regime, the non-linear small-x dynamics becomes +important. +Consequently, if compared to the collinear +factorization in which the distribution is determined by +vacuum collinear splitting, the shape of the θ-distribution +will be modified, due to a sizable initial transverse mo- +mentum qt of order the saturation scale Qs, see, the il- +lustrations in Fig. 1. Therefore, the nucleon EEC can +arXiv:2301.01788v1 [hep-ph] 4 Jan 2023 + +2 +ℰ(θ) +q+ = ξP, ξP > Q +θ +A +Q +} +qt ∼ ΛQCD +θ +A +ξ ∼ Q +P +qt ∼ Qs +ℰ(θ) +Q +} +FIG. 1. The fEEC(x, θ) in the collinear factorization (left) +and the CGC framework (right). Here Q represents the center +of mass energy of the partonic cross section. +be used to probe the gluon saturation phenomenon and +the small-x dynamics, as we will show in the rest of this +manuscript. +The measurement and the factorization theo- +rem. +We follow [57] to consider the unpolarized DIS +process l+A → l′+X in the Breit frame. We assume the +nucleus is moving along the +z-direction. We measure +the Bjorken xB = −q2 +2P ·q, the photon virtuality Q2 = −q2 +and the energy � +i Ei that deposits in a calorimeter at an +angle θ with respect to the beam, as shown in Fig. 2. Here +q = l′ −l is the momentum carried by the virtual photon. +We then measure the weighted cross section Σ(Q2, xB, θ) +defined as +Σ(Q2, xB, θ) = +� +i +� +dσ(xB, Q2, pi) Ei +EA +δ(θ2 − θ2 +i ) ,(3) +where EA is the energy carried by the incoming nucleus. +We note that the energy weight suppresses the soft con- +tributions, which is an important feature of the proposed +measurement and its resulting nucleon EEC. +θ +l +l′ +A +FIG. 2. The xB and Q2 measurement in DIS with a forward +detector that records the energy flow � +i Ei at the angle θ. +In order to probe the small-x dynamics, we are partic- +ularly interested in the scenario in which xB ≪ 0.1, and +we place the detector in the far-forward region such that +Qθ ≪ Q while Qθ ∼ Qs ≫ ΛQCD. At this point, we em- +phasize that the measurement involves neither additional +hadron tagging nor jet clustering, and in contrast to the +TMD which restricts the events in the small qt region, +this approach is inclusive and does not veto events. It +weights the full cross section by the energy recorded at a +certain angle θ, therefore the probe is as inclusive as the +DIS but with additional control via θ. +When θQ ≫ ΛQCD, the weighted cross section can be +calculated perturbatively in the collinear factorization. +More interestingly, when Qθ ≪ Q, it has been shown +that the Σ(Q2, xB, θ) fulfils the factorized form [57] +Σ(Q2, xB, θ) = +� dx +x ˆσi,DIS +�xB +x , Q +� +fi,EEC(x, θ) , (4) +where ˆσi,DIS is the fully inclusive partonic DIS cross +section. +fi,EEC is the nucleon EEC in Eq. (1). +The +θ-dependence enters entirely through the nucleon EEC +fEEC(x, θ), +and therefore the θ distribution of the +Σ(Q2, xB, θ) probes the nucleon EEC when θ is small. +We note that fEEC satisfies the same collinear evolution +as the collinear PDFs [57] +dfi,EEC(x, θ) +d ln µ += Pij ⊗ fj,EEC , +(5) +as required by dΣ/d ln µ = 0, and since dˆσi,DIS/d ln µ = +−Pji ⊗ ˆσj,DIS. Here the convolution in the momentum +fraction is defined as f ⊗ g(x) ≡ +� 1 +x +dz +z f +� x +z +� +g(z). It is +clear from the evolution that there is no perturbative +Sudakov suppression in fEEC, due to the absence of the +soft contribution in the collinear factorization eliminated +by the energy weight [57]. +ξ(1 − z)P +xi = ξz +ξP +θ +l +l′ +A +FIG. 3. +The collinear splitting that initiates the DIS process +and a daughter parton that hits the detector at θ ≪ 1. The +momentum fractions are shown. We abbreviates P + with P +in this work for simplicity notation. +The factorization theorem in Eq. (4) and Eq. (2) can +be easily understood by considering the leading contribu- +tion shown in Fig. 3, where a parton out of the nucleus A +with momentum ξP splits into a parton with momentum +fraction (1 − z)ξ that hits the detector at θ, and an in- +ternal line with fraction zξ and virtuality t = − +⃗k2 +t +1−z that +initiates the partonic inclusive DIS process. Here z is the +momentum fraction with respect to the incoming parton + +3 +and kt = 1 +2ξ(1 − z)P θ is the transverse momentum of +the final state parton. In the vacuum, the splitting is +described by the leading order vacuum collinear splitting +kernel 1 +t P (0) +ij . Since θQ ≪ Q, the chance for the radia- +tions from the hard interaction to reach the calorimeter +vanishes as θ → 0. It is then found that in the small θ +limit, +Σ(Q2, xB, θ) = +� dxi +xi +ˆσi,DIS +� +Q2, xB +xi +� +× +� +dξdz 1 +θ2 δ(xi − ξz)(1 − z)ξP (0) +ij (z) fj/A(ξ) , +(6) +which, aftet performing the z integration, gives +Σ(Q2, xB, θ) = +� dxi +xi +ˆσi,DIS +� +Q2, xB +xi +� +× 1 +θ2 +� dξ +ξ +� +1 − xi +ξ +� +P (0) +ij +�xi +ξ +� � +ξfj/A(ξ) +� +. +(7) +This produces the factorized form in Eq. (4) and Eq. (2) +by identifying the leading order matching coefficient +I(0) +ij (ξ, θ) = +1 +θ2 (1 − ξ)P (0) +ij (ξ). +If xB ≪ 0.1, the gluon density is overwhelmingly large +and the leading contribution to the Σ(Q2, xB, θ) is com- +ing from +Σ(Q2, xB, θ) = +� +q +4πα2e2 +q +Q4 +fq,EEC(xB, θ) , +(8) +with +fq,EEC(x, θ) += αsTR +2πθ2 +� 1 +x +dξ +ξ (1 − ξ)(ξ2 + (1 − ξ)2) +�x +ξ fg +�x +ξ +�� +.(9) +The collinear factorization predicts a +1 +θ2 -scaling behav- +ior at O(αs). For very small θ, the scaling rule could +receive corrections from both the evolution of the fEEC +in Eq. (5) and non-perturbative effects. But for generic +small θ, these effects are mild and therefore θ2Σ will be +insensitive to the values of θ, up to O(θQ) power correc- +tions. Furthermore, since the energy weight kills the soft +contribution, to all orders there will be no perturbative +Sudakov suppression in the small θ region in the collinear +factorization [57], as is clear from Eq. (5). Such a feature +will be modified by the small-x dynamics as we will show. +The nucleon EEC in the small-x regime. In the +small-x region, the gluon density grows as 1 +x and becomes +overwhelmingly important and has to be resummed to all +orders. To realize such resummation in fEEC, we invoke +the CGC effective theory framework and follow the strat- +egy in [94–96] to write the nucleon EEC in terms of the +CGC dipole distribution 1. By evaluating the diagrams +1 The complete calculation using the full dipole amplitude ψγ∗→q¯q +T,L +FIG. 4. The leading contribution to fq,EEC(x, θ) in the small- +x region, where the double line represents the gauge link and +the gluon requires momentum g+ = xgP and gt ∼ Qs ∼ θQ. +in Fig. 4, we find in the leading logarithmic (LL) approx- +imation +fq,EEC(xB, θ) = NCS⊥ +8π4 +� +d2⃗gt +× +� 1 +ξcut +dξ +ξ Aqg (ξ, θ,⃗gt) Fg,xB(⃗gt) , +(10) +where S⊥ is the averaged transverse area of the tar- +get nucleus and gt ∼ Qs ∼ θQ is the transverse mo- +mentum transfer. +Fg,xF = +� d2⃗r +4π2 e−i⃗gt·⃗rtS(2) +xF (⃗rt) is the +CGC dipole distribution evaluated at the scale xF , where +S(2) +xF (⃗rt) = +1 +NC ⟨Tr[W(⃗rt)W †(⃗0)]⟩xF . xF +Q +xB is the rapidity +scale/boundary that separates the fast moving modes be- +ing integrated out and the active slow moving partons in +the CGC effective frame work. In this work, we default +to the natural choice xF = xB. +1−ξ +ξ Q is the momentum +“+”-component that enters the detector. ξcut is deter- +mined by requiring the momentum of the active quark +does not exceed the rapidity boundary. Here the coeffi- +cient Aqg is given by +Aqg(ξ, θ,⃗gt) = 1 +θ2 (1 − ξ)⃗k2 +t (⃗kt − ⃗gt)2 +× +����� +⃗kt +ξ⃗k2 +t + (1 − ξ)(⃗kt − ⃗gt)2 − +⃗kt − ⃗gt +(⃗kt − ⃗gt)2 +����� +2 +, (11) +with kt defined as kt = 1−ξ +ξ +Q +2 θ, should be of order Qs. +It is easy to show that if gt ∼ Qs ≪ Qθ, Eq. (10) +reduces to the +1 +θ2 -scaling behavior of the collinear fac- +torization in Eq. (9). On the other hand, if θQ ≪ Qs, +for γ∗ → q¯q is presented in the Supplemental Material. Both +approaches agree in the small θ limit. + +4 +Eq. (10) scales as θ0. We thus expect that in CGC, the +θ2Σ will be independent of the θ for θQ ≫ Qs, however, +contrary to the collinear factorization, suppressed when +θQ ≪ Qs. Meanwhile the θ region between these two +limits provide the opportunity to estimate the saturation +scale Qs. +Numerics. Now we study the numerical impacts of +the small-x dynamics on the shape of the θ2Σ(Q2, xB, θ) +distribution from Eq. (10), compared with the collinear +prediction. +We are particularly interested in the re- +gion θ ≪ 1 where the θ distribution probes direcly the +fEEC(x, θ), see Eq. (4). +For the small-x dipole distri- +bution S(2) +xF (⃗rt), we use both the MV model with rcBK +running [12, 13, 18, 97–104] and the GBW model [105]. +As for the MV model with rcBK running, we adopt +the MV-like model [106] as the initial condition, whose +form is S(2) +x0 (⃗rt) = exp +� +− (r2 +t Q2 +s0)γ +4 +ln +� +1 +Λrt + e +�� +, where +we choose x0 = 0.01, γ = 1.119, Λ = 0.241 GeV, Q2 +s0 = +A1/30.168 GeV2 with A the atomic number. +We use +the solution to the LL BK evolution with αs run- +ning [98, 101, 106] of the dipole distribution to evolve the +dipole distribution from x0 to xF . In our calculation, we +use the result fitted from the HERA data for the trans- +verse area of the nucleus S⊥ [19]. The GBW model is +implemented using S(2) +xF (⃗rt) = exp +� +− 1 +4r2 +t Q2 +s(xF ) +� +, where +Q2 +s(xF ) = AN(x0/xF )λ GeV2 and we use x0 = 2.24 × +10−4, λ = 0.27 and AN = 1 for the proton while AN = 5 +for the Au [107]. +FIG. 5. +Normalized θ2Σ(Q2, xB, θ) distribution for proton and Au, with xB = 0.003, for both Q2 = 25 GeV2 , √s = 105 GeV +(left panel) and Q2 = 100 GeV2 , √s = 318 GeV (right panel). Green dots are from Pythia82 simulation, in which we demand +0.00295 < xB < 0.00305, and 25GeV2 ≤ Q2 < 35GeV2 for the left panel while 100GeV2 ≤ Q2 < 110GeV2 for the right. +In +Fig. +5, +we +show +the +CGC +predictions +for +θ2Σ(Q2, xB, θ) as a function of θ. Since we are only in- +terested in the shape, we normalized the distribution by +� θmax +θmin dθθ2Σ. We fixed xB = 3 × 10−3 and choose Q2 = +25 GeV2, √s = 105 GeV (left panel) and Q2 = 100 GeV2, +√s = 318 GeV (right panel). +We present predictions +from CGC for both proton (in purple lines) and Au (in +orange), by the MV model with rcBK running and GBW +model. We see both models predict similar shapes in the +θ spectrum, in which the small-θ region is suppressed. +They are impressively different from the collinear expec- +tations (in red lines and green dots). In the figure, the +collinear predictions (red lines) are made out of the com- +plete fixed order αs calculation, without the Qθ ≪ Q +approximation, using CT18A [108] and EPPS21 [109] PDF +sets for proton and Au, respectively. +To validate our +collinear calculation and to estimate the size of the evo- +lution effect in Eq. (5), we also run a Pythia82 simu- +lation [110] for the proton case, where the LL resum- +mation is performed. We see that for large θ, the fixed +order calculations agree well with the the Pythia simu- +lation, while for small θ values, the resummation effects +could be sizable but does not suppress the small-θ region +due to the absence of the perturbative Sudakov factor in +fEEC in the collinear factorization. The collinear predic- +tion for the Au follows closely the proton’s. The notable +difference demonstrates that the fEEC(x, θ) could serve +as a clean probe of the small-x phenomenon. For com- +parison, we also show the predictions from the full CGC +calculation derived in the Supplemental Material using +the GBW model in purple circles. +In Fig. 5, the proton spectrum turns into a plateau +for large values of θ, which is expected from Eq. (10) +when Qθ ≫ Qs. We can define a turning point around +which the slope of the distribution starts to switch its +monotonicity. The turning point allows us to estimate +the size of the saturation scale Qs. For instance, from + +5 +the left panel of Fig. 5, the turning point for the proton is +roughly around θ ∼ 0.15−0.2 and thus Qs ∼ θQ ∼ 0.75− +1.0 GeV which is consistent with the values of ΛQCD. And +we estimate the saturation scale for the Au will be around +θ ∼ 0.4 − 0.5 and thus Qs ∼ θQ ∼ 2 − 2.5 GeV. The +right panel of Fig. 5 is similar to the left, but with Q2 = +100 GeV2. Since Q2 is larger, the distribution enters the +plateau earlier as expected. Now the turning point for +the Au is around θ ∼ 0.2−0.3 which again indicates that +Qs ∼ 2 − 3 GeV, consistent with the Q2 = 25 GeV2 case. +We can further introduce the nuclear modification fac- +tor RpA = A−1ΣA(Q2,xB,θ) +Σp(Q2,xB,θ) +, which helps to reduce the sys- +tematics. In the collinear factorization, for θQ ≫ ΛQCD, +the θ distribution is determined by the matching coeffi- +cient Iij as predicted by Eq. (8), which is independent of +the incoming nucleus species. Thus taking the ratio RAp +reduces the impacts from perturbative higher order cor- +rections as well as possible non-perturbative hadroniza- +tion effects, and the collinear factorization predicts the +RAp insensitive to the θ values, as showed explicitly as +red lines in Fig. 6. +FIG. 6. RpA as a function of θ, with xB = 3 × 10−3 using the +MV model with rcBK running and collinear factorization. +. +Once again, the small-x formalism changes the pattern +as we observed in Fig. 6, where the modification factor +RpA is suppressed in the small θ region, while converges +toward around unity as θ becomes large and Qθ ≫ Qs. +Conclusions. In this manuscript, we have proposed +the nucleon energy-energy correlator (nucleon EEC) as a +new probe of the gluon saturation phenomenon in DIS at +the future electron-ion colliders. In particular, we have +shown that the θ-shape of the nucleon EEC fEEC(x, θ) +behaves differently in the collinear factorization theorem +and the CGC formalism. The drastic difference is due to +the intrinsic transverse momentum of order Qs induced +by the non-linear small-x dynamics. We thus expect the +fEEC to complement the other standard small-x processes +and offer a great opportunity to pin down the onset of +the gluon saturation phenomenon in eA-collisions. +The advantage of the nucleon EEC probe, as com- +pared to other standard small-x processes, is that it is +fully inclusive, involving no fragmentation functions and +jet clustering. Therefore the observable is expected to +be clean both theoretically and experimentally. Exten- +sions to other observables that are induced by the in- +trinsic transverse dynamics of the nucleon/nucleus shall +follow. For polarized hadron beam, we can study the spin +asymmetry by adding the azimuthal dependence to the +energy operator E and we expect different asymmetries +to be predicted in the collinear and CGC frameworks. +We hope that the results presented in this manuscript +will motivate carrying out the proposed measurement at +the current and future electron-ion facilitieis, meanwhile +stimulate further applications of the nucleon EEC in nu- +clear structure studies. +Acknowledgement. We are grateful to Farid Salazar, +Hongxi Xing, Jian Zhou for useful discussions. +This +work is supported by the Natural Science Foundation of +China under contract No. 12175016 (X. L.), No. 11975200 +(H. X. Z.), and the Office of Science of the U.S. De- +partment of Energy under Contract No. +DE-AC02- +05CH11231 (F.Y.). +∗ xiliu@bnu.edu.cn +† fyuan@lbl.gov +‡ zhuhx@zju.edu.cn +[1] L. V. Gribov, E. M. Levin, and M. G. Ryskin, Phys. +Rept. 100, 1 (1983). +[2] A. H. Mueller and J.-w. Qiu, Nucl. Phys. 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Commun. 191, 159 +(2015), 1410.3012. + +Supplemental Materials for “Nucleon Energy Correlators for the Color Glass +Condensate” +Hao-Yu Liu,1 Xiaohui Liu,1, ∗ Ji-Chen Pan,2, 3 Feng Yuan,4, † and Hua Xing Zhu5, ‡ +1Center of Advanced Quantum Studies, Department of Physics, +Beijing Normal University, Beijing, 100875, China +2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China +3School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China +4Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA +5Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, 310027, China +(Dated: January 3, 2023) +Here we present the Σ(Q2, xB, θ) +Σ(Q2, xB, θ) = +� +i +� +dσ(xB, Q2, pi) Ei +EA +δ(θ2 − θ2 +i ) , +(1) +within the CGC formalism without the small θ approximation. The calculation is achieved by evaluating amplitude +for a virtual photon γ∗ splitting into the q¯q pair, as shown in Fig. 1. Each of the quarks will enter the detector to +contribute to the energy deposit Ei, which we assume it is always the ¯q. The contribution from the q is obtained by +multiplying a symmetric factor at this order. +FIG. 1. Dipole amplitude for γ∗ → q¯q. The orange block represents the CGC Wilson line. +We work in the Breit frame, in which q = (0, 0, 0, −Q) for the virtual photon, and P = +Q +2xB (1, 0, 0, 1) for the nucleus. +The calculation is straightforward which gives +Σ(Q2, xB, θ) = +� +λ=l,t +fλΣγ∗ +λ (Q2, xB, θ) , +(2) +where we sum over the virtual photon polarization with the flux ft = 1 − y + y2 +2 for the transverse and fl = 1 − y +longitudinal polarization, respectively, with y = +Q2 +xBs the inelasticity. Here for the transverse photon contribution, we +find +Σγ∗ +t (Q2, xB, θ) = +� +q +2Ncα2e2 +q +π2xBQ2 S⊥ +� +dzd2⃗kt +d2⃗lt +(2π)2 Fg,xB(⃗lt) +� +z2 + (1 − z)2� +����� +⃗kt +⃗k2 +t + ∆2 − +⃗kt −⃗lt +(⃗kt −⃗lt)2 + ∆2 +����� +2 +× +�⃗k2 +t + (1 − z)2Q2 +(1 − z)Q +xB +Q +� +1 +2θδ +� +θ − tan−1 +2kt(1 − z)Q +k2 +t − (1 − z)2Q2 +� +θ +�⃗k2 +t + (1 − z)2Q2 +(1 − z)Q +< xB +Q +xB +� +,(3) +where k is the momentum for ¯q and 1 − z = k− +Q the momentum fraction with respect to the photon. We defined +∆2 = z(1−z)Q2. In the last line, the first term is the energy weight, derived using the on-shell condition k+k−−⃗k2 +t = 0, +normalized to the incoming nucleus energy Q/xB in the Breit frame. The second term defines the θ angle with respect +∗ xiliu@bnu.edu.cn +† fyuan@lbl.gov +‡ zhuhx@zju.edu.cn + +2 +to the z-axis, with θ < π/2 for kz > 0 while θ > π/2 for kz < 0. The last θ function ensures that 2k0 can not exceed +the incoming parton momentum xBP = xB +Q +xB [1]. As for the longitudinal polarization, we have +Σγ∗ +l (Q2, xB, θ) = +� +q +2Ncα2e2 +q +π2xBQ2 S⊥ +� +dzd2⃗kt +d2⃗lt +(2π)2 Fg,xB(⃗lt)4z2(1 − z)2Q2 +����� +1 +⃗k2 +t + ∆2 − +1 +(⃗kt −⃗lt)2 + ∆2 +����� +2 +× +�⃗k2 +t + (1 − z)2Q2 +(1 − z)Q +xB +Q +� +1 +2θδ +� +θ − tan−1 +2kt(1 − z)Q +k2 +t − (1 − z)2Q2 +� +θ +�⃗k2 +t + (1 − z)2Q2 +(1 − z)Q +< xB +Q +xB +� +. +(4) +Now we consider the small-θ limit. As θ → 0, the contribution is dominated by 1 − z ∼ k2 +t +Q2 . We define the variable +ξ through (1 − z)Q = +ξ +1−ξ +k2 +t +Q and only keep the leading contribution in z → 1 limit to find +Σ(Q2, xB, θ) = +� +q +4πα2e2 +q +Q4 +NcS⊥ +8π4 +1 +θ2 +� dξ +ξ d2⃗lt(1 − ξ)k2 +t (kt − lt)2 +����� +⃗kt +ξ⃗k2 +t + (1 − ξ)(⃗kt −⃗lt)2 − +⃗kt −⃗lt +(⃗kt −⃗lt)2 +����� +2 +×θ +�1 − ξ +ξ +< 1 +� +Fg,xB(⃗lt) , +(5) +where kt = 1−ξ +ξ +Q +2 θ. We thus identify fEEC(xB, θ) at this order in the small-x formalism as +fq,EEC(xB, θ) = NcS⊥ +8π4 +1 +θ2 +� 1 +ξcut +dξ +ξ d2⃗lt(1 − ξ)k2 +t (kt − lt)2 +����� +⃗kt +ξ⃗k2 +t + (1 − ξ)(⃗kt −⃗lt)2 − +⃗kt −⃗lt +(⃗kt −⃗lt)2 +����� +2 +Fg,xB(⃗lt) . +(6) +The ξcut is determined from the last θ-function in Eq. (5). +[1] G. Beuf, Phys. Rev. D 96, 074033 (2017), 1708.06557. + diff --git a/TNAzT4oBgHgl3EQf0v5s/content/tmp_files/load_file.txt b/TNAzT4oBgHgl3EQf0v5s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ceb212280139518d0676fc2e12c5abc4be291b51 --- /dev/null +++ b/TNAzT4oBgHgl3EQf0v5s/content/tmp_files/load_file.txt @@ -0,0 +1,1084 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf,len=1083 +page_content='Nucleon Energy Correlators for the Color Glass Condensate Hao-Yu Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='1 Xiaohui Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ∗ Ji-Chen Pan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='2 Feng Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' † and Hua Xing Zhu4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ‡ 1Center of Advanced Quantum Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 100875,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China 2Institute of High Energy Physics and School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China 3Nuclear Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' USA 4Zhejiang Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 310027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China We demonstrate the recently proposed nucleon energy-energy correlator (nucleon EEC) fEEC(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' θ) can unveil the gluon saturation in the small-x regime in eA collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The novelty of this probe is that it is fully inclusive just like the deep-inelastic scattering (DIS), with no requirements of jets or hadrons, but still provides an evident portal to the small-x dynamics through the shape of the θ-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We find that the saturation prediction is significantly different from the expectation of the collinear factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Small-x gluon saturation [1–6] has been one of the central focuses in nuclear physics com- munity in recent years and will be a major research area in the future Electron Ion Collider (EIC) [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' An effec- tive field theory called color-glass-condensate (CGC) [4– 6] has been established to compute the hadronic and nuclear structure functions in deep inelastic scattering (DIS) at small values of Bjorken-xB [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The CGC predicts the gluon saturation with a characteristic scale Qs, as a consequence of the small-x nonlinear dynamics governed by the BK-JIMWLK equation [12–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The sat- uration scale Qs represents the typical size of the gluon transverse momentum inside the nucleus and grows as the momentum fraction x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' For large nucleus and small-x, typically Qs > ΛQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Previous experiments from DIS in ep collisions at HERA and hadron productions in pA collisions at RHIC and LHC have shown some evidence of gluon saturation at small-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' With the planned EIC in the horizon, this physics will be explored in a systematic manner with un- precedented precision [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Extensive studies have been carried out for the EIC experiments, including the in- clusive DIS structure functions at small-xB [18–20] and the azimuthal correlations of di-jet/di-hadron/photon- jet/lepton-jet in the inclusive or diffractive processes [21– 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' These processes are considered as promising chan- nels to look for the gluon saturation in eA collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In this manuscript, we present a novel approach to probe the gluon saturation in eA collisions in terms of the nucleon energy-energy correlator (nucleon EEC) re- cently proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [57], which is an extension of the EEC [58, 59] to the nucleon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The EEC is the vacuum expectation of a set of final state correlators to reformulate jet substructures [60–84], while the nucleon EEC is the nucleon expectation of the initial-final state correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The latter encodes the partonic angular dis- tribution induced by the intrinsic transverse momentum within the nucleon [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Therefore we expect the fea- tures of the gluon saturation, especially the saturation scale Qs that measures the size of the intrinsic trans- verse momentum, should be naturally imprinted in the nucleon EEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Our numeric results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 5 and 6 will show that the saturation predictions have distinguished behaviors as compared to those from the collinear fac- torization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' From this comparison, we can further deduce the saturation scales in ep and eA collisions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The quark contribution to the nucleon EEC in the mo- mentum space is defined as fq,EEC(x, θ) = � dy− 4πEA e−ixP y−γ+⟨A|¯χ(y−) E(θ)χ(0)|A⟩ , (1) where x is the momentum fraction that initiates a scat- tering process, meanwhile we measure the energy de- posit in a detector at a given angle θ from the ini- tial state radiation and the remnants through the en- ergy operator E(θ) = limr→∞ � ∞ 0 dtT0⃗n(t,⃗nr)r2 [90–93], E(θ)|X⟩ = � i∈X Eiδ(θ2 i −θ2)|X⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The measured energy deposit is normalized to the energy EA carried by the nucleus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here, χ is the gauge invariant collinear quark field [85–89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The gluon EEC can be defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' When θEA ∼ ΛQCD, the fEEC probes the intrinsic trans- verse dynamics of the nucleus A through the operator E(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In the collinear factorization, it has been shown that when θEA ≫ ΛQCD, the fq,EEC(x, θ) can be further fac- torized as [57] fi,EEC(x, θ) = � dξ ξ Iij �x ξ , θ � � ξfj/A (ξ) � , (2) where fj/A(ξ) is the collinear PDF, and Iij is the match- ing coefficient found to be solely determined by the vac- uum collinear splitting functions [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' As the values of x decreases, the fq,EEC receives dra- matically enhanced contributions from the low x gluon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In this regime, the non-linear small-x dynamics becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Consequently, if compared to the collinear factorization in which the distribution is determined by vacuum collinear splitting, the shape of the θ-distribution will be modified, due to a sizable initial transverse mo- mentum qt of order the saturation scale Qs, see, the il- lustrations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Therefore, the nucleon EEC can arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='01788v1 [hep-ph] 4 Jan 2023 2 ℰ(θ) q+ = ξP, ξP > Q θ A Q } qt ∼ ΛQCD θ A ξ ∼ Q P qt ∼ Qs ℰ(θ) Q } FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The fEEC(x, θ) in the collinear factorization (left) and the CGC framework (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here Q represents the center of mass energy of the partonic cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' be used to probe the gluon saturation phenomenon and the small-x dynamics, as we will show in the rest of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The measurement and the factorization theo- rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We follow [57] to consider the unpolarized DIS process l+A → l′+X in the Breit frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We assume the nucleus is moving along the +z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We measure the Bjorken xB = −q2 2P ·q, the photon virtuality Q2 = −q2 and the energy � i Ei that deposits in a calorimeter at an angle θ with respect to the beam, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here q = l′ −l is the momentum carried by the virtual photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We then measure the weighted cross section Σ(Q2, xB, θ) defined as Σ(Q2, xB, θ) = � i � dσ(xB, Q2, pi) Ei EA δ(θ2 − θ2 i ) ,(3) where EA is the energy carried by the incoming nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We note that the energy weight suppresses the soft con- tributions, which is an important feature of the proposed measurement and its resulting nucleon EEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' θ l l′ A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The xB and Q2 measurement in DIS with a forward detector that records the energy flow � i Ei at the angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In order to probe the small-x dynamics, we are partic- ularly interested in the scenario in which xB ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='1, and we place the detector in the far-forward region such that Qθ ≪ Q while Qθ ∼ Qs ≫ ΛQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' At this point, we em- phasize that the measurement involves neither additional hadron tagging nor jet clustering, and in contrast to the TMD which restricts the events in the small qt region, this approach is inclusive and does not veto events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' It weights the full cross section by the energy recorded at a certain angle θ, therefore the probe is as inclusive as the DIS but with additional control via θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' When θQ ≫ ΛQCD, the weighted cross section can be calculated perturbatively in the collinear factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' More interestingly, when Qθ ≪ Q, it has been shown that the Σ(Q2, xB, θ) fulfils the factorized form [57] Σ(Q2, xB, θ) = � dx x ˆσi,DIS �xB x , Q � fi,EEC(x, θ) , (4) where ˆσi,DIS is the fully inclusive partonic DIS cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' fi,EEC is the nucleon EEC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The θ-dependence enters entirely through the nucleon EEC fEEC(x, θ), and therefore the θ distribution of the Σ(Q2, xB, θ) probes the nucleon EEC when θ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We note that fEEC satisfies the same collinear evolution as the collinear PDFs [57] dfi,EEC(x, θ) d ln µ = Pij ⊗ fj,EEC , (5) as required by dΣ/d ln µ = 0, and since dˆσi,DIS/d ln µ = −Pji ⊗ ˆσj,DIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here the convolution in the momentum fraction is defined as f ⊗ g(x) ≡ � 1 x dz z f � x z � g(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' It is clear from the evolution that there is no perturbative Sudakov suppression in fEEC, due to the absence of the soft contribution in the collinear factorization eliminated by the energy weight [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ξ(1 − z)P xi = ξz ξP θ l l′ A FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The collinear splitting that initiates the DIS process and a daughter parton that hits the detector at θ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The momentum fractions are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We abbreviates P + with P in this work for simplicity notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The factorization theorem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (2) can be easily understood by considering the leading contribu- tion shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 3, where a parton out of the nucleus A with momentum ξP splits into a parton with momentum fraction (1 − z)ξ that hits the detector at θ, and an in- ternal line with fraction zξ and virtuality t = − ⃗k2 t 1−z that initiates the partonic inclusive DIS process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here z is the momentum fraction with respect to the incoming parton 3 and kt = 1 2ξ(1 − z)P θ is the transverse momentum of the final state parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In the vacuum, the splitting is described by the leading order vacuum collinear splitting kernel 1 t P (0) ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Since θQ ≪ Q, the chance for the radia- tions from the hard interaction to reach the calorimeter vanishes as θ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' It is then found that in the small θ limit, Σ(Q2, xB, θ) = � dxi xi ˆσi,DIS � Q2, xB xi � × � dξdz 1 θ2 δ(xi − ξz)(1 − z)ξP (0) ij (z) fj/A(ξ) , (6) which, aftet performing the z integration, gives Σ(Q2, xB, θ) = � dxi xi ˆσi,DIS � Q2, xB xi � × 1 θ2 � dξ ξ � 1 − xi ξ � P (0) ij �xi ξ � � ξfj/A(ξ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (7) This produces the factorized form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (4) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (2) by identifying the leading order matching coefficient I(0) ij (ξ, θ) = 1 θ2 (1 − ξ)P (0) ij (ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' If xB ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='1, the gluon density is overwhelmingly large and the leading contribution to the Σ(Q2, xB, θ) is com- ing from Σ(Q2, xB, θ) = � q 4πα2e2 q Q4 fq,EEC(xB, θ) , (8) with fq,EEC(x, θ) = αsTR 2πθ2 � 1 x dξ ξ (1 − ξ)(ξ2 + (1 − ξ)2) �x ξ fg �x ξ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (9) The collinear factorization predicts a 1 θ2 -scaling behav- ior at O(αs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' For very small θ, the scaling rule could receive corrections from both the evolution of the fEEC in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (5) and non-perturbative effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' But for generic small θ, these effects are mild and therefore θ2Σ will be insensitive to the values of θ, up to O(θQ) power correc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Furthermore, since the energy weight kills the soft contribution, to all orders there will be no perturbative Sudakov suppression in the small θ region in the collinear factorization [57], as is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Such a feature will be modified by the small-x dynamics as we will show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The nucleon EEC in the small-x regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In the small-x region, the gluon density grows as 1 x and becomes overwhelmingly important and has to be resummed to all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' To realize such resummation in fEEC, we invoke the CGC effective theory framework and follow the strat- egy in [94–96] to write the nucleon EEC in terms of the CGC dipole distribution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' By evaluating the diagrams 1 The complete calculation using the full dipole amplitude ψγ∗→q¯q T,L FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The leading contribution to fq,EEC(x, θ) in the small- x region, where the double line represents the gauge link and the gluon requires momentum g+ = xgP and gt ∼ Qs ∼ θQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 4, we find in the leading logarithmic (LL) approx- imation fq,EEC(xB, θ) = NCS⊥ 8π4 � d2⃗gt × � 1 ξcut dξ ξ Aqg (ξ, θ,⃗gt) Fg,xB(⃗gt) , (10) where S⊥ is the averaged transverse area of the tar- get nucleus and gt ∼ Qs ∼ θQ is the transverse mo- mentum transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Fg,xF = � d2⃗r 4π2 e−i⃗gt·⃗rtS(2) xF (⃗rt) is the CGC dipole distribution evaluated at the scale xF , where S(2) xF (⃗rt) = 1 NC ⟨Tr[W(⃗rt)W †(⃗0)]⟩xF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' xF Q xB is the rapidity scale/boundary that separates the fast moving modes be- ing integrated out and the active slow moving partons in the CGC effective frame work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In this work, we default to the natural choice xF = xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 1−ξ ξ Q is the momentum “+”-component that enters the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ξcut is deter- mined by requiring the momentum of the active quark does not exceed the rapidity boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here the coeffi- cient Aqg is given by Aqg(ξ, θ,⃗gt) = 1 θ2 (1 − ξ)⃗k2 t (⃗kt − ⃗gt)2 × ����� ⃗kt ξ⃗k2 t + (1 − ξ)(⃗kt − ⃗gt)2 − ⃗kt − ⃗gt (⃗kt − ⃗gt)2 ����� 2 , (11) with kt defined as kt = 1−ξ ξ Q 2 θ, should be of order Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' It is easy to show that if gt ∼ Qs ≪ Qθ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (10) reduces to the 1 θ2 -scaling behavior of the collinear fac- torization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' On the other hand, if θQ ≪ Qs, for γ∗ → q¯q is presented in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Both approaches agree in the small θ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 4 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (10) scales as θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We thus expect that in CGC, the θ2Σ will be independent of the θ for θQ ≫ Qs, however, contrary to the collinear factorization, suppressed when θQ ≪ Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Meanwhile the θ region between these two limits provide the opportunity to estimate the saturation scale Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Numerics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Now we study the numerical impacts of the small-x dynamics on the shape of the θ2Σ(Q2, xB, θ) distribution from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (10), compared with the collinear prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We are particularly interested in the re- gion θ ≪ 1 where the θ distribution probes direcly the fEEC(x, θ), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' For the small-x dipole distri- bution S(2) xF (⃗rt), we use both the MV model with rcBK running [12, 13, 18, 97–104] and the GBW model [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' As for the MV model with rcBK running, we adopt the MV-like model [106] as the initial condition, whose form is S(2) x0 (⃗rt) = exp � − (r2 t Q2 s0)γ 4 ln � 1 Λrt + e �� , where we choose x0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='01, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='119, Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='241 GeV, Q2 s0 = A1/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='168 GeV2 with A the atomic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We use the solution to the LL BK evolution with αs run- ning [98, 101, 106] of the dipole distribution to evolve the dipole distribution from x0 to xF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In our calculation, we use the result fitted from the HERA data for the trans- verse area of the nucleus S⊥ [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The GBW model is implemented using S(2) xF (⃗rt) = exp � − 1 4r2 t Q2 s(xF ) � , where Q2 s(xF ) = AN(x0/xF )λ GeV2 and we use x0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='24 × 10−4, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='27 and AN = 1 for the proton while AN = 5 for the Au [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Normalized θ2Σ(Q2, xB, θ) distribution for proton and Au, with xB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='003, for both Q2 = 25 GeV2 , √s = 105 GeV (left panel) and Q2 = 100 GeV2 , √s = 318 GeV (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Green dots are from Pythia82 simulation, in which we demand 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='00295 < xB < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='00305, and 25GeV2 ≤ Q2 < 35GeV2 for the left panel while 100GeV2 ≤ Q2 < 110GeV2 for the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 5, we show the CGC predictions for θ2Σ(Q2, xB, θ) as a function of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Since we are only in- terested in the shape, we normalized the distribution by � θmax θmin dθθ2Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We fixed xB = 3 × 10−3 and choose Q2 = 25 GeV2, √s = 105 GeV (left panel) and Q2 = 100 GeV2, √s = 318 GeV (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We present predictions from CGC for both proton (in purple lines) and Au (in orange), by the MV model with rcBK running and GBW model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We see both models predict similar shapes in the θ spectrum, in which the small-θ region is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' They are impressively different from the collinear expec- tations (in red lines and green dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In the figure, the collinear predictions (red lines) are made out of the com- plete fixed order αs calculation, without the Qθ ≪ Q approximation, using CT18A [108] and EPPS21 [109] PDF sets for proton and Au, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' To validate our collinear calculation and to estimate the size of the evo- lution effect in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (5), we also run a Pythia82 simu- lation [110] for the proton case, where the LL resum- mation is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We see that for large θ, the fixed order calculations agree well with the the Pythia simu- lation, while for small θ values, the resummation effects could be sizable but does not suppress the small-θ region due to the absence of the perturbative Sudakov factor in fEEC in the collinear factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The collinear predic- tion for the Au follows closely the proton’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The notable difference demonstrates that the fEEC(x, θ) could serve as a clean probe of the small-x phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' For com- parison, we also show the predictions from the full CGC calculation derived in the Supplemental Material using the GBW model in purple circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 5, the proton spectrum turns into a plateau for large values of θ, which is expected from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (10) when Qθ ≫ Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We can define a turning point around which the slope of the distribution starts to switch its monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The turning point allows us to estimate the size of the saturation scale Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' For instance, from 5 the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 5, the turning point for the proton is roughly around θ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='15−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='2 and thus Qs ∼ θQ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='75− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='0 GeV which is consistent with the values of ΛQCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' And we estimate the saturation scale for the Au will be around θ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='5 and thus Qs ∼ θQ ∼ 2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 5 is similar to the left, but with Q2 = 100 GeV2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Since Q2 is larger, the distribution enters the plateau earlier as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Now the turning point for the Au is around θ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='3 which again indicates that Qs ∼ 2 − 3 GeV, consistent with the Q2 = 25 GeV2 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We can further introduce the nuclear modification fac- tor RpA = A−1ΣA(Q2,xB,θ) Σp(Q2,xB,θ) , which helps to reduce the sys- tematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In the collinear factorization, for θQ ≫ ΛQCD, the θ distribution is determined by the matching coeffi- cient Iij as predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (8), which is independent of the incoming nucleus species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Thus taking the ratio RAp reduces the impacts from perturbative higher order cor- rections as well as possible non-perturbative hadroniza- tion effects, and the collinear factorization predicts the RAp insensitive to the θ values, as showed explicitly as red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' RpA as a function of θ, with xB = 3 × 10−3 using the MV model with rcBK running and collinear factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Once again, the small-x formalism changes the pattern as we observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 6, where the modification factor RpA is suppressed in the small θ region, while converges toward around unity as θ becomes large and Qθ ≫ Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In this manuscript, we have proposed the nucleon energy-energy correlator (nucleon EEC) as a new probe of the gluon saturation phenomenon in DIS at the future electron-ion colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In particular, we have shown that the θ-shape of the nucleon EEC fEEC(x, θ) behaves differently in the collinear factorization theorem and the CGC formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The drastic difference is due to the intrinsic transverse momentum of order Qs induced by the non-linear small-x dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We thus expect the fEEC to complement the other standard small-x processes and offer a great opportunity to pin down the onset of the gluon saturation phenomenon in eA-collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The advantage of the nucleon EEC probe, as com- pared to other standard small-x processes, is that it is fully inclusive, involving no fragmentation functions and jet clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Therefore the observable is expected to be clean both theoretically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Exten- sions to other observables that are induced by the in- trinsic transverse dynamics of the nucleon/nucleus shall follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' For polarized hadron beam, we can study the spin asymmetry by adding the azimuthal dependence to the energy operator E and we expect different asymmetries to be predicted in the collinear and CGC frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We hope that the results presented in this manuscript will motivate carrying out the proposed measurement at the current and future electron-ion facilitieis, meanwhile stimulate further applications of the nucleon EEC in nu- clear structure studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We are grateful to Farid Salazar, Hongxi Xing, Jian Zhou for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' This work is supported by the Natural Science Foundation of China under contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 12175016 (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 11975200 (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ), and the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' De- partment of Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' DE-AC02- 05CH11231 (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ∗ xiliu@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='cn † fyuan@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='gov ‡ zhuhx@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='cn [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Gribov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Levin, and M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' D 96, 094017 (2017), 1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='07328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [21] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Dominguez, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Xiao, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Mueller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Xiao, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Yuan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' D 88, 114010 (2013), 1308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='2993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Metz and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Zhou, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Lappi, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Skokov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 115, 252301 (2015), 1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='04438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' D 104, 114032 (2021), 2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Boussarie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Grabovsky, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Szymanowski, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Wallon, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 128, 202001 (2022), 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='06353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [44] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Iancu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Mueller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Triantafyllopoulos, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Wei, JHEP 10, 103 (2022), 2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='06268.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rezaeian, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' B 758, 373 (2016), 1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='07452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [47] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' M¨antysaari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Xiao, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' D 89, 074037 (2014), 1403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='2413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [50] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Bergabo and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Venu- gopalan, JHEP 01, 052 (2021), 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='04372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [55] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Tong, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' C 82, 413 (2022), 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='12462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [110] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Sj¨ostrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=', Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 191, 159 (2015), 1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='3012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Supplemental Materials for “Nucleon Energy Correlators for the Color Glass Condensate” Hao-Yu Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='1 Xiaohui Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ∗ Ji-Chen Pan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 3 Feng Yuan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' † and Hua Xing Zhu5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' ‡ 1Center of Advanced Quantum Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 100875,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China 2Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China 3School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China 4Nuclear Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' USA 5Zhejiang Institute of Modern Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Zhejiang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 310027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' China (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 2023) Here we present the Σ(Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' xB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' θ) Σ(Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' xB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' θ) = � i � dσ(xB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' pi) Ei EA δ(θ2 − θ2 i ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (1) within the CGC formalism without the small θ approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The calculation is achieved by evaluating amplitude for a virtual photon γ∗ splitting into the q¯q pair, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Each of the quarks will enter the detector to contribute to the energy deposit Ei, which we assume it is always the ¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The contribution from the q is obtained by multiplying a symmetric factor at this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Dipole amplitude for γ∗ → q¯q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The orange block represents the CGC Wilson line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We work in the Breit frame, in which q = (0, 0, 0, −Q) for the virtual photon, and P = Q 2xB (1, 0, 0, 1) for the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The calculation is straightforward which gives Σ(Q2, xB, θ) = � λ=l,t fλΣγ∗ λ (Q2, xB, θ) , (2) where we sum over the virtual photon polarization with the flux ft = 1 − y + y2 2 for the transverse and fl = 1 − y longitudinal polarization, respectively, with y = Q2 xBs the inelasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Here for the transverse photon contribution, we find Σγ∗ t (Q2, xB, θ) = � q 2Ncα2e2 q π2xBQ2 S⊥ � dzd2⃗kt d2⃗lt (2π)2 Fg,xB(⃗lt) � z2 + (1 − z)2� ����� ⃗kt ⃗k2 t + ∆2 − ⃗kt −⃗lt (⃗kt −⃗lt)2 + ∆2 ����� 2 × �⃗k2 t + (1 − z)2Q2 (1 − z)Q xB Q � 1 2θδ � θ − tan−1 2kt(1 − z)Q k2 t − (1 − z)2Q2 � θ �⃗k2 t + (1 − z)2Q2 (1 − z)Q < xB Q xB � ,(3) where k is the momentum for ¯q and 1 − z = k− Q the momentum fraction with respect to the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We defined ∆2 = z(1−z)Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' In the last line, the first term is the energy weight, derived using the on-shell condition k+k−−⃗k2 t = 0, normalized to the incoming nucleus energy Q/xB in the Breit frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The second term defines the θ angle with respect ∗ xiliu@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='cn † fyuan@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='gov ‡ zhuhx@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='cn 2 to the z-axis, with θ < π/2 for kz > 0 while θ > π/2 for kz < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' The last θ function ensures that 2k0 can not exceed the incoming parton momentum xBP = xB Q xB [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' As for the longitudinal polarization, we have Σγ∗ l (Q2, xB, θ) = � q 2Ncα2e2 q π2xBQ2 S⊥ � dzd2⃗kt d2⃗lt (2π)2 Fg,xB(⃗lt)4z2(1 − z)2Q2 ����� 1 ⃗k2 t + ∆2 − 1 (⃗kt −⃗lt)2 + ∆2 ����� 2 × �⃗k2 t + (1 − z)2Q2 (1 − z)Q xB Q � 1 2θδ � θ − tan−1 2kt(1 − z)Q k2 t − (1 − z)2Q2 � θ �⃗k2 t + (1 − z)2Q2 (1 − z)Q < xB Q xB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (4) Now we consider the small-θ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' As θ → 0, the contribution is dominated by 1 − z ∼ k2 t Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We define the variable ξ through (1 − z)Q = ξ 1−ξ k2 t Q and only keep the leading contribution in z → 1 limit to find Σ(Q2, xB, θ) = � q 4πα2e2 q Q4 NcS⊥ 8π4 1 θ2 � dξ ξ d2⃗lt(1 − ξ)k2 t (kt − lt)2 ����� ⃗kt ξ⃗k2 t + (1 − ξ)(⃗kt −⃗lt)2 − ⃗kt −⃗lt (⃗kt −⃗lt)2 ����� 2 ×θ �1 − ξ ξ < 1 � Fg,xB(⃗lt) , (5) where kt = 1−ξ ξ Q 2 θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' We thus identify fEEC(xB, θ) at this order in the small-x formalism as fq,EEC(xB, θ) = NcS⊥ 8π4 1 θ2 � 1 ξcut dξ ξ d2⃗lt(1 − ξ)k2 t (kt − lt)2 ����� ⃗kt ξ⃗k2 t + (1 − ξ)(⃗kt −⃗lt)2 − ⃗kt −⃗lt (⃗kt −⃗lt)2 ����� 2 Fg,xB(⃗lt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (6) The ξcut is determined from the last θ-function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Beuf, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content=' D 96, 074033 (2017), 1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} +page_content='06557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNAzT4oBgHgl3EQf0v5s/content/2301.01788v1.pdf'} diff --git a/TNFJT4oBgHgl3EQfMCzl/content/tmp_files/2301.11472v1.pdf.txt b/TNFJT4oBgHgl3EQfMCzl/content/tmp_files/2301.11472v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..30dbca121ecc48a3283be42a5bebf0b2d4bddfb2 --- /dev/null +++ b/TNFJT4oBgHgl3EQfMCzl/content/tmp_files/2301.11472v1.pdf.txt @@ -0,0 +1,1242 @@ +arXiv:2301.11472v1 [stat.ME] 27 Jan 2023 +A Flexible Zero-Inflated +Conway–Maxwell–Poisson Regression Model +for Spatiotemporal Data of US Vaccine +Refusal +Bokgyeong Kang1, John Hughes2, and Murali Haran1 +1Department of Statistics, Pennsylvania State University +2College of Health, Lehigh University +Abstract +Vaccination is widely acknowledged as one of the most effective tools for pre- +venting disease. +However, there has been a rise in parental refusal and delay of +childhood vaccination in recent years in the United States. This trend undermines +the maintenance of herd immunity and elevates the likelihood of outbreaks of vaccine- +preventable diseases. Our aim is to identify demographic or socioeconomic character- +istics associated with vaccine refusal, which could help public health professionals and +medical providers develop interventions targeted to concerned parents. We examine +US county-level vaccine refusal data for patients under five years of age collected on a +monthly basis during the period 2012–2015. These data exhibit challenging features: +zero inflation, spatial dependence, seasonal variation, spatially-varying dispersion, +and a large sample size (approximately 3,000 counties per month). We propose a +flexible zero-inflated Conway–Maxwell–Poisson (ZICOMP) regression model that ad- +dresses these challenges. Because ZICOMP models have an intractable normalizing +function, it is challenging to do Bayesian inference for these models. We propose +a new hybrid Monte Carlo algorithm that permits efficient sampling and provides +asymptotically exact estimates of model parameters. +Keywords: High-dimensional spatial model; Spatially-varying dispersion; Vaccine refusal; +Zero inflation; Bayesian spatial filtering; Exchange algorithm. +1 + +1 +Introduction +Childhood vaccination coverage has remained high in the United States. However, due to +recent outbreaks of childhood vaccine-preventable diseases in the United States, there is still +increased public health concern regarding the upward trend of vaccine refusal and hesitancy. +The Centers for Disease Control and Prevention (CDC) reported that 1,274 individual +cases of measles were confirmed across 31 states in 2019. This is the greatest number of +cases reported in a single year since 1992. And the incidence of pertussis has been steadily +increasing since it reached its lowest point in 1976. A significant proportion of the cases were +found to be among unvaccinated or undervaccinated individuals. Communities with higher +vaccine exemption rates have higher incidence rates of the diseases (Phadke et al., 2016; +Patel et al., 2019). It is important to identify communities’ demographic or socioeconomic +characteristics associated with vaccine refusal. This could help public health professionals +and medical providers develop interventions targeted to concerned communities. +There is a large literature on vaccine uptake for childhood infections in the United +States. Most studies have relied on the CDC’s National Immunization Survey (NIS) data or +school vaccination exemption records. These studies have been limited in spatial resolution +or scale. +The NIS data are representative and comparable across states but have low +spatial resolution (state-level) and low response rates (Smith et al., 2004; Salmon et al., +2006; Frieden et al., 2014; Hill et al., 2015). School vaccination exemption records have +finer spatial resolution and high response rates but are available for only a small number +of states (Zipfel et al., 2020). A limited number of studies have used claims data where +vaccination status were assessed using ICD-9 codes (McCarthy et al., 2013; Glanz et al., +2013; Lieu et al., 2015). However, these studies have been small in scale. +In this manuscript we examine large-scale high-resolution childhood vaccine refusal data +(Kang et al., 2022). These data represent county-level monthly incidence of childhood vac- +cine refusal across the United States from 2012 to 2015 obtained from data managed by +IMS Health. The dataset includes county-level demographic and socioeconomic character- +istics that are hypothesized determinants of vaccine refusal. Kang et al. (2022) aggregated +the data by year to remove seasonality. To avoid information loss due to aggregation, we +2 + +use the original data. These data exhibit challenging features. First, there is an excess of +zeros. Over 79% of the counties recorded no refusals in any given month. Second, vac- +cine refusal has been shown to exhibit seasonal variation since high incidences of vaccine +refusal are observed during the first few months of school. Third, vaccine refusal behavior +has been found to be spatially clustered. We observe that the high incidences of vaccine +refusal are clustered. Fourth, the dispersion of the refusal counts may vary over space. Our +exploratory analysis suggests that some counties may be under-dispersed and some may be +over-dispersed. Finally, the sample size is large. There are approximately 3,000 observa- +tions in any given month. This makes Markov chain Monte Carlo (MCMC) algorithms for +these data computationally demanding due to large-matrix operations and slow mixing of +spatial random effects. These challenges motivate the development of a new, flexible model +and an efficient computational method. +In this article we propose a new zero-inflated Conway–Maxwell–Poisson (ZICOMP) +regression model and several computational methods that address the above mentioned +challenges. We include spatially dependent random effects that account for the spatial +clustering of refusal and for spatially-varying dispersion. The spatial effects are modeled +via a Bayesian spatial filtering (BSF) approach (Hughes, 2017), which represents the ran- +dom effects with linear combinations of spatially patterned basis functions. +This BSF +method addresses key computational challenges by reducing the dimension of and corre- +lation among the spatial random effects. Previous studies have used information criteria +or cross-validation for choosing the best basis vectors. However, these approaches require +fitting a model multiple times for various subsets of the basis vectors, which can be compu- +tationally expensive. We propose to choose suitable basis vectors using a reversible jump +MCMC (Green, 1995; Godsill, 2012) algorithm. Bayesian inference for the class of ZICOMP +models is challenging due to an intractable normalizing function. We propose an efficient +new hybrid Monte Carlo algorithm that also yields asymptotically exact inference, i.e., we +sample from a Markov chain whose stationary distribution is exactly equal to the target +distribution. +The remainder of this article is organized as follows. +In Section 2 we describe the +3 + +vaccine refusal data. In Section 3 we specify our ZICOMP model. In Section 4 we present +our hybrid Monte Carlo algorithm. In Section 5 we discuss simulation experiments for our +proposed model and computational approach. In Section 6 we analyze the vaccine refusal +data. We conclude in Section 7. +2 +Vaccine refusal data +Reports for vaccine refusal among patients under five years of age were obtained from a +database of U.S. medical claims managed by IMS Health on a monthly basis from 2012 +to 2015. Claims were submitted from both private and government insurance providers, +and data were aggregated according to U.S. five-digit ZIP codes. The reported data cover +Table 1: Description of the covariate variables. +Variable +Description +Measurement variables +Physician-patient interactions +Number of physician-patient interactions +Health insurance +Proportion of people with health insurance +Pediatrician reporting +Rate at which a pediatrician voluntarily reports +non-billable diagnoses +Demographic or socioeconomic variables +Household size +Average number of individuals living in a single +household +Religious congregations +Per capita number of congregations of religions +historically opposed to vaccination +Limited English proficiency +Proportion of people who are not proficient in +English +Private school +Proportion of children who attend private school +High income +Proportion of people in the upper 20% quantile +of income in the US +Same area +Proportion of people living in the same county +one year prior +State law leniency +Exemption law effectiveness index +State autism +Among families with more than 1 child, the +proportion with a current or past diagnosis +of autism +4 + +0.00 +0.25 +0.50 +0.75 +Percent +0 +2 +6 +4 +1 +8 +5 +7 +3 +9 +10 +Number of refusal cases +(a) Partial histogram of observations +2000 +3000 +4000 +5000 +Number of cases across space +Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec +Month +Year +2012 +2013 +2014 +2015 +(b) Monthly vaccine refusals +(c) Observations in Dec 2015 +0 +1 +2 to 4 +5 to 14 +15 to 49 +50 to 215 +(d) Ratio of mean to variance +0.0 to 0.5 +0.5 to 1.0 +1.0 to 5.0 +5.0 to 10.0 +10.0 to 15.0 +15.0 to 25.0 +Figure 1: (a) Partial histogram of refusal incidence (up to 10 cases). (b) Monthly incidence +of refusal aggregated across space. (c) Refusal incidence observed in December 2015. (d) +Ratio of sample mean to variance of positive counts across time for each county. +126,049 cases of vaccine refusal for children under the age of five across 2,470,596,759 +physician-patient interactions. Vaccine refusal was identified with the International Classi- +fication of Disease, Ninth Revision (ICD-9) code and sub-codes for “vaccination not carried +out” (V64.0). Thus a vaccine refusal represents a case when a patient was not immunized +due to philosophical or religious reasons. +There are four conditions, all of which would have to be met for a vaccine refusal to be +captured in the database: (1) an individual seeks pediatric health care from a provider, (2) +the individual is insured, (3) the provider uses the claims database, and (4) the provider +reports the vaccine refusal. The set of covariate variables includes factors representing these +measurement mechanisms (top of Table 1) and county-level demographic or socioeconomic +factors that are hypothesized determinants of vaccine refusal (bottom of Table 1). The +5 + +data set contains records for approximately 3,000 counties for each month. All explanatory +variables were centered and standardized for use in the model. +Several salient features of the data must be considered when formulating a statistical +model. First, the observations are potentially zero-inflated. Figure 1 (a) shows the partial +histogram of observed outcomes. +Approximately 86% of outcomes are zeros, and over +79% of the counties reported no refusals in a given month. Some of the zeros may have +been produced by imperfect detection of refusal due to spatial variation in healthcare +access and insurance rates. An appropriate model should account for imperfect detection. +Second, childhood vaccine refusal has been shown to exhibit seasonal variation. This is +mainly caused by state-mandated school entry immunization requirements in the United +States. Figure 1 (b) displays the monthly incidence of refusal aggregated across space. It +is observed that the incidence tends to rise during the first few months of the school year. +An appropriate model should accommodate this seasonal pattern. Third, vaccine refusal +behavior has been shown to exhibit spatial clustering. Figure 1 (c) shows the outcomes +observed in December 2015. It is observed that counties with high incidence are clustered. +Similar results are observed for the other months. This indicates that our model should +provide spatial smoothing and borrowing of information among adjacent regions. Forth, the +data are potentially over- and under-dispersed. Figure 1 (d) shows the ratio of sample mean +to variance of positive counts across time for each county. Some counties have ratio values +less than 1, which implies potential over-dispersion. Some have ratio values greater than +1, indicating possible under-dispersion. Thus a suitable model should accommodate both +over- and under-dispersed, and should permit the dispersion to vary over space. Finally, +the data size is large—approximately 3,000 observations in any given month. Our model +and inferential approach should handle this large data set efficiently. +6 + +3 +Zero-inflated Conway–Maxwell–Poisson regression +model +In this section we introduce a new model and several computational innovations that to- +gether address the challenging features of the data described in Section 2. We begin with +a description of the well known Conway-Maxwell-Poisson (COMP) distribution and build +upon it to first handle zero-inflated spatial regression and spatially varying dispersion. +Then we incorporate spatial filtering to greatly reduce the computational burden. Finally, +we complete our model specification by introducing indicator variables that facilitate basis- +vector selection via reversible jump MCMC. +3.1 +Conway–Maxwell–Poisson distribution +The Conway–Maxwell–Poisson (COMP) distribution (Conway and Maxwell, 1962) is a two- +parameter generalization of the Poisson distribution that allows for under-dispersion (vari- +ance less than the mean), equi-dispersion (variance equals the mean), and over-dispersion +(variance greater than the mean). A count variable Y is said to follow the COMP(λ, ν) +distribution if Y ’s probability mass function (pmf) is +P(Y = y) = +1 +c(λ, ν) +λy +(y!)ν , +where λ > 0 is a generalization of the Poisson rate parameter, ν ≥ 0 is the dispersion +parameter, and c(λ, ν) = �∞ +z=0 λz/(z!)ν is a normalizing function. There is under-dispersion +when ν > 1; there is equi-dispersion when ν = 1; and there is over-dispersion when 0 +≤ ν < 1. The COMP distribution contains three classical count distributions as special +cases: Poisson (ν = 1), geometric (ν = 0, λ < 1, and success probability 1−λ), and Bernoulli +(ν = ∞ and success probability λ/(1 + λ)). We use a reparameterization introduced by +Guikema and Goffelt (2008) that substitutes η = λ1/ν to approximate the center of the +COMP distribution. A variable Y is said to follow the COMPη(η, ν) distribution if Y ’s +7 + +pmf is +P(Y = y) = +1 +cη(η, ν) +�ηy +y! +�ν +, +with cη(η, ν) = �∞ +z=0 (ηz/z!)ν. The mode of the this distribution is ⌊η⌋. Shmueli et al. +(2005) approximated the mean and variance by +E(Y ) ≈ η + 1 +2ν − 1 +2, +V(Y ) ≈ η +ν . +(1) +These approximations are close for a wide range of η and ν. +3.2 +ZICOMP regression model +A zero-inflated model (Lambert, 1992) comprises two processes: a binary process and a +count process. The binary process is a point mass at zero and accounts for excess zeros. +The count process follows a distribution that can explain the remaining zeros and positive +observations. In the context of our application, examining vaccine refusal incidence, the +binary process classifies the entire data set into two groups: a missing data group and a +detected refusal group. The missing data group contains zeros, and those zeros represent +the fact that some people refused vaccination but those refusals were not captured in the +database. The detected refusal group consists of positive observations and the remaining +zeros, and those zeros mean there were no refusals. +The observations in the detected +refusal group are described by the count process. For the count process we use the COMP +distribution so we can account for under- and over-dispersion in the data. +Modeling these two processes together leads to the zero-inflated Conway–Maxwell– +Poisson (ZICOMP) distribution, which has pmf +P(Yst = yst) = (1 − πst)1{wst=0,yst=0} + πstp(yst; ηst, νs)1{wst=1} +s = 1, . . . , n; t = 1, . . . , T, +where yst denotes the observed response at spatial location s and time t, p(·; ηst, νs) is the +8 + +pmf of the COMPη(ηst, νs) distribution, πst is the probability of yst being in the detected +refusal group, and wst ∼ Bernoulli(πst) indicates whether yst is in the detected group. We +model the parameters as follows: +logit(πst) = X⊤ +stβ1 +log(ηst) = X⊤ +stβ2 + Vs + M⊤ +t ζ +log(νs) = α + Ws, +where Xst is a vector consisting of an intercept and covariate values for location s and time +t, β1 and β2 are regression coefficients, Vs and Ws are spatial random effects for location +s, Mt is a vector of 11 dummy variables for month fixed effects, ζ are their coefficients, +and α is an intercept for dispersion. The dispersion is constant across space if Ws = 0 for +s = 1, . . . , n. We assume that the fields of spatial random effects are independent between +the approximate modes η and dispersions ν, i.e., V = (V1, . . . , Vn)⊤ is independent of +W = (W1, . . . , Wn)⊤. +The conditional autoregressive (CAR) model is a traditional choice of prior distribution +for spatial random effects (Besag, 1974). The CAR priors for the spatial random fields V +and W are given by +V | κ ∼ Normaln +� +0, Q−1/κ +� +(2) +W | τ ∼ Normaln +� +0, Q−1/τ +� +, +(3) +where κ and τ are smoothing parameters and Q = diag(A1) − ρA is a precision matrix +where A is the adjacency matrix of the underlying graph, 1 is the conformable vector +of 1s, and ρ ∈ [0, 1) indicates the strength of spatial correlation (ρ = 0 implies spatial +independence while ρ near 1 implies strong spatial correlation). Note that Q intuitively +accounts for both dependences (Vi and Vj, i ̸= j, are independent given their neighbors if +and only if (Q)ij = (Q)ji = 0 if and only if locations i and j are not adjacent) and prior +uncertainty (uncertainty about Vi is inversely proportional to the number of neighbors of +location i: (Q)ii = (A)i1 where (A)i denotes the ith row of A). Markov chain Monte Carlo +9 + +(MCMC) is a standard approach for estimating the random effects. However, in this setting +MCMC is computationally burdensome for a large sample size n, owing to costly (O(n3)) +evaluations of n-dimensional multivariate normal likelihood functions at each iteration. +Additionally, strong correlations among the spatial random effects can lead to slow mixing +chains (Haran, 2011). Using the CAR model for our data is impractical since there are n ≈ +3,000 counties in a given month. +3.3 +ZICOMP with spatial filtering +To reduce the dimensions of and the correlations among the spatial random effects, we +reparameterize the spatial random effects as linear combinations of basis vectors: +V = Bγ∗ +W = Bδ∗, +where B is a basis matrix the columns of which are q ≪ n basis vectors, and γ∗ and δ∗ are +basis coefficients. There are two common choices of B: (1) in restricted spatial regression +(RSR), the spatial basis vectors are constrained to be orthogonal to the fixed-effects predic- +tors (cf. Reich et al., 2006; Hughes and Haran, 2013); and (2) in Bayesian spatial filtering +(BSF), the spatial basis vectors are constrained to be orthogonal to the intercept (Hughes, +2017). The RSR models have been found to yield low coverage rates for the regression co- +efficients (Hanks et al., 2015; Khan and Calder, 2020; Zimmerman and Hoef, 2021). Our +simulation experiments showed that the BSF parameterization does not adversely impact +inference for the regression coefficients, and so we employ the BSF approach. +Specifically, the BSF basis matrix B comprises selected eigenvectors of F = (I − +11⊤/n)A(I − 11⊤/n), where I is the n × n identity matrix and 1 is the n-dimensional +vector of 1s. The eigenvectors of F comprise all possible mutually distinct patterns that +can arise on the graph and hence can be used to model a spatial random field. The positive +(negative) eigenvalues of F correspond to varying degrees of attractive (repulsive) spatial +dependence. An eigenvector’s pattern has finer scale with decreasing magnitude of the +10 + +corresponding eigenvalue. In other words, B can accommodate spatial structure at multi- +ple scales. We henceforth assume that B comprises the first q ≪ n basis vectors since we +expect neighboring observations to be similar (i.e., we discard all of the repulsive patterns). +The priors for the basis coefficients are given by +γ∗ | κ ∼ Normalq +� +0, Q−1 +B /κ +� +δ∗ | τ ∼ Normalq +� +0, Q−1 +B /τ +� +, +where QB = B⊤QB. This allows for Bγ∗ and Bδ∗ to stand in for the CAR effects in +(2) and (3), respectively. The number q of basis vectors has been decided via information +criteria or cross-validation. However, these approaches require fitting the model multiple +times for various choices of q. This can be too computationally demanding for big data. +We use a reversible jump MCMC approach to allow for automatic selection of suitable +basis vectors. We introduce latent variables indicating whether basis vectors are included +in the model. +Let γ∗ = (γ∗ +1, . . . , γ∗ +q)⊤ and δ∗ = (δ∗ +1, . . . δ∗ +q)⊤. +For j = 1, . . . , q, we +reparameterize the basis coefficients γ∗ +j and δ∗ +j as +γ∗ +j = γjIγj +δ∗ +j = δjIδj, +where Iγj and Iδj are 1 if the jth basis vector is suitable, and they are 0 otherwise. Let γ = +(γ1, . . . , γq)⊤, δ = (δ1, . . . , δq)⊤, Iγ = diag[(Iγ1, . . . , Iγq)⊤], and Iδ = diag[(Iδ1, . . . , Iδq)⊤]. +11 + +Then our proposed model is given by +yst + + + += 0 +w.p. 1 − πst +∼ COMPη(ηst, νs) +w.p. πst, +logit(πst) = X⊤ +stβ1 +log(ηst) = X⊤ +stβ2 + B⊤ +s γIγ + M⊤ +t ζ +log(νs) = α + B⊤ +s δIδ, +s = 1, . . . , n; t = 1, . . . , T, +where Bs is the q-dimensional vector of basis function values corresponding to location s. +4 +A hybrid Monte Carlo algorithm for our ZICOMP +model +In this section we propose a hybrid Monte Carlo algorithm that combines several Monte +Carlo algorithms to provide asymptotically exact estimates of our model parameters. Stan- +dard MCMC cannot be used for our model due to the intractable normalizing function of +the COMP distribution. We employ an exchange algorithm (Murray et al., 2006) that was +developed for carrying out inference in the presence of intractable normalizing functions. +We introduce a proposal distribution that allows for the algorithm to perform better for +zero-inflated models. +4.1 +Computational challenge +Let θ = (w, β1, β2, ζ, α, γ, δ, Iγ, Iδ, κ, τ) be the collection of model parameters. The joint +posterior distribution of θ is given by +π(θ | y) ∝ p(θ)L(θ | y) += p(θ) +n +� +s=1 +T� +t=1 +(1 − πst)1−wst +� +πst +cη(ηst, νs) +�ηyst +st +yst! +�νs�wst +, +12 + +where p(θ) denotes a prior density and L(θ | y) represents the likelihood function for our +model. Let θi be a subset of the parameters and θ−i denote the rest of the parameters. +The full conditional posterior distribution of θi is given by +π(θi | θ−i, y) ∝ p(θi)L(θi | θ−i, y), +where p(θi) is a prior density and L(θi | θ−i, y) is obtained by removing all terms not +involving θi from L(θ | y). Consider the full conditional π(θi | θ−i, y). The Metropolis- +Hastings (MH) algorithm proposes θ′ +i from q(· | θi) and accepts θ′ +i with probability +α(θ′ +i | θi) = min +� +1, p(θ′ +i)L(θ′ +i | θ−i, y)q(θi | θ′ +i) +p(θi)L(θi | θ−i, y)q(θ′ +i | θi) +� +, +at each step of the algorithm. Gibbs sampling is a special case of the MH algorithm where +we can generate samples exactly from π(θi | θ−i, y). We use an MH update for β1 and +Gibbs updates for smoothing parameters κ and τ. +Standard MCMC cannot be used for the other parameters, however. Consider param- +eters besides β1, κ, and τ. Then the full conditional of θi is given by +π(θi | θ−i, y) ∝ p(θi) +n +� +s=1 +T� +t=1 +(1 − πst)1−wst +� +πst +cη(ηst, νs) +�ηyst +st +yst! +�νs�wst +. +Let h(y | θ) = �n +s=1 +�T +t=1 (ηyst +st /yst!)νswst be an unnormalized likelihood and r(θ) = �n +s=1 +�T +t=1 (1 − πst)1−wst {πst/cη(ηst, νs)}wst be its normalizing constant. The normalizing con- +stant r(θ) is intractable since cη(ηst, νs) is an infinite sum. The MH acceptance probability +becomes +α(θ′ +i | θi) = min +� +1, p(θ′ +i)h(y | θ′)r(θ)q(θi | θ′ +i) +p(θi)h(y | θ)r(θ′)q(θ′ +i | θi) +� +, +where θ′ = (θ′ +i, θ−i). The intractable normalizing constant r(θ) does not cancel out in the +acceptance probability. Thus standard MCMC techniques cannot be applied. We sidestep +this problem by introducing an auxiliary variable as described in the following section. +13 + +4.2 +An exchange algorithm for our ZICOMP model +Murray et al. (2006) introduced an auxiliary variable z that follows h(z | θ′)/r(θ′) so +that the intractable terms cancel out in the MH acceptance probability. Consider the full +conditional π(θi | θ−i, y). The exchange algorithm proceeds as follows: given θi, +1. propose θ′ +i ∼ q(· | θi), +2. generate an auxiliary variable exactly from the probability model at θ′: z ∼ h(·|θ′) +r(θ′) , and +3. accept θ′ +i with probability +α = min +� +1, p(θ′ +i)h(y | θ′)❍❍❍ +r(θ)h(z | θ)❍❍❍ +r(θ′)q(θi | θ′ +i) +p(θi)h(y | θ)❍❍❍ +r(θ′)h(z | θ′)❍❍❍ +r(θ)q(θ′ +i | θi) +� +. +We see that the normalizing constants cancel in the acceptance probability. We note that +the exchange algorithm provides asymptotically exact estimates of model parameters. We +use the exchange algorithm for β2, ζ, α, γ, δ, Iγ, and Iδ. A fast rejection sampling scheme +for COMP distributions (Chanialidis et al., 2018; Benson and Friel, 2021) can be used for +generating an auxiliary variable in Step 2. +However, we cannot use this method for the detection indicator variables w. If yst > 0, +then wst = 1 with probability 1, by definition. If yst = 0, then we observe either missing +data (implying wst = 0) or a detected zero (implying wst = 1). Conditional on yst = 0, the +full conditional of θi = wst is given by +π(wst | θ−i, yst = 0) ∝ p(wst)(1 − πst)1−wst +� +πst +cη(ηst, νs) +�wst +, +where p(wst) is a prior density. Given wst = 1, the full conditional is proportional to the +intractable COMP normalizing function cη(ηst, νs). +We propose w′ +st from the swapping +distribution q(· | wst) = δ(1 − wst − ·), where δ denotes the Dirac delta function. Suppose +we generate an auxiliary variable according to +zst + + + += 0 +if w′ +st = 0 +∼ COMPη(ηst, νs) +if w′ +st = 1. +14 + +Suppose wst = 0. Then the algorithm proposes w′ +st = 1, generates an auxiliary variable zst +∼ COMPη(ηst, νs), and accepts w′ +st = 1 with probability +α(w′ +st = 1 | wst = 0) = min + + +1, +p(w′ +st) +πst +❳❳❳❳ +cη(ηst,νs)δ(zst) +p(wst)(1 − πst) +1 +❳❳❳❳ +cη(ηst,νs) +� +ηzst +st +zst! +�νs + + + , += min + + +1, +p(w′ +st)πstδ(zst) +p(wst)(1 − πst) +� +ηzst +st +zst! +�νs + + + . +The acceptance probability α(w′ +st = 1 | wst = 0) = 0 whenever zst > 0. +In practice, +when there is severe over-dispersion, the probability of accepting w′ +st = 1 becomes very +small, leading to an impractical algorithm. We can address this problem by introducing +the following mixture distribution for the auxiliary variable: +zst ∼ + + + +NB(ηst, νs) +if w′ +st = 0 +COMPη(ηst, νs) +if w′ +st = 1, +(4) +where NB(a, b) denotes the negative binomial distribution with mean a and dispersion b. +To simplify the description of our acceptance ratio, let +g(zst | θ′) = +� +Γ(zst + νs) +Γ(zst + 1)Γ(νs) +� +νs +ηst + νs +�νs � +ηst +ηst + νs +�zst�1−w′ +st �ηzst +st +zst! +�νsw′ +st +, +where θ′ = (w′ +st, θ−i). An exchange algorithm with the proposal distribution (4) for the +auxiliary variable accepts w′ +st with probability +α(w′ +st | wst) = min +� +1, p(w′ +st)(1 − πst)1−w′ +stπw′ +st +st g(zst | θ) +p(wst)(1 − πst)1−wstπwst +st g(zst | θ′) +� +. +We found that this algorithm performed well in our simulation experiments. +4.3 +A hybrid Monte Carlo algorithm +Here we summarize our hybrid Monte Carlo algorithm. The algorithm proceeds as follows. +15 + +1. Use an exchange algorithm for wst for s = 1, . . . , n and t = 1, . . . , T. The proposal for +the auxiliary variable is given in (4). +2. Use a MH update for β1. +3. Use an exchange algorithm for β2, ζ, α, γ, and δ. The proposal for the auxiliary variable +is our ZICOMP model. +4. Use an exchange algorithm for Iγj and Iδj for j = 1, . . . , q. The proposal for the auxiliary +variable is our ZICOMP model. +5. Do Gibbs updates for κ and τ. +The proposed algorithm generates a Markov chain whose stationary distribution is +exactly equal to the posterior distribution of interest. We generate the auxiliary variable in +parallel in Steps 1, 3, and 4. It can be computationally demanding to update Iγj and Iδj for +j = 1, . . . , q at every iteration for big data. To speed up computation, we randomly select m +basis vectors and update only those indicator variables for a given iteration. Alternatively, +we could update all variables at every kth iteration. To our knowledge, no existing theory +suggests that the latter approach is asymptotically exact. But we found that this method +produces faster convergence than the former method and correctly chooses the true basis +vectors. And so we use the latter method in the sequel. +5 +Applications to simulated data +Here we apply our ZICOMP model to data simulated from (i) a full model with spatial and +temporal effects and spatially-varying dispersion, (ii) a model with constant dispersion, +i.e., δ = 0, (iii) a model with fixed effects only, i.e., γ = δ = 0, and (iv) a model with +covariate effects only, i.e., γ = δ = ζ = 0. An aim of the simulation experiments is to assess +how our model and computational approach perform in the context of data generated from +models with varying degrees of spatial and temporal dependence. The underlying graph +for the data is the 30 × 30 lattice. Our design matrix is Xt = [1 x1 x2] for t = 1, . . . , T, +where x1 = (x1,1, . . . , x1,900)⊤ and x2 = (x2,1, . . . , x2,900)⊤ are the x- and y-coordinates of +16 + +the vertices. We restrict the coordinates of the vertices to the unit square. We use the first +25 eigenvectors of F to simulate data for our study, i.e., dim(γ) = dim(δ) = 25 and B is +900 × 25. We simulate T = 24 observations per vertex for a total of N = 900 × 24 = +21,600 observations. +We assume independent N(0, 100I) priors for the fixed effects β1, β2, ζ, and α. We +assign gamma priors with shape parameter equal to 0.001 and rate parameter equal to 1,000 +to the smoothing parameters κ and τ. We assume that wst ∼ Bernoulli(0.5) for s = 1, . . . , n +and t = 1, . . . , T. We assign independent Bernoulli(0.1) priors to the basis vector indicator +variables Iγj and Iδj for j = 1, . . . , q. This prior is appealing since it corresponds to the +prior belief that the fixed effects are sufficient to account for data, and this prior can prevent +our method from producing artifactual spatial structure in the posterior. +We fit our ZICOMP model by using our hybrid Monte Carlo algorithm illustrated in +Section 4.3. We generate posterior sample paths of length at least 2 million in all cases +to ensure that the Monte Carlo standard errors calculated by the batch means method +(Jones et al., 2006; Flegal et al., 2008) are sufficiently small. We use normal proposals for +β1, β2, ζ, and α and adapt proposal covariance matrices using the Log-Adaptive Proposal +algorithm (Shaby and Wells, 2011). We use swapping proposals for wst, Iγj, and Iδj. +5.1 +Data simulated from the full model +We create a data set by first setting κ = τ = 1 and simulating random effects according to +γ ∼ Normal25(0, Q−1 +B ) and δ ∼ Normal25(0, Q−1 +B ). We generate response values under the +following model: +logit(πst) = X⊤ +stβ1 +log(ηst) = X⊤ +stβ2 + B⊤ +s γ + M⊤ +t ζ +log(νs) = α + B⊤ +s δ, +s = 1, . . . , 900; t = 1, . . . , 24, +17 + +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +x2 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +x2 +0.00 +0.25 +0.50 +0.75 +1.00 +0.00 +0.25 +0.50 +0.75 +1.00 +x2 +Truth +(a) Detection probability +Truth +(b) Approximate mode +Truth +(c) Dispersion +Estimate +Estimate +Estimate +0.00 +0.25 +0.50 +0.75 +1.00 +x1 +0.2 +0.4 +0.6 +0.8 +π +0.00 +0.25 +0.50 +0.75 +1.00 +x1 +5 +10 +15 +20 +25 +η +0.00 +0.25 +0.50 +0.75 +1.00 +x1 +0.6 +0.9 +1.2 +1.5 +ν +Figure 2: (a) True and estimated values of the detection probability πst for t = 1. (b) True +and estimated values of approximate mode ηst for t = 1. (c) True and estimated values of +dispersion νs. +where β1 = (0, −3, 2)⊤, β2 = (2, −0.5, 1)⊤, ζ = (0, 0, 0, 0, 0, 0, 0.1, 0.2, 0.5, 0.4, 0.3)⊤, and +α = −0.3. We fit our ZICOMP model with 50 eigenvectors to the simulated data set. We +use more basis vectors than the truth to assess how our basis vector selection approach +performs. +We see that the estimated posterior medians of the model parameters are close to the +true values. All of their 95% highest posterior density (HPD) intervals cover the true values. +To validate the performance more thoroughly, we apply our model to 100 simulated data +sets and estimate coverage rates and type I and II error rates based on 95% HPD intervals. +We observe 90.8–99.0% for the coverage rates of all parameters, 1.0–9.2% for type I error +rates of parameters whose true values are zeros, and 0.0–2.0% for type II error rates of +parameters whose true values are nonzero. This shows that our approach performs reliably. +18 + +For the spatial effects in η, we observe that 22 basis vectors have estimated posterior +probabilities of 0.5 and above. For the spatial effects in ν, we see that 13 basis vectors +have estimated posterior probabilities of at least 0.5. The selected basis vectors are all true +basis vectors. The unselected ones among the true basis vectors are found to have true +basis coefficient values close to 0. This shows that our methodology performs reliably in +selecting important basis vectors. +Figure 2 presents maps of true and estimated values for the detection probability πst, +approximate mode ηst, and dispersion νs for t = 1. +Similar results are observed for +t = 2, . . . , 24. We see that the estimated spatial patterns closely mirror the true spatial +distributions. This shows that our approach recovers well the underlying spatial patterns +in the data. +5.2 +Data simulated from simpler models +We simulate a data set from each of the following models. +(ii) Model with constant dispersion: +logit(πst) = X⊤ +stβ1 +log(ηst) = X⊤ +stβ2 + B⊤ +s γ + M⊤ +t ζ +log(νs) = α +(iii) Model with fixed effects only: +logit(πst) = X⊤ +stβ1 +log(ηst) = X⊤ +stβ2 + M⊤ +t ζ +log(νs) = α +(iv) Model with covariate effects only: +logit(πst) = X⊤ +stβ1 +log(ηst) = X⊤ +stβ2 +log(νs) = α +s = 1, . . . , 900; t = 1, . . . , 24, +where γ ∼ Normal25(0, Q−1 +B /κ). We fit our ZICOMP model with 50 eigenvectors to these +three simulated data sets. The aim of this simulation experiment is to see if our method- +19 + +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +Standardized RQR +−5.0 +−2.5 +0.0 +2.5 +5.0 +7.5 +Standardized RQR +ZICOMP +ZIP +(a) Residual plot +ZICOMP +ZIP +(b) Quantile−quantile plot +−30 +−20 +−10 +0 +−15 +−10 +−5 +0 +5 +Log of estimate +−2.5 +0.0 +2.5 +−2.5 +0.0 +2.5 +Standard normal quantile +Figure 3: (a) Residual plots for the RQRs stemming from our ZICOMP model and the +ZIP model. (b) Quantile-quantile plots for our ZICOMP model and the ZIP model. Our +ZICOMP model fits the data much better than the ZIP model. +ology produces artifactual spatial or temporal structures in the posterior. +We observe that the estimated posterior medians of the model parameters are close to +the simulated true values. All of their 95% HPD intervals cover the true values. When the +true model is Model (ii), the fitted model correctly suggests that there is evident spatial +dependence in η while there is no spatial variation in ν. For the spatial effects in η, we +observe that 23 basis vectors have estimated posterior probabilities of 0.5 and above. For +the spatial effects in ν, none of the basis vectors have estimated posterior probabilities of +0.5 and above. +When the true models are Model (ii) and (iv), the fitted models correctly indicate that +there is spatial variation in neither η nor ν. We observe that none of the basis vectors +have estimated posterior probabilities of 0.5 and above. For the data simulated from Model +(iv), all of the 95% HPD intervals for ζ cover 0. This correctly suggests that there is no +temporal effect evident in the data. In summary, our method effectively avoids overfitting +and provides accurate inference for all parameters. +6 +Application to vaccine refusal data +Now we apply our proposed methodology to the vaccine refusal data described in Section 2. +We use 400 basis vectors so the fit can accommodate a rich spatial structure, if necessary. +We use the same priors and proposal distributions as those given in Section 5. +20 + +(a) Observed number of refusals +0 +1 +2 +4 +14 +48 +215 +(b) Expected number of refusals +0 +1 +2 +4 +14 +48 +215 +(c) Spatial effects in refusal +-15 +-10 +-5 +0 +5 +(d) Dispersion +0.0 +0.2 +1.0 +1.3 +5.3 +5.4 +Figure 4: +(a) The observed number of vaccine refusal cases in December 2015. +(b) +Estimates of the mean number of cases in December 2015. (c) Estimates of the spatial +effects in refusal. +Negative effects in red are associated with decrease in refusal while +positive effects in green are associated with increase in refusal. (d) Estimates of dispersion. +The red means over-dispersion while the blue mean under-dispersion. Note: The reason +for our choice of scale is that the data are zero-inflated and only a few counties have large +counts. +For comparison we also fit a zero-inflated Poisson (ZIP) regression model. The ZIP +model describes the count process using the Poisson distribution with means equal to +µst = exp(X⊤ +stβ2 + B⊤ +s γ + M⊤ +t ζ) and assumes equi-dispersion. We compare our ZICOMP +model and the ZIP model via residual diagnosis. For these count data, the distribution of +the standardized residuals is far from normal even though the model is well specified. And +so we use randomized quantile residuals (RQRs), which have been shown to have low type I +error rates and high statistical power for detecting model misspecification for count models, +including zero-inflated models (Dunn and Smyth, 1996; Feng et al., 2020). Figure 3 shows +residual plots for the RQRs stemming from our ZICOMP model fit and the ZIP model fit. +The ZIP model produces infinite values of RQR that were excluded from the plots. We +21 + +State autism +State law leniency +Same area +High income +Private school +Limited English +Religious congregation +Household size +Pediatrician reporting +Health insurance +log(Interaction) +−0.25 +0.00 +0.25 +0.50 +0.752.4 +2.6 +95% HPD interval +Covariate variable +(a) Detection +State autism +State law leniency +Same area +High income +Private school +Limited English +Religious congregation +Household size +Pediatrician reporting +Health insurance +log(Interaction) +−0.3 +0.0 +0.3 +4.2 +4.5 4.75 +5 +95% HPD interval +Covariate variable +(b) Refusal cases +Figure 5: Estimated posterior medians (shaded dots or triangles) and 95% HPD intervals +(horizontal solid or dashed bars) for covariate coefficients. The shaded dot and horizontal +solid bar represent that the HPD interval does not include zero. The triangle and horizontal +dashed bar represent that the HPD interval includes zero. +see that our ZICOMP model fits the data much better than the ZIP model. This suggests +that we should allow for flexibility in modeling the dispersion of the vaccine refusal data. +Figure 4 (b) shows the estimated mean incidences of refusal under perfect detection in +December 2015. To obtain the mean estimates, we simulate 10,000 response values from +the fitted model and average them for each county. The spatial pattern of the mean is +similar to the pattern for the observed counts presented in Figure 4 (a). We observe high +incidences of refusal, on average, in the Northwest, Southwest, and Northeast regions of +the United States, Florida, and the area around Lake Michigan. +Figure 4 (c) presents the estimates of the spatial effects in vaccine refusal. Positive +effects (in green) are associated with increased refusal while negative effects (in red) are +associated with decreased refusal. This spatial pattern may have been produced by some +unobserved spatially-structured variables that are associated with refusal. Alternatively, it +may have been produced by social influence, in which vaccination behavior is contagious +and diffuses between neighboring areas, producing refusal clusters (Alvarez−Zuzek et al., +2022). +Figure 4 (c) displays the estimates of dispersion. We find evidence of spatial variation +in the dispersion. Some areas have over-dispersed counts and some have under-dispersed +22 + +counts. For the month effects, we used January for a reference month. We find that people +are less likely to refuse to vaccinate their children in March to August while people are +more likely to refuse to vaccinate their children in September to December, compared to +January. +Figure 5 (a) displays the estimated posterior medians and 95% HPD intervals for the re- +gression coefficients for detection of vaccine refusal. We find that all healthcare-related mea- +surement variables are predictive of refusal. Physician-patient interaction is overwhelmingly +predictive. Among demographic or socioeconomic variables, we observe that communities +with small household sizes, religions historically opposed to vaccination, limited proficiency +in English, high rates of private school attendance, high incomes, lack of continuity of care, +high leniency in the state’s vaccination laws, and low incidence of autism are likely to have +reported refusals. +Figure 5 (b) presents the estimates and 95% HPD intervals for the regression coefficients +for refusal cases. Given perfect detection, we see that communities with increased access +to care, high insurance coverage, high likelihood of physician reporting, small household +sizes, groups historically opposed to vaccination, low rates of private school attendance, +high incomes, high leniency in state vaccination laws, and low incidence of autism are more +likely to refuse to vaccinate their children. +7 +Discussion +In this article we proposed a new, flexible ZICOMP regression model for examining the +occurrence of childhood vaccine refusal in the United States. We also proposed several +computational approaches that provide computational efficiency in carrying out Bayesian +inference for our model. Our methodology has several attractive features. First, it ad- +dresses potential zero inflation relative to a standard count distribution. This allows us +to account for imperfect detection of refusal cases and infer the refusal distribution under +perfect detection. Our model also accounts for the underlying spatial pattern in vaccine +refusal while correctly accommodating spatially-varying dispersion, which could not be +done using previous models. Our approach could be useful in many disciplines, such as +23 + +ecology, agriculture, criminology, medicine, and public health studies where zero-inflated +spatial data are commonly encountered (cf. Ratcliffe and Mccord, 2007; Neelon et al., 2016; +Lyashevska et al., 2016). +The vaccine refusal analysis revealed several important findings. First, communities +that have religions historically opposed to vaccination, have high incomes, and live in states +with permissive vaccination laws are more likely to have high incidence of refusal. These as- +sociations have been reported in earlier studies (cf. Omer et al., 2006; Salmon et al., 2015; +McKee and Bohannon, 2016). We also found that communities with large family sizes, high +rates of private school attendance, and high incidence of autism are more likely to have low +incidence of refusal. Our analysis indicated that vaccine refusal exhibits spatial dependence +that is not explained by the set of our covariates. This spatial dependence may have been +produced by some unobserved spatially-structured variables. This hypothesis means that +the clustering in vaccine refusal only reflects spatial clustering in underlying drivers. Alter- +natively, the spatial dependence may have been caused by diffusion of vaccination behavior +between neighboring areas. Identifying the source of clustering is important for effectively +alleviating the clustering and reducing the risk of disease outbreaks (Alvarez−Zuzek et al., +2022). Our findings suggest that refusal behavior may be contagious. +We conclude with a few crucial caveats. We used observational data and therefore can +infer only associations between vaccine refusal behavior and the covariates. Readers inter- +ested in developing interventions for improving public health would expect to be informed +about causal effects. When we interpret our results, consideration must be taken to prevent +the ecological inference fallacy. We carry out statistical inference at the county-level and +try to infer ecological factors on vaccine refusal rather than individual factors. Our Markov +chains mix slowly, which inspired us to use reparameterization techniques. However, we +still need to generate long sample paths to ensure convergence of the chains. +Further +improvement in computing may be an interesting topic for future research. +24 + +References +Alvarez−Zuzek, L. G., Zipfel, C., and Bansal, S. (2022). Spatial clustering in vaccination +hesitancy: the role of social influence and social selection. PLoS Computational Biology. +Benson, A. and Friel, N. (2021). 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Scientific Data, 7:1–7. +29 + diff --git a/TNFJT4oBgHgl3EQfMCzl/content/tmp_files/load_file.txt b/TNFJT4oBgHgl3EQfMCzl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d210b2648f5d05872593c2604c8b7a4ccb749802 --- /dev/null +++ b/TNFJT4oBgHgl3EQfMCzl/content/tmp_files/load_file.txt @@ -0,0 +1,1102 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf,len=1101 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='11472v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='ME] 27 Jan 2023 A Flexible Zero-Inflated Conway–Maxwell–Poisson Regression Model for Spatiotemporal Data of US Vaccine Refusal Bokgyeong Kang1, John Hughes2, and Murali Haran1 1Department of Statistics, Pennsylvania State University 2College of Health, Lehigh University Abstract Vaccination is widely acknowledged as one of the most effective tools for pre- venting disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, there has been a rise in parental refusal and delay of childhood vaccination in recent years in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This trend undermines the maintenance of herd immunity and elevates the likelihood of outbreaks of vaccine- preventable diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our aim is to identify demographic or socioeconomic character- istics associated with vaccine refusal, which could help public health professionals and medical providers develop interventions targeted to concerned parents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We examine US county-level vaccine refusal data for patients under five years of age collected on a monthly basis during the period 2012–2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' These data exhibit challenging features: zero inflation, spatial dependence, seasonal variation, spatially-varying dispersion, and a large sample size (approximately 3,000 counties per month).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We propose a flexible zero-inflated Conway–Maxwell–Poisson (ZICOMP) regression model that ad- dresses these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Because ZICOMP models have an intractable normalizing function, it is challenging to do Bayesian inference for these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We propose a new hybrid Monte Carlo algorithm that permits efficient sampling and provides asymptotically exact estimates of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Keywords: High-dimensional spatial model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Spatially-varying dispersion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Vaccine refusal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Zero inflation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Bayesian spatial filtering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Exchange algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 1 1 Introduction Childhood vaccination coverage has remained high in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, due to recent outbreaks of childhood vaccine-preventable diseases in the United States, there is still increased public health concern regarding the upward trend of vaccine refusal and hesitancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The Centers for Disease Control and Prevention (CDC) reported that 1,274 individual cases of measles were confirmed across 31 states in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This is the greatest number of cases reported in a single year since 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' And the incidence of pertussis has been steadily increasing since it reached its lowest point in 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' A significant proportion of the cases were found to be among unvaccinated or undervaccinated individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Communities with higher vaccine exemption rates have higher incidence rates of the diseases (Phadke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' It is important to identify communities’ demographic or socioeconomic characteristics associated with vaccine refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This could help public health professionals and medical providers develop interventions targeted to concerned communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' There is a large literature on vaccine uptake for childhood infections in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Most studies have relied on the CDC’s National Immunization Survey (NIS) data or school vaccination exemption records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' These studies have been limited in spatial resolution or scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The NIS data are representative and comparable across states but have low spatial resolution (state-level) and low response rates (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Salmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Frieden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' School vaccination exemption records have finer spatial resolution and high response rates but are available for only a small number of states (Zipfel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' A limited number of studies have used claims data where vaccination status were assessed using ICD-9 codes (McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Glanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Lieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, these studies have been small in scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In this manuscript we examine large-scale high-resolution childhood vaccine refusal data (Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' These data represent county-level monthly incidence of childhood vac- cine refusal across the United States from 2012 to 2015 obtained from data managed by IMS Health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The dataset includes county-level demographic and socioeconomic character- istics that are hypothesized determinants of vaccine refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Kang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (2022) aggregated the data by year to remove seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' To avoid information loss due to aggregation, we 2 use the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' These data exhibit challenging features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' First, there is an excess of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Over 79% of the counties recorded no refusals in any given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Second, vac- cine refusal has been shown to exhibit seasonal variation since high incidences of vaccine refusal are observed during the first few months of school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Third, vaccine refusal behavior has been found to be spatially clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We observe that the high incidences of vaccine refusal are clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Fourth, the dispersion of the refusal counts may vary over space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our exploratory analysis suggests that some counties may be under-dispersed and some may be over-dispersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Finally, the sample size is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' There are approximately 3,000 observa- tions in any given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This makes Markov chain Monte Carlo (MCMC) algorithms for these data computationally demanding due to large-matrix operations and slow mixing of spatial random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' These challenges motivate the development of a new, flexible model and an efficient computational method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In this article we propose a new zero-inflated Conway–Maxwell–Poisson (ZICOMP) regression model and several computational methods that address the above mentioned challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We include spatially dependent random effects that account for the spatial clustering of refusal and for spatially-varying dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The spatial effects are modeled via a Bayesian spatial filtering (BSF) approach (Hughes, 2017), which represents the ran- dom effects with linear combinations of spatially patterned basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This BSF method addresses key computational challenges by reducing the dimension of and corre- lation among the spatial random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Previous studies have used information criteria or cross-validation for choosing the best basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, these approaches require fitting a model multiple times for various subsets of the basis vectors, which can be compu- tationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We propose to choose suitable basis vectors using a reversible jump MCMC (Green, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Godsill, 2012) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Bayesian inference for the class of ZICOMP models is challenging due to an intractable normalizing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We propose an efficient new hybrid Monte Carlo algorithm that also yields asymptotically exact inference, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', we sample from a Markov chain whose stationary distribution is exactly equal to the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The remainder of this article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In Section 2 we describe the 3 vaccine refusal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In Section 3 we specify our ZICOMP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In Section 4 we present our hybrid Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In Section 5 we discuss simulation experiments for our proposed model and computational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In Section 6 we analyze the vaccine refusal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We conclude in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 2 Vaccine refusal data Reports for vaccine refusal among patients under five years of age were obtained from a database of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' medical claims managed by IMS Health on a monthly basis from 2012 to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Claims were submitted from both private and government insurance providers, and data were aggregated according to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' five-digit ZIP codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The reported data cover Table 1: Description of the covariate variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Measurement variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Physician-patient interactions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Number of physician-patient interactions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Health insurance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Proportion of people with health insurance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Pediatrician reporting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Rate at which a pediatrician voluntarily reports ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='non-billable diagnoses ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Demographic or socioeconomic variables ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Household size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Average number of individuals living in a single ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='household ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Religious congregations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Per capita number of congregations of religions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='historically opposed to vaccination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Limited English proficiency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Proportion of people who are not proficient in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='English ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Private school ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Proportion of children who attend private school ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='High income ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Proportion of people in the upper 20% quantile ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='of income in the US ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Same area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Proportion of people living in the same county ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='one year prior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='State law leniency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Exemption law effectiveness index ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='State autism ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='Among families with more than 1 child,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' the proportion with a current or past diagnosis of autism 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 Percent 0 2 6 4 1 8 5 7 3 9 10 Number of refusal cases (a) Partial histogram of observations 2000 3000 4000 5000 Number of cases across space Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month Year 2012 2013 2014 2015 (b) Monthly vaccine refusals (c) Observations in Dec 2015 0 1 2 to 4 5 to 14 15 to 49 50 to 215 (d) Ratio of mean to variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 to 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 Figure 1: (a) Partial histogram of refusal incidence (up to 10 cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (b) Monthly incidence of refusal aggregated across space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (c) Refusal incidence observed in December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (d) Ratio of sample mean to variance of positive counts across time for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 126,049 cases of vaccine refusal for children under the age of five across 2,470,596,759 physician-patient interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Vaccine refusal was identified with the International Classi- fication of Disease, Ninth Revision (ICD-9) code and sub-codes for “vaccination not carried out” (V64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Thus a vaccine refusal represents a case when a patient was not immunized due to philosophical or religious reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' There are four conditions, all of which would have to be met for a vaccine refusal to be captured in the database: (1) an individual seeks pediatric health care from a provider, (2) the individual is insured, (3) the provider uses the claims database, and (4) the provider reports the vaccine refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The set of covariate variables includes factors representing these measurement mechanisms (top of Table 1) and county-level demographic or socioeconomic factors that are hypothesized determinants of vaccine refusal (bottom of Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The 5 data set contains records for approximately 3,000 counties for each month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' All explanatory variables were centered and standardized for use in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Several salient features of the data must be considered when formulating a statistical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' First, the observations are potentially zero-inflated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 1 (a) shows the partial histogram of observed outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Approximately 86% of outcomes are zeros, and over 79% of the counties reported no refusals in a given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Some of the zeros may have been produced by imperfect detection of refusal due to spatial variation in healthcare access and insurance rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' An appropriate model should account for imperfect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Second, childhood vaccine refusal has been shown to exhibit seasonal variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This is mainly caused by state-mandated school entry immunization requirements in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 1 (b) displays the monthly incidence of refusal aggregated across space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' It is observed that the incidence tends to rise during the first few months of the school year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' An appropriate model should accommodate this seasonal pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Third, vaccine refusal behavior has been shown to exhibit spatial clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 1 (c) shows the outcomes observed in December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' It is observed that counties with high incidence are clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Similar results are observed for the other months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This indicates that our model should provide spatial smoothing and borrowing of information among adjacent regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Forth, the data are potentially over- and under-dispersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 1 (d) shows the ratio of sample mean to variance of positive counts across time for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Some counties have ratio values less than 1, which implies potential over-dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Some have ratio values greater than 1, indicating possible under-dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Thus a suitable model should accommodate both over- and under-dispersed, and should permit the dispersion to vary over space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Finally, the data size is large—approximately 3,000 observations in any given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our model and inferential approach should handle this large data set efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 6 3 Zero-inflated Conway–Maxwell–Poisson regression model In this section we introduce a new model and several computational innovations that to- gether address the challenging features of the data described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We begin with a description of the well known Conway-Maxwell-Poisson (COMP) distribution and build upon it to first handle zero-inflated spatial regression and spatially varying dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Then we incorporate spatial filtering to greatly reduce the computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Finally, we complete our model specification by introducing indicator variables that facilitate basis- vector selection via reversible jump MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='1 Conway–Maxwell–Poisson distribution The Conway–Maxwell–Poisson (COMP) distribution (Conway and Maxwell, 1962) is a two- parameter generalization of the Poisson distribution that allows for under-dispersion (vari- ance less than the mean), equi-dispersion (variance equals the mean), and over-dispersion (variance greater than the mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' A count variable Y is said to follow the COMP(λ, ν) distribution if Y ’s probability mass function (pmf) is P(Y = y) = 1 c(λ, ν) λy (y!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' )ν , where λ > 0 is a generalization of the Poisson rate parameter, ν ≥ 0 is the dispersion parameter, and c(λ, ν) = �∞ z=0 λz/(z!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' )ν is a normalizing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' There is under-dispersion when ν > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' there is equi-dispersion when ν = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' and there is over-dispersion when 0 ≤ ν < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The COMP distribution contains three classical count distributions as special cases: Poisson (ν = 1), geometric (ν = 0, λ < 1, and success probability 1−λ), and Bernoulli (ν = ∞ and success probability λ/(1 + λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use a reparameterization introduced by Guikema and Goffelt (2008) that substitutes η = λ1/ν to approximate the center of the COMP distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' A variable Y is said to follow the COMPη(η, ν) distribution if Y ’s 7 pmf is P(Y = y) = 1 cη(η, ν) �ηy y!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' �ν , with cη(η, ν) = �∞ z=0 (ηz/z!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=')ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The mode of the this distribution is ⌊η⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Shmueli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (2005) approximated the mean and variance by E(Y ) ≈ η + 1 2ν − 1 2, V(Y ) ≈ η ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (1) These approximations are close for a wide range of η and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2 ZICOMP regression model A zero-inflated model (Lambert, 1992) comprises two processes: a binary process and a count process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The binary process is a point mass at zero and accounts for excess zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The count process follows a distribution that can explain the remaining zeros and positive observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In the context of our application, examining vaccine refusal incidence, the binary process classifies the entire data set into two groups: a missing data group and a detected refusal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The missing data group contains zeros, and those zeros represent the fact that some people refused vaccination but those refusals were not captured in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The detected refusal group consists of positive observations and the remaining zeros, and those zeros mean there were no refusals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The observations in the detected refusal group are described by the count process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For the count process we use the COMP distribution so we can account for under- and over-dispersion in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Modeling these two processes together leads to the zero-inflated Conway–Maxwell– Poisson (ZICOMP) distribution, which has pmf P(Yst = yst) = (1 − πst)1{wst=0,yst=0} + πstp(yst;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' ηst, νs)1{wst=1} s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , T, where yst denotes the observed response at spatial location s and time t, p(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' ηst, νs) is the 8 pmf of the COMPη(ηst, νs) distribution, πst is the probability of yst being in the detected refusal group, and wst ∼ Bernoulli(πst) indicates whether yst is in the detected group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We model the parameters as follows: logit(πst) = X⊤ stβ1 log(ηst) = X⊤ stβ2 + Vs + M⊤ t ζ log(νs) = α + Ws, where Xst is a vector consisting of an intercept and covariate values for location s and time t, β1 and β2 are regression coefficients, Vs and Ws are spatial random effects for location s, Mt is a vector of 11 dummy variables for month fixed effects, ζ are their coefficients, and α is an intercept for dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The dispersion is constant across space if Ws = 0 for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We assume that the fields of spatial random effects are independent between the approximate modes η and dispersions ν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', V = (V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , Vn)⊤ is independent of W = (W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , Wn)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The conditional autoregressive (CAR) model is a traditional choice of prior distribution for spatial random effects (Besag, 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The CAR priors for the spatial random fields V and W are given by V | κ ∼ Normaln � 0, Q−1/κ � (2) W | τ ∼ Normaln � 0, Q−1/τ � , (3) where κ and τ are smoothing parameters and Q = diag(A1) − ρA is a precision matrix where A is the adjacency matrix of the underlying graph, 1 is the conformable vector of 1s, and ρ ∈ [0, 1) indicates the strength of spatial correlation (ρ = 0 implies spatial independence while ρ near 1 implies strong spatial correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Note that Q intuitively accounts for both dependences (Vi and Vj, i ̸= j, are independent given their neighbors if and only if (Q)ij = (Q)ji = 0 if and only if locations i and j are not adjacent) and prior uncertainty (uncertainty about Vi is inversely proportional to the number of neighbors of location i: (Q)ii = (A)i1 where (A)i denotes the ith row of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Markov chain Monte Carlo 9 (MCMC) is a standard approach for estimating the random effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, in this setting MCMC is computationally burdensome for a large sample size n, owing to costly (O(n3)) evaluations of n-dimensional multivariate normal likelihood functions at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Additionally, strong correlations among the spatial random effects can lead to slow mixing chains (Haran, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Using the CAR model for our data is impractical since there are n ≈ 3,000 counties in a given month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3 ZICOMP with spatial filtering To reduce the dimensions of and the correlations among the spatial random effects, we reparameterize the spatial random effects as linear combinations of basis vectors: V = Bγ∗ W = Bδ∗, where B is a basis matrix the columns of which are q ≪ n basis vectors, and γ∗ and δ∗ are basis coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' There are two common choices of B: (1) in restricted spatial regression (RSR), the spatial basis vectors are constrained to be orthogonal to the fixed-effects predic- tors (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Reich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Hughes and Haran, 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' and (2) in Bayesian spatial filtering (BSF), the spatial basis vectors are constrained to be orthogonal to the intercept (Hughes, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The RSR models have been found to yield low coverage rates for the regression co- efficients (Hanks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Khan and Calder, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Zimmerman and Hoef, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our simulation experiments showed that the BSF parameterization does not adversely impact inference for the regression coefficients, and so we employ the BSF approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Specifically, the BSF basis matrix B comprises selected eigenvectors of F = (I − 11⊤/n)A(I − 11⊤/n), where I is the n × n identity matrix and 1 is the n-dimensional vector of 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The eigenvectors of F comprise all possible mutually distinct patterns that can arise on the graph and hence can be used to model a spatial random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The positive (negative) eigenvalues of F correspond to varying degrees of attractive (repulsive) spatial dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' An eigenvector’s pattern has finer scale with decreasing magnitude of the 10 corresponding eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In other words, B can accommodate spatial structure at multi- ple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We henceforth assume that B comprises the first q ≪ n basis vectors since we expect neighboring observations to be similar (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', we discard all of the repulsive patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The priors for the basis coefficients are given by γ∗ | κ ∼ Normalq � 0, Q−1 B /κ � δ∗ | τ ∼ Normalq � 0, Q−1 B /τ � , where QB = B⊤QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This allows for Bγ∗ and Bδ∗ to stand in for the CAR effects in (2) and (3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The number q of basis vectors has been decided via information criteria or cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, these approaches require fitting the model multiple times for various choices of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This can be too computationally demanding for big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use a reversible jump MCMC approach to allow for automatic selection of suitable basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We introduce latent variables indicating whether basis vectors are included in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Let γ∗ = (γ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , γ∗ q)⊤ and δ∗ = (δ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' δ∗ q)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , q, we reparameterize the basis coefficients γ∗ j and δ∗ j as γ∗ j = γjIγj δ∗ j = δjIδj, where Iγj and Iδj are 1 if the jth basis vector is suitable, and they are 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Let γ = (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , γq)⊤, δ = (δ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , δq)⊤, Iγ = diag[(Iγ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , Iγq)⊤], and Iδ = diag[(Iδ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , Iδq)⊤].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 11 Then our proposed model is given by yst \uf8f1 \uf8f2 \uf8f3 = 0 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 1 − πst ∼ COMPη(ηst, νs) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' πst, logit(πst) = X⊤ stβ1 log(ηst) = X⊤ stβ2 + B⊤ s γIγ + M⊤ t ζ log(νs) = α + B⊤ s δIδ, s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , T, where Bs is the q-dimensional vector of basis function values corresponding to location s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 4 A hybrid Monte Carlo algorithm for our ZICOMP model In this section we propose a hybrid Monte Carlo algorithm that combines several Monte Carlo algorithms to provide asymptotically exact estimates of our model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Stan- dard MCMC cannot be used for our model due to the intractable normalizing function of the COMP distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We employ an exchange algorithm (Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2006) that was developed for carrying out inference in the presence of intractable normalizing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We introduce a proposal distribution that allows for the algorithm to perform better for zero-inflated models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='1 Computational challenge Let θ = (w, β1, β2, ζ, α, γ, δ, Iγ, Iδ, κ, τ) be the collection of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The joint posterior distribution of θ is given by π(θ | y) ∝ p(θ)L(θ | y) = p(θ) n � s=1 T� t=1 (1 − πst)1−wst � πst cη(ηst, νs) �ηyst st yst!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' �νs�wst , 12 where p(θ) denotes a prior density and L(θ | y) represents the likelihood function for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Let θi be a subset of the parameters and θ−i denote the rest of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The full conditional posterior distribution of θi is given by π(θi | θ−i, y) ∝ p(θi)L(θi | θ−i, y), where p(θi) is a prior density and L(θi | θ−i, y) is obtained by removing all terms not involving θi from L(θ | y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Consider the full conditional π(θi | θ−i, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The Metropolis- Hastings (MH) algorithm proposes θ′ i from q(· | θi) and accepts θ′ i with probability α(θ′ i | θi) = min � 1, p(θ′ i)L(θ′ i | θ−i, y)q(θi | θ′ i) p(θi)L(θi | θ−i, y)q(θ′ i | θi) � , at each step of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Gibbs sampling is a special case of the MH algorithm where we can generate samples exactly from π(θi | θ−i, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use an MH update for β1 and Gibbs updates for smoothing parameters κ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Standard MCMC cannot be used for the other parameters, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Consider param- eters besides β1, κ, and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Then the full conditional of θi is given by π(θi | θ−i, y) ∝ p(θi) n � s=1 T� t=1 (1 − πst)1−wst � πst cη(ηst, νs) �ηyst st yst!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' �νs�wst .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Let h(y | θ) = �n s=1 �T t=1 (ηyst st /yst!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' )νswst be an unnormalized likelihood and r(θ) = �n s=1 �T t=1 (1 − πst)1−wst {πst/cη(ηst, νs)}wst be its normalizing constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The normalizing con- stant r(θ) is intractable since cη(ηst, νs) is an infinite sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The MH acceptance probability becomes α(θ′ i | θi) = min � 1, p(θ′ i)h(y | θ′)r(θ)q(θi | θ′ i) p(θi)h(y | θ)r(θ′)q(θ′ i | θi) � , where θ′ = (θ′ i, θ−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The intractable normalizing constant r(θ) does not cancel out in the acceptance probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Thus standard MCMC techniques cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We sidestep this problem by introducing an auxiliary variable as described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2 An exchange algorithm for our ZICOMP model Murray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (2006) introduced an auxiliary variable z that follows h(z | θ′)/r(θ′) so that the intractable terms cancel out in the MH acceptance probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Consider the full conditional π(θi | θ−i, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The exchange algorithm proceeds as follows: given θi, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' propose θ′ i ∼ q(· | θi), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' generate an auxiliary variable exactly from the probability model at θ′: z ∼ h(·|θ′) r(θ′) , and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' accept θ′ i with probability α = min � 1, p(θ′ i)h(y | θ′)❍❍❍ r(θ)h(z | θ)❍❍❍ r(θ′)q(θi | θ′ i) p(θi)h(y | θ)❍❍❍ r(θ′)h(z | θ′)❍❍❍ r(θ)q(θ′ i | θi) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We see that the normalizing constants cancel in the acceptance probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We note that the exchange algorithm provides asymptotically exact estimates of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use the exchange algorithm for β2, ζ, α, γ, δ, Iγ, and Iδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' A fast rejection sampling scheme for COMP distributions (Chanialidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Benson and Friel, 2021) can be used for generating an auxiliary variable in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, we cannot use this method for the detection indicator variables w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' If yst > 0, then wst = 1 with probability 1, by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' If yst = 0, then we observe either missing data (implying wst = 0) or a detected zero (implying wst = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Conditional on yst = 0, the full conditional of θi = wst is given by π(wst | θ−i, yst = 0) ∝ p(wst)(1 − πst)1−wst � πst cη(ηst, νs) �wst , where p(wst) is a prior density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Given wst = 1, the full conditional is proportional to the intractable COMP normalizing function cη(ηst, νs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We propose w′ st from the swapping distribution q(· | wst) = δ(1 − wst − ·), where δ denotes the Dirac delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Suppose we generate an auxiliary variable according to zst \uf8f1 \uf8f2 \uf8f3 = 0 if w′ st = 0 ∼ COMPη(ηst, νs) if w′ st = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 14 Suppose wst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Then the algorithm proposes w′ st = 1, generates an auxiliary variable zst ∼ COMPη(ηst, νs), and accepts w′ st = 1 with probability α(w′ st = 1 | wst = 0) = min \uf8f1 \uf8f2 \uf8f31, p(w′ st) πst ❳❳❳❳ cη(ηst,νs)δ(zst) p(wst)(1 − πst) 1 ❳❳❳❳ cη(ηst,νs) � ηzst st zst!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' �νs \uf8fc \uf8fd \uf8fe , = min \uf8f1 \uf8f2 \uf8f31, p(w′ st)πstδ(zst) p(wst)(1 − πst) � ηzst st zst!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' �νs \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The acceptance probability α(w′ st = 1 | wst = 0) = 0 whenever zst > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In practice, when there is severe over-dispersion, the probability of accepting w′ st = 1 becomes very small, leading to an impractical algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We can address this problem by introducing the following mixture distribution for the auxiliary variable: zst ∼ \uf8f1 \uf8f2 \uf8f3 NB(ηst, νs) if w′ st = 0 COMPη(ηst, νs) if w′ st = 1, (4) where NB(a, b) denotes the negative binomial distribution with mean a and dispersion b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' To simplify the description of our acceptance ratio, let g(zst | θ′) = � Γ(zst + νs) Γ(zst + 1)Γ(νs) � νs ηst + νs �νs � ηst ηst + νs �zst�1−w′ st �ηzst st zst!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' �νsw′ st , where θ′ = (w′ st, θ−i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' An exchange algorithm with the proposal distribution (4) for the auxiliary variable accepts w′ st with probability α(w′ st | wst) = min � 1, p(w′ st)(1 − πst)1−w′ stπw′ st st g(zst | θ) p(wst)(1 − πst)1−wstπwst st g(zst | θ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We found that this algorithm performed well in our simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3 A hybrid Monte Carlo algorithm Here we summarize our hybrid Monte Carlo algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The algorithm proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Use an exchange algorithm for wst for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , n and t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The proposal for the auxiliary variable is given in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Use a MH update for β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Use an exchange algorithm for β2, ζ, α, γ, and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The proposal for the auxiliary variable is our ZICOMP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Use an exchange algorithm for Iγj and Iδj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The proposal for the auxiliary variable is our ZICOMP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Do Gibbs updates for κ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The proposed algorithm generates a Markov chain whose stationary distribution is exactly equal to the posterior distribution of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We generate the auxiliary variable in parallel in Steps 1, 3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' It can be computationally demanding to update Iγj and Iδj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , q at every iteration for big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' To speed up computation, we randomly select m basis vectors and update only those indicator variables for a given iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Alternatively, we could update all variables at every kth iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' To our knowledge, no existing theory suggests that the latter approach is asymptotically exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' But we found that this method produces faster convergence than the former method and correctly chooses the true basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' And so we use the latter method in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 5 Applications to simulated data Here we apply our ZICOMP model to data simulated from (i) a full model with spatial and temporal effects and spatially-varying dispersion, (ii) a model with constant dispersion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', δ = 0, (iii) a model with fixed effects only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', γ = δ = 0, and (iv) a model with covariate effects only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', γ = δ = ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' An aim of the simulation experiments is to assess how our model and computational approach perform in the context of data generated from models with varying degrees of spatial and temporal dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The underlying graph for the data is the 30 × 30 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our design matrix is Xt = [1 x1 x2] for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , T, where x1 = (x1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , x1,900)⊤ and x2 = (x2,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , x2,900)⊤ are the x- and y-coordinates of 16 the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We restrict the coordinates of the vertices to the unit square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use the first 25 eigenvectors of F to simulate data for our study, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', dim(γ) = dim(δ) = 25 and B is 900 × 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We simulate T = 24 observations per vertex for a total of N = 900 × 24 = 21,600 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We assume independent N(0, 100I) priors for the fixed effects β1, β2, ζ, and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We assign gamma priors with shape parameter equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='001 and rate parameter equal to 1,000 to the smoothing parameters κ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We assume that wst ∼ Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5) for s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , n and t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We assign independent Bernoulli(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='1) priors to the basis vector indicator variables Iγj and Iδj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This prior is appealing since it corresponds to the prior belief that the fixed effects are sufficient to account for data, and this prior can prevent our method from producing artifactual spatial structure in the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We fit our ZICOMP model by using our hybrid Monte Carlo algorithm illustrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We generate posterior sample paths of length at least 2 million in all cases to ensure that the Monte Carlo standard errors calculated by the batch means method (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Flegal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2008) are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use normal proposals for β1, β2, ζ, and α and adapt proposal covariance matrices using the Log-Adaptive Proposal algorithm (Shaby and Wells, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use swapping proposals for wst, Iγj, and Iδj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='1 Data simulated from the full model We create a data set by first setting κ = τ = 1 and simulating random effects according to γ ∼ Normal25(0, Q−1 B ) and δ ∼ Normal25(0, Q−1 B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We generate response values under the following model: logit(πst) = X⊤ stβ1 log(ηst) = X⊤ stβ2 + B⊤ s γ + M⊤ t ζ log(νs) = α + B⊤ s δ, s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , 900;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , 24, 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 x2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 x2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 x2 Truth (a) Detection probability Truth (b) Approximate mode Truth (c) Dispersion Estimate Estimate Estimate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='8 π 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 x1 5 10 15 20 25 η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 x1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 ν Figure 2: (a) True and estimated values of the detection probability πst for t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (b) True and estimated values of approximate mode ηst for t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (c) True and estimated values of dispersion νs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' where β1 = (0, −3, 2)⊤, β2 = (2, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5, 1)⊤, ζ = (0, 0, 0, 0, 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3)⊤, and α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We fit our ZICOMP model with 50 eigenvectors to the simulated data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use more basis vectors than the truth to assess how our basis vector selection approach performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We see that the estimated posterior medians of the model parameters are close to the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' All of their 95% highest posterior density (HPD) intervals cover the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' To validate the performance more thoroughly, we apply our model to 100 simulated data sets and estimate coverage rates and type I and II error rates based on 95% HPD intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We observe 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='8–99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0% for the coverage rates of all parameters, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2% for type I error rates of parameters whose true values are zeros, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0% for type II error rates of parameters whose true values are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This shows that our approach performs reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 18 For the spatial effects in η, we observe that 22 basis vectors have estimated posterior probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For the spatial effects in ν, we see that 13 basis vectors have estimated posterior probabilities of at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The selected basis vectors are all true basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The unselected ones among the true basis vectors are found to have true basis coefficient values close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This shows that our methodology performs reliably in selecting important basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 2 presents maps of true and estimated values for the detection probability πst, approximate mode ηst, and dispersion νs for t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Similar results are observed for t = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We see that the estimated spatial patterns closely mirror the true spatial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This shows that our approach recovers well the underlying spatial patterns in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2 Data simulated from simpler models We simulate a data set from each of the following models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (ii) Model with constant dispersion: logit(πst) = X⊤ stβ1 log(ηst) = X⊤ stβ2 + B⊤ s γ + M⊤ t ζ log(νs) = α (iii) Model with fixed effects only: logit(πst) = X⊤ stβ1 log(ηst) = X⊤ stβ2 + M⊤ t ζ log(νs) = α (iv) Model with covariate effects only: logit(πst) = X⊤ stβ1 log(ηst) = X⊤ stβ2 log(νs) = α s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , 900;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' , 24, where γ ∼ Normal25(0, Q−1 B /κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We fit our ZICOMP model with 50 eigenvectors to these three simulated data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The aim of this simulation experiment is to see if our method- 19 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 Standardized RQR −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 Standardized RQR ZICOMP ZIP (a) Residual plot ZICOMP ZIP (b) Quantile−quantile plot −30 −20 −10 0 −15 −10 −5 0 5 Log of estimate −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 Standard normal quantile Figure 3: (a) Residual plots for the RQRs stemming from our ZICOMP model and the ZIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (b) Quantile-quantile plots for our ZICOMP model and the ZIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our ZICOMP model fits the data much better than the ZIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' ology produces artifactual spatial or temporal structures in the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We observe that the estimated posterior medians of the model parameters are close to the simulated true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' All of their 95% HPD intervals cover the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' When the true model is Model (ii), the fitted model correctly suggests that there is evident spatial dependence in η while there is no spatial variation in ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For the spatial effects in η, we observe that 23 basis vectors have estimated posterior probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For the spatial effects in ν, none of the basis vectors have estimated posterior probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' When the true models are Model (ii) and (iv), the fitted models correctly indicate that there is spatial variation in neither η nor ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We observe that none of the basis vectors have estimated posterior probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For the data simulated from Model (iv), all of the 95% HPD intervals for ζ cover 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This correctly suggests that there is no temporal effect evident in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' In summary, our method effectively avoids overfitting and provides accurate inference for all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 6 Application to vaccine refusal data Now we apply our proposed methodology to the vaccine refusal data described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use 400 basis vectors so the fit can accommodate a rich spatial structure, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We use the same priors and proposal distributions as those given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 20 (a) Observed number of refusals 0 1 2 4 14 48 215 (b) Expected number of refusals 0 1 2 4 14 48 215 (c) Spatial effects in refusal 15 10 5 0 5 (d) Dispersion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='4 Figure 4: (a) The observed number of vaccine refusal cases in December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (b) Estimates of the mean number of cases in December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (c) Estimates of the spatial effects in refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Negative effects in red are associated with decrease in refusal while positive effects in green are associated with increase in refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (d) Estimates of dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The red means over-dispersion while the blue mean under-dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Note: The reason for our choice of scale is that the data are zero-inflated and only a few counties have large counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For comparison we also fit a zero-inflated Poisson (ZIP) regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The ZIP model describes the count process using the Poisson distribution with means equal to µst = exp(X⊤ stβ2 + B⊤ s γ + M⊤ t ζ) and assumes equi-dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We compare our ZICOMP model and the ZIP model via residual diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For these count data, the distribution of the standardized residuals is far from normal even though the model is well specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' And so we use randomized quantile residuals (RQRs), which have been shown to have low type I error rates and high statistical power for detecting model misspecification for count models, including zero-inflated models (Dunn and Smyth, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 3 shows residual plots for the RQRs stemming from our ZICOMP model fit and the ZIP model fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The ZIP model produces infinite values of RQR that were excluded from the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We 21 State autism State law leniency Same area High income Private school Limited English Religious congregation Household size Pediatrician reporting Health insurance log(Interaction) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='6 95% HPD interval Covariate variable (a) Detection State autism State law leniency Same area High income Private school Limited English Religious congregation Household size Pediatrician reporting Health insurance log(Interaction) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content='75 5 95% HPD interval Covariate variable (b) Refusal cases Figure 5: Estimated posterior medians (shaded dots or triangles) and 95% HPD intervals (horizontal solid or dashed bars) for covariate coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The shaded dot and horizontal solid bar represent that the HPD interval does not include zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The triangle and horizontal dashed bar represent that the HPD interval includes zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' see that our ZICOMP model fits the data much better than the ZIP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This suggests that we should allow for flexibility in modeling the dispersion of the vaccine refusal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 4 (b) shows the estimated mean incidences of refusal under perfect detection in December 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' To obtain the mean estimates, we simulate 10,000 response values from the fitted model and average them for each county.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The spatial pattern of the mean is similar to the pattern for the observed counts presented in Figure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We observe high incidences of refusal, on average, in the Northwest, Southwest, and Northeast regions of the United States, Florida, and the area around Lake Michigan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 4 (c) presents the estimates of the spatial effects in vaccine refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Positive effects (in green) are associated with increased refusal while negative effects (in red) are associated with decreased refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This spatial pattern may have been produced by some unobserved spatially-structured variables that are associated with refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Alternatively, it may have been produced by social influence, in which vaccination behavior is contagious and diffuses between neighboring areas, producing refusal clusters (Alvarez−Zuzek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 4 (c) displays the estimates of dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We find evidence of spatial variation in the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Some areas have over-dispersed counts and some have under-dispersed 22 counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' For the month effects, we used January for a reference month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We find that people are less likely to refuse to vaccinate their children in March to August while people are more likely to refuse to vaccinate their children in September to December, compared to January.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 5 (a) displays the estimated posterior medians and 95% HPD intervals for the re- gression coefficients for detection of vaccine refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We find that all healthcare-related mea- surement variables are predictive of refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Physician-patient interaction is overwhelmingly predictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Among demographic or socioeconomic variables, we observe that communities with small household sizes, religions historically opposed to vaccination, limited proficiency in English, high rates of private school attendance, high incomes, lack of continuity of care, high leniency in the state’s vaccination laws, and low incidence of autism are likely to have reported refusals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Figure 5 (b) presents the estimates and 95% HPD intervals for the regression coefficients for refusal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Given perfect detection, we see that communities with increased access to care, high insurance coverage, high likelihood of physician reporting, small household sizes, groups historically opposed to vaccination, low rates of private school attendance, high incomes, high leniency in state vaccination laws, and low incidence of autism are more likely to refuse to vaccinate their children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 7 Discussion In this article we proposed a new, flexible ZICOMP regression model for examining the occurrence of childhood vaccine refusal in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We also proposed several computational approaches that provide computational efficiency in carrying out Bayesian inference for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our methodology has several attractive features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' First, it ad- dresses potential zero inflation relative to a standard count distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This allows us to account for imperfect detection of refusal cases and infer the refusal distribution under perfect detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our model also accounts for the underlying spatial pattern in vaccine refusal while correctly accommodating spatially-varying dispersion, which could not be done using previous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our approach could be useful in many disciplines, such as 23 ecology, agriculture, criminology, medicine, and public health studies where zero-inflated spatial data are commonly encountered (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Ratcliffe and Mccord, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Neelon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Lyashevska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' The vaccine refusal analysis revealed several important findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' First, communities that have religions historically opposed to vaccination, have high incomes, and live in states with permissive vaccination laws are more likely to have high incidence of refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' These as- sociations have been reported in earlier studies (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Omer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Salmon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' McKee and Bohannon, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We also found that communities with large family sizes, high rates of private school attendance, and high incidence of autism are more likely to have low incidence of refusal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our analysis indicated that vaccine refusal exhibits spatial dependence that is not explained by the set of our covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This spatial dependence may have been produced by some unobserved spatially-structured variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' This hypothesis means that the clustering in vaccine refusal only reflects spatial clustering in underlying drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Alter- natively, the spatial dependence may have been caused by diffusion of vaccination behavior between neighboring areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Identifying the source of clustering is important for effectively alleviating the clustering and reducing the risk of disease outbreaks (Alvarez−Zuzek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our findings suggest that refusal behavior may be contagious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We conclude with a few crucial caveats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We used observational data and therefore can infer only associations between vaccine refusal behavior and the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Readers inter- ested in developing interventions for improving public health would expect to be informed about causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' When we interpret our results, consideration must be taken to prevent the ecological inference fallacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' We carry out statistical inference at the county-level and try to infer ecological factors on vaccine refusal rather than individual factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Our Markov chains mix slowly, which inspired us to use reparameterization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' However, we still need to generate long sample paths to ensure convergence of the chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' Further improvement in computing may be an interesting topic for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' 24 References Alvarez−Zuzek, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', Zipfel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=', and Bansal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNFJT4oBgHgl3EQfMCzl/content/2301.11472v1.pdf'} +page_content=' (2022).' metadata={'source': 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Olmo1, ∗ and P. J. Porf´ırio2, † +1Departamento de F´ısica Te´orica and IFIC, +Centro Mixto Universitat de Val`encia–CSIC. Universitat +de Val`encia, Burjassot-46100, Val`encia, Spain +2Departamento de F´ısica, Universidade Federal da Para´ıba, +Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, Para´ıba, Brazil +(Dated: January 6, 2023) +Abstract +It has been recently claimed in [1] that an action of the Einstein-Palatini form plus a torsionless +Pontryagin term (multiplied by a constant) represents a counterexample to the conclusions of [2], +namely, that Lovelock gravity is the only case in which the metric and metric-affine formulations +of gravity are equivalent. However, given that the Pontryagin term (multiplied by a constant) can +be written as a total D-divergence, it is a textbook matter to realise that the addition of such (or +any other) D-divergence only affects at the boundary, leaving invariant the field equations and its +solutions, which are those of GR `a la Palatini. We thus conclude that the example provided in [1] +is not a valid counterexample of [2]. +∗Electronic address: gonzalo.olmo@uv.es +†Electronic address: pporfirio@fisica.ufpb.br +1 + +The torsionless metric-affine action in D = 2n dimensions considered in [1], +S = +ˆ +dDx +�√−ggabRab(Γ) + 1 +nθǫa1a2...aDRi1 +i2a1a2(Γ)Ri2 +i3a3a4(Γ)...Rin +i1aD−1aD(Γ) +� +, +(1) +can be written when θ is constant as +S = +ˆ +dDx +�√−ggabRab(Γ) + θ∂aKa� +, +(2) +with Ka as in Eq.(2) of [1]. The addition of this fancy total derivative (or of any other) +does not have any effect on the field equations [3]. +Since the scalar gabRab(Γ) is just +the linear in curvature term of Lovelock gravity, the action (1) can still be regarded as +included in the equivalence class of Lovelock theories, up to the addition of irrelevant total +divergences. Thus, the example presented in [1] cannot be regarded as a counterexample of +the conclusions of [2]. +If the parameter θ in (1) is promoted to the status of field, one obtains that the connection +field equations are given by +−∇c +�√−ggi2b� ++∇d +�√−ggi2d� +δb +c−2 (∇dθ) ǫdba3...aD−1aDRi2 +i3a3a4(Γ)...Rin +caD−1aD(Γ) = 0 , (3) +which naturally boil down to those of GR when θ is a constant. In the non-constant case, +taking b = c in Eq.(3), one finds ∇d +�√−ggi2d� += 0. Plugging this result back into Eq.(3), +one arrives at +∇c +�√−ggi2b� ++ 2 (∇dθ) ǫdba3...aD−1aDRi2 +i3a3a4(Γ)...Rin +caD−1aD(Γ) = 0. +(4) +In particular, when D = 4, Eq.(3) boils down to +∇c +�√−ggab� += −2 (∇dθ) ǫdba3a4Ra +ca3a4(Γ), +(5) +which is similar to the connection equation obtained either in [4] or in [5] for the torsion- +less case and by disregarding the homothetic curvature terms. The resulting equations are +obviously different from those found in the metric formulation and lead to different phe- +nomenology [6, 7]. Thus, when θ ̸= constant, the conclusions of [2] for the action (1) hold +as well. +Before concluding, we note that [2] neither provided an algorithm to find all the solutions +for the connection in the metric-affine formulation of Lovelock theories nor ruled out the +possible existence of solutions different from the Levi-Civita one [8]. +2 + +Acknowledgments. This work is supported by the Spanish Grant PID2020-116567GB- +C21 funded by MCIN/AEI/10.13039/501100011033, the project PROMETEO/2020/079 +(Generalitat Valenciana), and by the European Union’s Horizon 2020 research and inno- +vation programme under the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740. PJP +would like to thank CAPES for financial support. +[1] U. Lindstr¨om and ¨O. Sarıo˘glu, Phys. Lett. B 836 (2023) 137619, arXiv:2211.12327 [gr-qc]. +[2] Q. +Exirifard +and +M. +M. +Sheikh-Jabbari, +Phys. +Lett. +B +661, +158-161 +(2008) +doi:10.1016/j.physletb.2008.02.012 [arXiv:0705.1879 [hep-th]]. +[3] L. D. Landau and E. M. Lifschits, “The Classical Theory of Fields”, Pergamon Press, 1975. +[4] F. Sulantay, M. Lagos and M. Ba˜nados, [arXiv:2211.08925 [gr-qc]]. +[5] S. Boudet, F. Bombacigno, G. J. Olmo and P. J. Porfirio, JCAP 05 (2022) no.05, 032, +[arXiv:2203.04000 [gr-qc]]. +[6] S. Boudet, F. Bombacigno, F. Moretti and G. J. Olmo, [arXiv:2209.14394 [gr-qc]]. +[7] F. Bombacigno, F. Moretti, S. Boudet and G. J. Olmo, [arXiv:2210.07673 [gr-qc]]. +[8] B. Janssen, +A. Jim´enez-Cano and J. A. Orejuela, +Phys. Lett. B 795, +42-48 (2019) +doi:10.1016/j.physletb.2019.06.002 [arXiv:1903.00280 [gr-qc]]. +3 + diff --git a/TtA0T4oBgHgl3EQfEP_A/content/tmp_files/load_file.txt b/TtA0T4oBgHgl3EQfEP_A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f50c6661f3705d0a4268486af5b41bd72095aa6a --- /dev/null +++ b/TtA0T4oBgHgl3EQfEP_A/content/tmp_files/load_file.txt @@ -0,0 +1,122 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf,len=121 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='02016v1 [gr-qc] 5 Jan 2023 Comment on “A comment on metric vs metric-affine gravity” Gonzalo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Olmo1, ∗ and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Porf´ırio2, † 1Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universitat de Val`encia–CSIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Universitat de Val`encia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Burjassot-46100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Val`encia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Spain 2Departamento de F´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Universidade Federal da Para´ıba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Caixa Postal 5008,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' 58051-970,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Jo˜ao Pessoa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Para´ıba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Brazil (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' 2023) Abstract It has been recently claimed in [1] that an action of the Einstein-Palatini form plus a torsionless Pontryagin term (multiplied by a constant) represents a counterexample to the conclusions of [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' that Lovelock gravity is the only case in which the metric and metric-affine formulations of gravity are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' However, given that the Pontryagin term (multiplied by a constant) can be written as a total D-divergence, it is a textbook matter to realise that the addition of such (or any other) D-divergence only affects at the boundary, leaving invariant the field equations and its solutions, which are those of GR `a la Palatini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' We thus conclude that the example provided in [1] is not a valid counterexample of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' ∗Electronic address: gonzalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='olmo@uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='es †Electronic address: pporfirio@fisica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='ufpb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='br 1 The torsionless metric-affine action in D = 2n dimensions considered in [1], S = ˆ dDx �√−ggabRab(Γ) + 1 nθǫa1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='aDRi1 i2a1a2(Γ)Ri2 i3a3a4(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='Rin i1aD−1aD(Γ) � , (1) can be written when θ is constant as S = ˆ dDx �√−ggabRab(Γ) + θ∂aKa� , (2) with Ka as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' (2) of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' The addition of this fancy total derivative (or of any other) does not have any effect on the field equations [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Since the scalar gabRab(Γ) is just the linear in curvature term of Lovelock gravity, the action (1) can still be regarded as included in the equivalence class of Lovelock theories, up to the addition of irrelevant total divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Thus, the example presented in [1] cannot be regarded as a counterexample of the conclusions of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' If the parameter θ in (1) is promoted to the status of field, one obtains that the connection field equations are given by −∇c �√−ggi2b� +∇d �√−ggi2d� δb c−2 (∇dθ) ǫdba3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='aD−1aDRi2 i3a3a4(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='Rin caD−1aD(Γ) = 0 , (3) which naturally boil down to those of GR when θ is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' In the non-constant case, taking b = c in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' (3), one finds ∇d �√−ggi2d� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Plugging this result back into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' (3), one arrives at ∇c �√−ggi2b� + 2 (∇dθ) ǫdba3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='aD−1aDRi2 i3a3a4(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='Rin caD−1aD(Γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' (4) In particular, when D = 4, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' (3) boils down to ∇c �√−ggab� = −2 (∇dθ) ǫdba3a4Ra ca3a4(Γ), (5) which is similar to the connection equation obtained either in [4] or in [5] for the torsion- less case and by disregarding the homothetic curvature terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' The resulting equations are obviously different from those found in the metric formulation and lead to different phe- nomenology [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Thus, when θ ̸= constant, the conclusions of [2] for the action (1) hold as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Before concluding, we note that [2] neither provided an algorithm to find all the solutions for the connection in the metric-affine formulation of Lovelock theories nor ruled out the possible existence of solutions different from the Levi-Civita one [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' 2 Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' This work is supported by the Spanish Grant PID2020-116567GB- C21 funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='13039/501100011033, the project PROMETEO/2020/079 (Generalitat Valenciana), and by the European Union’s Horizon 2020 research and inno- vation programme under the H2020-MSCA-RISE-2017 Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' FunFiCO-777740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' PJP would like to thank CAPES for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' Lindstr¨om and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='002 [arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content='00280 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} +page_content=' 3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtA0T4oBgHgl3EQfEP_A/content/2301.02016v1.pdf'} diff --git a/TtE5T4oBgHgl3EQfbA8f/content/tmp_files/2301.05592v1.pdf.txt b/TtE5T4oBgHgl3EQfbA8f/content/tmp_files/2301.05592v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6ec6e4970704cadbd58486537eee0d4e3a5b5df --- /dev/null +++ b/TtE5T4oBgHgl3EQfbA8f/content/tmp_files/2301.05592v1.pdf.txt @@ -0,0 +1,482 @@ +1 + +Voltage-Controlled Magnon Transistor via Tunning Interfacial Exchange Coupling +Y. Z. Wang#, T. Y. Zhang#, J. Dong, P. Chen, C. H. Wan*, G. Q. Yu, X. F. Han* +1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of +Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China +2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of +Sciences, Beijing 100049, China +3Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China +*Email: xfhan@iphy.ac.cn; wancaihua@iphy.ac.cn +Abstract: Magnon transistors that can effectively regulate magnon transport by an electric field are +desired for magnonics which aims to provide a Joule-heating free alternative to the conventional +electronics owing to the electric neutrality of magnons (the key carriers of spin-angular momenta +in the magnonics). However, also due to their electric neutrality, magnons have no access to +directly interact with an electric field and it is thus difficult to manipulate magnon transport by +voltages straightforwardly. Here, we demonstrated a gate voltage (𝑉g) applied on a nonmagnetic +metal/magnetic insulator (NM/MI) interface that bended the energy band of the MI and then +modulated the possibility for conduction electrons in the NM to tunnel into the MI can +consequently enhance or weaken the spin-magnon conversion efficiency at the interface. A +voltage-controlled magnon transistor based on the magnon-mediated electric current drag (MECD) +effect in a Pt/Y3Fe5O12 (YIG)/Pt sandwich was then experimentally realized with 𝑉g modulating +the magnitude of the MECD signal. The obtained efficiency (the change ratio between the MECD +voltage at ±𝑉g) reached 10%/(MV/cm) at 300 K. This prototype of magnon transistor offers an +effective scheme to control magnon transport by a gate voltage. +Magnons as the collective excitation of a magnetically ordered lattice possess both spin-angular +momenta and phases but no charges [1], born an ideal information carrier for the Joule-heating- +free electronics [2,3]. In order to efficiently manipulate magnon transport, magnon transistors, as +an elementary brick for magnonics, are long-desired. Despite of great achievements in efficiently +exciting [4-10], propagating [11-13] and detecting [14-18] magnons, the electric neutrality of +magnons sets a high level of difficulty in controlling magnon transport by electric fields. + +2 + +Several magnon gating methods have been realized. The YIG/Au/YIG/Pt magnon valves [19] and +YIG/NiO/YIG/Pt magnon junctions [20-22] are gated by an external magnetic field (H). Large +(small) spin-Seebeck voltage was output by setting the two YIG layers into the parallel +(antiparallel) state by H currently, though spin-orbit torques (SOT) are potentially used to gate the +magnon valves/junctions in the future [23]. Another magnon-spin valve YIG/CoO/Co was also H- +gateable [24]. Its spin pumping voltage depends on the H-controlled parallel/antiparallel states +between YIG and Co. Another method is gating current. In the magnon transistor consisting of +three Pt stripes on top of a YIG film [25], a charge current in the leftmost Pt stripe excites a magnon +current in YIG via (i) the spin Hall effect (SHE) in Pt and (ii) the interfacial s-d coupling at the +Pt/YIG interface. The as-induced magnon current diffuses toward the rightmost Pt stripe where +the inverse process occurs, resulting in a detectable voltage. This phenomenon, featured by the +nonlocal electric induction across a MI with the help of magnons, is named as the magnon- +mediated electric current drag (MECD) effect [4,6]. A gating current flowing in the middle Pt strip +changes magnon density of YIG in the gate region and consequently modifies the MECD +efficiency. Gate voltage is advantageous in energy consumption. However, due to no direct +coupling of magnons with any electric fields, magnon transistors inherently controlled by Vg are +still missing. +Inspired by the model of Chen et al. [26,27], we realize the spin mixing conductance (𝐺↑↓) at a +NM/MI interface relies sensitively on the interfacial s-d exchange coupling. Here, we proposed a +voltage-gated magnon transistor (Fig.1(a)) where Vg across the NM/MI interface tilts downward +(upward) the energy band of the MI (Fig.1(b)), decreases (increases) the probability of electrons +penetrating into the MI (Fig.1(c)), thus weakens (strengthens) the spin-magnon conversion +efficiency at the interface and consequently changes the magnon excitation efficiency in the +magnon transistor. +We extended the model by including the Vg-induced band bending of MI via the Hamiltonian +𝐻MI = 𝑝2 2𝑚 +⁄ ++ 𝑉0 + Г𝐒 · 𝛔 + 𝑒𝑧𝑉g 𝑡 +⁄ +(1) +, where 𝑉0 is the energy barrier at the interface; Г𝐒 · 𝛔 describes the s-d coupling of electron spins +𝛔 in NM with localized moments 𝐒 in MI; 𝑒𝑧𝑉g 𝑡 +⁄ describes the conduction band bending by Vg +and t is the MI thickness (calculation details in Supplementary Materials). The predicted V0 and + +3 + +electric field E (𝐸 = 𝑉g 𝑡 +⁄ ) dependence of the real part of 𝐺↑↓ (Gr) was plotted in Fig.1(d) (taking +Fermi energy of NM 𝜀 = 5 eV and s-d coupling strength Г=0.5 eV for the Pt/YIG interface [4]). +The typical Gr-E curves at three V0 values (Fig.1(e)) suggest the positive Vg can efficiently increase +Gr and vice versa. The spin-magnon convertance at the NM/MI interface is proportional to Gr [28] +and thus the magnon current excited in MI can be modified by Vg as experimentally shown below. + +Fig.1. Mechanism of voltage-gated magnon transistor. (a) Schematics of the voltage-gated magnon transistor. A +spin current is generated by the spin Hall effect (SHE) in the bottom (B)-NM, which induces imbalanced spin +accumulation (𝜇𝑠) at the B-NM/MI interface. Due to the s-d exchange coupling at the interface, 𝜇𝑠 relaxes by +annihilating (generating) magnons in MI as 𝜇𝑠 has parallel (antiparallel) polarization to the magnetization of MI. +The excited magnon current was thus manipulated by Vg: positive (negative) Vg increases (decreases) its +magnitude. (b) Schematics of potential profile near the B-NM/MI interface under positive Vg. (c) Schematics of +probability |φ|2 at the B-NM/MI interface under positive and negative Vg. (d) The predicted V0 and E dependence +of Gr (the color scale bar in units of e2/ℏa2). (e) The E-dependence of Gr under V0=5.625, 5.675 and 5.725 eV +extracted from Fig.1(d). +The Vg-controlled magnon transistor was then experimentally achieved in a Pt(10)/YIG(80)/Pt(5 +nm) sandwich (details in Method and Supplementary Material) where Vg across the YIG was able +to tune the MECD effect. The measured voltage V along the top (T)-Pt electrode follows the + +4 + +coming 3 characteristics: (1) the angular dependence of 𝑉 = 𝑉drag cos 2𝜃 (𝜃 the angle between +spin polarization 𝛔 and magnetization M, Fig.S6(a)), (2) the linear dependence of 𝑉drag on the +input current (𝐼in) along the B-Pt electrode (Fig.S6(b,c)) and (3) the 𝑇5/2 temperature-dependence +(Fig.S6(d)), all coinciding with Ref.6&7 [7,8]. These features confirmed the MECD nature of the +measured voltage. The insulating property of YIG was also checked by 𝐼leak − 𝑉g curves +(Fig.S3(b)) with the leakage current 𝐼leak. 𝐼leak was independent on H, assuring the irrelevance of +the observed H-dependent V with 𝐼leak (Fig.S4). + +Fig.2. Voltage-controlled MECD effect. (a) The γ-dependence of ∆𝑉 with H rotated in the yoz plane and 𝐼in = +5 mA under 𝑉g = −5, 0 and + 5 V. The open circles (solid lines) are the experimental data (fitted curves by +∆𝑉 = 𝑉drag cos 2γ). (b) The 𝐼in-dependences of 𝑉drag under different 𝑉g and their linear fittings. (c) The 𝑉g- +dependence of magnon drag parameter α. Error bars for Device 1 and 2 are from the standard deviations of the +linear fittings of the 𝑉drag − 𝐼in relation and the ∆𝑉 = 𝑉drag cos 2γ fittings, respectively. The red line is the +hyperbolic tangent fitting of the 𝛼- Vg curve. (d) The γ-dependence of the difference in ∆𝑉 between 𝑉g = ±5 V. +The 𝑉g-controllability of the MECD effect is clearly shown in Fig.2. The MECD magnitude was +noticeably enhanced (weakened) under 𝑉g = +5 V (−5 V) (Fig.2(a)), which was further +confirmed by the slope change of the 𝑉drag − 𝐼in curves (Fig.2(b)). The magnon drag parameter + +V +ZtH +y +6 +I=linx +3 +0 +V +M +5V +9 +U +06 +180 +2/0 +360 +3 +5 +6 +(009) +(mA) +1.0 +0.5 +0. +10 +0 +62 +1 +0 +1 +3 +7. +5 +90 +180 +(6op)5 + +𝛼 was then calculated by +𝑉drag +𝑅T−Pt = 𝛼𝐼𝑖𝑛 . The 𝑉g -dependence of the extracted α (see method) +(Fig.2(c)) showed a clear change as 𝑉g = [−2 V, +2 V] and nearly saturated beyond the region. +The maximum 𝛼 tunability by 𝑉g ( +𝛼(𝑉g>+2𝑉)−𝛼(𝑉g<−2𝑉) +𝛼(𝑉g<−2𝑉) +) reached ~ 5% with 𝛼(𝑉g > ++2𝑉)~1.71 × 10−5 and 𝛼(𝑉g < −2𝑉)~1.63 × 10−5 . The 𝛼 -controllability by Vg was also +repeated in another Device 2. In order to trace the trend of the Vg-induced change in 𝛼, we fitted +the 𝛼-Vg curve by a hyperbolic tangent function 𝛼 = 𝑎 + 𝑏tanh(𝑐𝑉g) as shown by the red line in +Fig.2(c). Note that this fitting only mathematically impacts with |𝑏 𝑎 +⁄ | and c reflecting the +magnitude and saturation speed of the Vg-tunability, respectively. Here, for the 𝛼 -Vg curve +|𝑏 𝑎 +⁄ |=0.019 and c=-0.55 V-1. +In the following, we reveal the origin of the 𝑉g-tunability over the MECD effect. First, the 𝑉g- +dependence of the MECD effect cannot be caused by any magnon coupling possibilities with the +leakage current since Ileak increased divergently with the increase in |𝑉g| but 𝛼 nearly saturated +above ±2 V. Second, the resistance of T-Pt directly changed by Vg was negligibly small (<0.008%, +Fig.S8), also impossible to cause such significant change ~5% in the MCD signal. Third, though +negligibly small in garnets [29-31], the interfacial Dzyaloshinsky-Moriya interaction (DMI) may +introduce an additional magnon-drift velocity 𝐯DMI = 𝐳̂ × 𝐦̂ +2𝛾 +𝑀𝑠 𝐷 to influence magnon transport +with 𝐳̂ the interfacial normal, 𝐦̂ (𝑀𝑠) the magnetization direction (saturated magnetization), 𝛾 the +gyromagnetic ratio and 𝐷 a Vg-changeable parameter quantifying the DMI [32-34]. However, this +DMI mechanism, if any, would bring about a 360o period in the yoz rotation owing to the 𝐦̂- +dependence of 𝐯DMI. In stark contrast, the Δ𝑉+5 𝑉 − Δ𝑉−5 𝑉 vs 𝛾 curve (Fig.2d) shows a cos2𝛾 +symmetry (180o period), thus ruling out the DMI origin of the 𝑉g controllability. +To be more specific, the MECD effect can be explicitly expressed as below [4,6]: +𝐣e +T−Pt ∝ 𝜃SH +top𝜃SH +bottom𝐺S +s−m𝐺S +m−s𝛔 × (𝐌 × 𝐣e +B−Pt) +(2) +here, 𝐣e +T−Pt (𝐣e +B−Pt) is the induced (input) charge current density along the T-Pt (B-Pt) electrode, +𝜃SH +top(bottom) is the spin Hall angle of the top (bottom) Pt electrode, 𝐺S +s−m(𝐺S +m−s) is the effective +spin-magnon (magnon-spin) convertance at the B-Pt/YIG (YIG/T-Pt) interface, 𝛔 is the spin +polarization perpendicular to 𝐣e +T−Pt and M is the YIG magnetization. Ruling out the above 3 + +6 + +reasons, the MECD voltage can still be potentially manipulated by 𝑉g in the following scenarios: +(1) 𝑉g-induced changes in the effective magnetization of YIG, (2) the spin Hall angles (𝜃SH) of Pt +or (3) the spin-magnon conversion efficiency across the B-Pt/YIG or YIG/T-Pt interfaces. +Hereafter, we experimentally check their possibilities one-by-one. + +Fig.3. Schematics setups for (a) spin pumping measurement where the spin pumping voltage (𝑉SP) was picked +up along the B-Pt stripe with H perpendicular to the stripe and 𝑉g applied across the sandwich and for (e) SMR +measurement where the resistance change Δ𝑅B of the B-Pt stripe was measured with H rotated in the yoz plane. +(b) The H-dependence of the normalized 𝑉SP(𝐻)/𝑉SP +max under different rf frequencies (𝑓) and 𝑉g=-3.9, 0 and ++3.9 V. (c) The 𝐻-dependence of 𝑉SP at 𝑓 = 5 GHz and 𝑉g = −3.9, 0, +3.9 V. (Error bars from standard +deviation by fitting 𝑉SP − 𝐻 curves with the Lorentzian function.) (d) The 𝛾-dependences of the Δ𝑅B (open +circles) and their ∆𝑅B = Δ𝑅SMR cos 2γ fittings. (f) The resonance field (𝐻r ) dependence of f under 𝑉g = +−3.9, 0 and + 3.9 V (open circles) and their Kittle fittings. The 𝑉g-dependence of (g) the peak value of 𝑉SP − +𝐻 curve (𝑉SP +peak) under 𝑓 = 5 GHz and (h) the SMR ratio. (Error bars from standard deviation of the ∆𝑅B = +Δ𝑅SMR cos 2γ fittings.) Red lines in Fig.3(c&d) are the hyperbolic tangent fitting of the 𝑉SP +peak-Vg and SMR ratio- +Vg curves, respectively. +To investigate the 𝑉g-dependence of Ms, we conducted spin pumping experiments (experimental +details in Method). The spin pumping voltage VSP picked up in the B-Pt electrode at various Vg is +exhibited in Fig.3(a). The H-dependences of a normalized VSP at different f and 𝑉g show no +noticeable changes (variation<0.3%) in the resonance field (𝐻r) (Fig.3(b)) and the overlapped +Kittle fittings manifested no changes in the magnetization and anisotropy of YIG under Vg. +Interestingly, the magnitude of 𝑉SP +peak changed by Vg (Fig.3(c)). The tunability defined by + +(a) +3.8 +H/ +1 3.9 +: +12 + 3 +.355 +rf field ++o +3.53 +4.0 +00 + + +i" +1m +2.101 +2.5 +180 +360 +(e) +1.9 +3.5 +33 +13. +Z +TH +.3.3 ++6 +3 +38.3.2 +x +8.0 +.1 +1.3 +1.83. +3 + tkx7 + +𝑉SP +peak(𝑉g=+3.9 𝑉)−𝑉SP +peak(𝑉g=−3.9 𝑉) +𝑉SP +peak(𝑉g=−3.9 𝑉) + was also ~5 %. Moreover, the 𝑉SP +peak-Vg tendency seemed similar to +the Vdrag-Vg relation, with |𝑏 𝑎 +⁄ |=0.021 and c=-0.54 V-1 extracted from the hyperbolic tangent +fitting. We also tested VSP along the T-Pt stripe, which had ideally identical Hr but opposite polarity +with the B-Pt stripe (Fig.S7(a)). However, 𝑉SP +peak was not changed by Vg for the T-Pt detector +(Fig.S7(c)). Since spin currents were both pumped out from the sandwiched YIG, the different Vg- +controllability on VSP for the B-Pt and the T-Pt detectors strongly hinted an interfacial gating origin +instead of any bulk YIG reasons. +The following Vg-dependent spin Hall magnetoresistance (SMR) effect also supported this +interfacial claim. Since SMR originates from spin-transfer at interfaces and shunted by a thick Pt +layer, we fabricated another Pt(4)/YIG(80)/Pt(5 nm) sandwich. Its ΔRB-Pt-γ relation at various Vg +and the summarized Vg-dependence of the SMR ratio are shown in Fig.3(d,h). The similar +coefficients of |𝑏 𝑎 +⁄ |=0.022 and c=-0.51 V-1 were obtained from the hyperbolic tangent fitting (the +red line in Fig.3(h)), illustrating the Vg-tunability on the SMR ratio also followed the similar trend +as the Vg-dependence of 𝛼 and 𝑉SP +peak. The SMR effect in the T-Pt stripe was independent on Vg +(Fig.S7(b,d)). + +Fig.4. (a) The γ-dependence of ΔV(Vg=5V)-ΔV(Vg=-5V) under different T. (b) The T-dependence of the +difference in the magnon drag parameter Δα=α(Vg=5 V)-α(Vg=-5 V) between Vg=±5 V. The solid lines are +obtained by fitting data using ∆𝛼 = 𝐴𝑒−∆𝐸 kB𝑇 +⁄ +. (c) The calculated Vg-dependence of Gr by taking redistributed +voltage on the contact resistance (Rcontact) into consideration. The red line is the hyperbolic tangent fitting result. +After the above analysis we have narrowed possibility for the Vg-controlled MCD effect to (1) a +Vg-changeable spin Hall angle in B-Pt or (2) a Vg-controllable spin-magnon conversion efficiency +across the B-Pt/YIG interface. If the bulk spin Hall angle was modulated by Vg, we would not +expect a substantial difference between the B-Pt and T-Pt stripes since they were both textured in + +2.2 +6.38 +8 +1.0 +A::.136V +0.3 +0. +240 230 80 2/0 280 290 300 +(K8 + +the (111) orientation (Fig.S5). The Vg-independent resistivity of B-Pt (Fig.S8) did not support a +Vg-modulated spin Hall angle of the B-Pt as well [35]. +We further measured the Vg-controlled MECD effect at different T. The Vg-tunability over the +MECD effect was strongly depended on T from 240 K to 300 K (Fig.4(a)). The difference in α +under Vg=±5 V increased by a factor of 3.5 (from 0.6×10-7 at 240 K to 2.1×10-7 at 300 K) (Fig.4(b)). +This strong T-dependence cannot favor the possibility of a Vg-controlled intrinsic spin Hall +conductivity (𝜎SH +int) since the electronic structure of Pt varies little with T. Nevertheless, the strong +T-dependence can be naturally obtained as following. According to the spin-mixing conductance +model across a NM/MI interface [4,6,26,27,36], the spin-torque-transfer efficiency and the spin- +magnon convertance both depend on the s-d exchange coupling strength and thus probability of +electrons penetrating into the insulating YIG as evanescent states. The probability certainly +depends on the interface barrier (thus Vg) and also T since T determines the kinetic energy of +electrons in YIG. Supposing (1) ±5 V gating leads to the similar band bending at different T and +(2) the classic thermal activation theory holds, we would expect an exponential T-dependence +(Arrhenius law [37]) for the MECD coefficient. Fig.4(b) shows the fitting well matched the +experimental data and the caused difference in the effective tunneling barrier by ±Vg reached 0.13 +eV. Since the spin-mixing conductance depended on the s-d coupling in the same way as the spin- +magnon convertance, the SMR shared the same Vg-dependence as the MECD effect naturally. +Band bending at interfaces relies on charged defect density which pins the Fermi level and +influences bending degree, which probably accounts for the observation that a smoother and well- +crystallized B-Pt/YIG interface (evidenced by a sharper electron diffraction pattern at this region) +contributed to the Vg-controllability. +Now the above experimental data persuade us to attribute the Vg-controlled MECD effect to the +Vg-induced changes in the spin-magnon conductance across the B-Pt/YIG interface. However, the +measured Vg-α deviated from the theoretical prediction by the saturation trend at large Vg. We +attribute this deviation to the redistributed voltage on the contact resistance (Rcontact) since Ileak +increases divergently with Vg. In practice, we rewrote the Hamiltonian in YIG 𝐻MI = 𝑝2 2𝑚 +⁄ ++ +𝑉0 + Г𝐒 · 𝛔 − 𝑒𝑧 +𝑉g−𝐼leak∙𝑅contact +𝑡 +, considering the voltage dropped on Rcontact. The calculated Gr-Vg +relation (Fig.4(c)) using parameters for Pt/YIG: Fermi energy 𝜀 = 5 eV, 𝑉0 = 5.5 eV, Г=0.5 eV [4] +and 𝑅contact = 15 MΩ agrees well with experiment. Gr increased (decreased) with positive + +9 + +(negative) Vg and saturated at 𝑉g ≈ ±2 V . The calculated Gr change by Vg saturated at +13%/(MV/cm), also in a quantitative agreement with the experiment value ~10%/(MV/cm). The +calculated result can also be well fitted with the hyperbolic tangent function with |𝑏 𝑎 +⁄ |=0.041 and +c=-0.45 V-1, which was the reason why we had used the hyperbolic tangent fitting to +mathematically trace the Vg-dependences of α, 𝑉SP +peak and SMR ratio. +In summary we have experimentally demonstrated a field-effect magnon transistor based on the +MECD effect in the Pt/YIG/Pt sandwich. With the voltage-induced band bending of YIG, the +energy profile of the B-Pt/YIG interfacial barrier and consequently its spin-magnon convertance +was modulated. In this sense, the MECD effect was directly modulated by the gate voltage. Our +finding promises direct modulation of spin-magnon conversion by electric fields, which shows a +feasible pathway toward electrically controllable magnonics. +References: +[1] F. Bloch, Z. Angew. Phys. 61, 206 (1930). +[2] A. V. Chumak, A. A. Serga, and B. Hillebrands, Nat. Commun. 5, 4700 (2014). +[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). +[4] S. S. L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603 (2012). +[5] S. S. L. Zhang and S. Zhang, Phys. Rev. B 86, 214424 (2012). +[6] S. M. Rezende, R. L. Rodríguez-Suárez, R. O. Cunha, A. R. Rodrigues, F. L. A. Machado, G. A. +Fonseca Guerra, J. C. Lopez Ortiz, and A. Azevedo, Phys. Rev. B 89, 014416 (2014). +[7] H. Wu, C. H. Wan, X. Zhang, Z. H. 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Acta 386, 91 (2002). +Acknowledgements: This work was financial supported by the National Key Research and +Development Program of China [MOST Grant No. 2022YFA1402800], the National Natural +Science Foundation of China [NSFC, Grant No. 51831012, 12134017], and partially supported by +the Strategic Priority Research Program (B) of Chinese Academy of Sciences [CAS Grant No. +XDB33000000, Youth Innovation Promotion Association of CAS (2020008)]. +Contributions: X.F.H. led and was involved in all aspects of the project. Y.Z.W., J. D. and P. C. +deposited stacks and fabricated devices. Y.Z.W. and C.H.W. conducted magnetic and transport +property measurement. T.Y.Z., C.H.W. and Y.Z.W. contributed to modelling and theoretical +analysis. C.H.W., Y.Z.W., T. Y. Z., G. Q. Y. and X.F.H. wrote the paper. X.F.H. and C.H.W. +supervised and designed the experiments. All the authors contributed to data mining and analysis. +Conflict of Interests: The authors declare no competing interests. +Methods: +Sample Preparation: +The Si/SiO2//Pt(10)/Y3Fe5O12 (YIG)(80)/Pt(5 nm) heterostructures are deposited by ultrahigh +vacuum magnetron sputtering system (ULVAC-MPS-400-HC7) with a base pressure＜5×10-6 Pa. +The bottom Pt Hall bar (B-Pt) with dimensions of 20×200 μm2 was first fabricated on substrates + +11 + +by standard photolithography, followed by deposition of 80 nm YIG film. After deposition, a +high-temperature annealing was carried out in an oxygen atmosphere to improve the crystalline +quality of both YIG and Pt/YIG interface. Finally, another round of deposition and +photolithography was carried out to fabricate top Pt Hall bar (T-Pt) with same dimensions. The +terminal of T-Pt and B-Pt Hall bars are designed away from each other allow two Hall bars being +input and detected independently. For the spin pumping device, B-Pt and T-Pt are fabricated into +two independent stripes with dimensions of 10×360 μm2 by the above mentioned method. And an +80 nm Au co-planar wave guide (CPW) was deposited afterwards. The two Pt strips are placed in +the gap of the CPW. +Measurement of Magnetic Property: +The M-H hysteresis was measured with a vibrating sample magnetometer (VSM, MicroSense EZ- +9) with field applied parallel to the film plane (IP curve) or perpendicular to film plane (OOP +curve). +Measurement of Transport Property: +All the magnon mediate current drag (MCD) and spin magnetoresistance (SMR) test were carried +out in a physical property measurement system (PPMS-9 T, Quantum design) with magnetic field +up to 9 T and temperatures down to 1.8 K. During measurements, the input current was supplied +by a Keithley 2400 source-meter while a Keithley 2182 nanovoltmeter detected the corresponding +voltage. The gate voltage was provided by another Keithley 2400 across the Hall channel of T-Pt +and B-Pt (grounded) Hall bars. The magnetic field was fixed at 1 T and sample rotated in xoy, xoz +or yoz plane. +For angular dependent of MCD signal measurements, the input current (Iin) was applied in the long +axis of B-Pt Hall bar and the voltage signal (ΔV) was picked up alone the long axis of T-Pt Hall +bar. Then the magnitude of MCD voltage Vdrag under certain Iin was obtained by fitting the ∆𝑉 − 𝛾 +curves measured at different 𝐼in with ∆𝑉 = 𝑉drag cos 2𝛾. The magnon drag parameter was then +calculated by 𝛼 ≡ +𝑉drag +𝐼in𝑅T−Pt, where 𝑅T−Pt is the resistance of T-Pt electrode. + +12 + +And for SMR test the resistance of the B-Pt (T-Pt) electrode was measured by four-terminal +method, and the angular dependence of change in RPt (ΔRPt-γ) was well fitted by ∆𝑅Pt = +𝑅SMR cos 2𝛾 and the SMR ratio was thus obtained by |𝑅SMR 𝑅Pt +⁄ +|. +The spin pumping test was carried out at room temperature in a home-build electromagnet with +magnetic up to ~ 3500 Oe. A signal generator (ROHDE&SCHWARZ SMB 100A) supplies a +microwave signal modulated with a 1.172 kHz signal to CPW and the voltage signal was picked +up by a lock-in amplifier (Stanford SR830), while external magnetic field H applied perpendicular +to the direction that spin pumping voltage was picked up. To minimize interference, 𝑉g was +provided by dry batteries during spin pumping measurements. +Data availability: The data that support the findings of this study are available from the +corresponding author upon reasonable request. + diff --git a/TtE5T4oBgHgl3EQfbA8f/content/tmp_files/load_file.txt b/TtE5T4oBgHgl3EQfbA8f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c7566904d1615cb5ce97714d61623c866ceff266 --- /dev/null +++ b/TtE5T4oBgHgl3EQfbA8f/content/tmp_files/load_file.txt @@ -0,0 +1,706 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf,len=705 +page_content='1 Voltage-Controlled Magnon Transistor via Tunning Interfacial Exchange Coupling Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Wang#, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Zhang#, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Dong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Wan*, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Han* 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, China 2Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China 3Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China Email: xfhan@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' wancaihua@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='cn Abstract: Magnon transistors that can effectively regulate magnon transport by an electric field are desired for magnonics which aims to provide a Joule-heating free alternative to the conventional electronics owing to the electric neutrality of magnons (the key carriers of spin-angular momenta in the magnonics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' However, also due to their electric neutrality, magnons have no access to directly interact with an electric field and it is thus difficult to manipulate magnon transport by voltages straightforwardly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Here, we demonstrated a gate voltage (𝑉g) applied on a nonmagnetic metal/magnetic insulator (NM/MI) interface that bended the energy band of the MI and then modulated the possibility for conduction electrons in the NM to tunnel into the MI can consequently enhance or weaken the spin-magnon conversion efficiency at the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' A voltage-controlled magnon transistor based on the magnon-mediated electric current drag (MECD) effect in a Pt/Y3Fe5O12 (YIG)/Pt sandwich was then experimentally realized with 𝑉g modulating the magnitude of the MECD signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The obtained efficiency (the change ratio between the MECD voltage at ±𝑉g) reached 10%/(MV/cm) at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' This prototype of magnon transistor offers an effective scheme to control magnon transport by a gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Magnons as the collective excitation of a magnetically ordered lattice possess both spin-angular momenta and phases but no charges [1], born an ideal information carrier for the Joule-heating- free electronics [2,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In order to efficiently manipulate magnon transport, magnon transistors, as an elementary brick for magnonics, are long-desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Despite of great achievements in efficiently exciting [4-10], propagating [11-13] and detecting [14-18] magnons, the electric neutrality of magnons sets a high level of difficulty in controlling magnon transport by electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 2 Several magnon gating methods have been realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The YIG/Au/YIG/Pt magnon valves [19] and YIG/NiO/YIG/Pt magnon junctions [20-22] are gated by an external magnetic field (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Large (small) spin-Seebeck voltage was output by setting the two YIG layers into the parallel (antiparallel) state by H currently, though spin-orbit torques (SOT) are potentially used to gate the magnon valves/junctions in the future [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Another magnon-spin valve YIG/CoO/Co was also H- gateable [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Its spin pumping voltage depends on the H-controlled parallel/antiparallel states between YIG and Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Another method is gating current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In the magnon transistor consisting of three Pt stripes on top of a YIG film [25], a charge current in the leftmost Pt stripe excites a magnon current in YIG via (i) the spin Hall effect (SHE) in Pt and (ii) the interfacial s-d coupling at the Pt/YIG interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The as-induced magnon current diffuses toward the rightmost Pt stripe where the inverse process occurs, resulting in a detectable voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' This phenomenon, featured by the nonlocal electric induction across a MI with the help of magnons, is named as the magnon- mediated electric current drag (MECD) effect [4,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' A gating current flowing in the middle Pt strip changes magnon density of YIG in the gate region and consequently modifies the MECD efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Gate voltage is advantageous in energy consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' However, due to no direct coupling of magnons with any electric fields, magnon transistors inherently controlled by Vg are still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Inspired by the model of Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' [26,27], we realize the spin mixing conductance (𝐺↑↓) at a NM/MI interface relies sensitively on the interfacial s-d exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Here, we proposed a voltage-gated magnon transistor (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1(a)) where Vg across the NM/MI interface tilts downward (upward) the energy band of the MI (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1(b)), decreases (increases) the probability of electrons penetrating into the MI (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1(c)), thus weakens (strengthens) the spin-magnon conversion efficiency at the interface and consequently changes the magnon excitation efficiency in the magnon transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' We extended the model by including the Vg-induced band bending of MI via the Hamiltonian 𝐻MI = 𝑝2 2𝑚 ⁄ + 𝑉0 + Г𝐒 · 𝛔 + 𝑒𝑧𝑉g 𝑡 ⁄ (1) , where 𝑉0 is the energy barrier at the interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Г𝐒 · 𝛔 describes the s-d coupling of electron spins 𝛔 in NM with localized moments 𝐒 in MI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 𝑒𝑧𝑉g 𝑡 ⁄ describes the conduction band bending by Vg and t is the MI thickness (calculation details in Supplementary Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The predicted V0 and 3 electric field E (𝐸 = 𝑉g 𝑡 ⁄ ) dependence of the real part of 𝐺↑↓ (Gr) was plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1(d) (taking Fermi energy of NM 𝜀 = 5 eV and s-d coupling strength Г=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 eV for the Pt/YIG interface [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The typical Gr-E curves at three V0 values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1(e)) suggest the positive Vg can efficiently increase Gr and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The spin-magnon convertance at the NM/MI interface is proportional to Gr [28] and thus the magnon current excited in MI can be modified by Vg as experimentally shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Mechanism of voltage-gated magnon transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (a) Schematics of the voltage-gated magnon transistor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' A spin current is generated by the spin Hall effect (SHE) in the bottom (B)-NM, which induces imbalanced spin accumulation (𝜇𝑠) at the B-NM/MI interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Due to the s-d exchange coupling at the interface, 𝜇𝑠 relaxes by annihilating (generating) magnons in MI as 𝜇𝑠 has parallel (antiparallel) polarization to the magnetization of MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The excited magnon current was thus manipulated by Vg: positive (negative) Vg increases (decreases) its magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (b) Schematics of potential profile near the B-NM/MI interface under positive Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (c) Schematics of probability |φ|2 at the B-NM/MI interface under positive and negative Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (d) The predicted V0 and E dependence of Gr (the color scale bar in units of e2/ℏa2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (e) The E-dependence of Gr under V0=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='625, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='675 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='725 eV extracted from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The Vg-controlled magnon transistor was then experimentally achieved in a Pt(10)/YIG(80)/Pt(5 nm) sandwich (details in Method and Supplementary Material) where Vg across the YIG was able to tune the MECD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The measured voltage V along the top (T)-Pt electrode follows the 4 coming 3 characteristics: (1) the angular dependence of 𝑉 = 𝑉drag cos 2𝜃 (𝜃 the angle between spin polarization 𝛔 and magnetization M, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S6(a)), (2) the linear dependence of 𝑉drag on the input current (𝐼in) along the B-Pt electrode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S6(b,c)) and (3) the 𝑇5/2 temperature-dependence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S6(d)), all coinciding with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='6&7 [7,8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' These features confirmed the MECD nature of the measured voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The insulating property of YIG was also checked by 𝐼leak − 𝑉g curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S3(b)) with the leakage current 𝐼leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 𝐼leak was independent on H, assuring the irrelevance of the observed H-dependent V with 𝐼leak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Voltage-controlled MECD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (a) The γ-dependence of ∆𝑉 with H rotated in the yoz plane and 𝐼in = 5 mA under 𝑉g = −5, 0 and + 5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The open circles (solid lines) are the experimental data (fitted curves by ∆𝑉 = 𝑉drag cos 2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (b) The 𝐼in-dependences of 𝑉drag under different 𝑉g and their linear fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (c) The 𝑉g- dependence of magnon drag parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Error bars for Device 1 and 2 are from the standard deviations of the linear fittings of the 𝑉drag − 𝐼in relation and the ∆𝑉 = 𝑉drag cos 2γ fittings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The red line is the hyperbolic tangent fitting of the 𝛼- Vg curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (d) The γ-dependence of the difference in ∆𝑉 between 𝑉g = ±5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The 𝑉g-controllability of the MECD effect is clearly shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The MECD magnitude was noticeably enhanced (weakened) under 𝑉g = +5 V (−5 V) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2(a)), which was further confirmed by the slope change of the 𝑉drag − 𝐼in curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The magnon drag parameter V ZtH y 6 I=linx 3 0 V M 5V 9 U 06 180 2/0 360 3 5 6 (009) (mA) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 10 0 62 1 0 1 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 5 90 180 (6op)5 𝛼 was then calculated by 𝑉drag 𝑅T−Pt = 𝛼𝐼𝑖𝑛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The 𝑉g -dependence of the extracted α (see method) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2(c)) showed a clear change as 𝑉g = [−2 V, +2 V] and nearly saturated beyond the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The maximum 𝛼 tunability by 𝑉g ( 𝛼(𝑉g>+2𝑉)−𝛼(𝑉g<−2𝑉) 𝛼(𝑉g<−2𝑉) ) reached ~ 5% with 𝛼(𝑉g > +2𝑉)~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='71 × 10−5 and 𝛼(𝑉g < −2𝑉)~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='63 × 10−5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The 𝛼 -controllability by Vg was also repeated in another Device 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In order to trace the trend of the Vg-induced change in 𝛼, we fitted the 𝛼-Vg curve by a hyperbolic tangent function 𝛼 = 𝑎 + 𝑏tanh(𝑐𝑉g) as shown by the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Note that this fitting only mathematically impacts with |𝑏 𝑎 ⁄ | and c reflecting the magnitude and saturation speed of the Vg-tunability, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Here, for the 𝛼 -Vg curve |𝑏 𝑎 ⁄ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='019 and c=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='55 V-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In the following, we reveal the origin of the 𝑉g-tunability over the MECD effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' First, the 𝑉g- dependence of the MECD effect cannot be caused by any magnon coupling possibilities with the leakage current since Ileak increased divergently with the increase in |𝑉g| but 𝛼 nearly saturated above ±2 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Second, the resistance of T-Pt directly changed by Vg was negligibly small (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='008%, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S8), also impossible to cause such significant change ~5% in the MCD signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Third, though negligibly small in garnets [29-31], the interfacial Dzyaloshinsky-Moriya interaction (DMI) may introduce an additional magnon-drift velocity 𝐯DMI = 𝐳̂ × 𝐦̂ 2𝛾 𝑀𝑠 𝐷 to influence magnon transport with 𝐳̂ the interfacial normal, 𝐦̂ (𝑀𝑠) the magnetization direction (saturated magnetization), 𝛾 the gyromagnetic ratio and 𝐷 a Vg-changeable parameter quantifying the DMI [32-34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' However, this DMI mechanism, if any, would bring about a 360o period in the yoz rotation owing to the 𝐦̂- dependence of 𝐯DMI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In stark contrast, the Δ𝑉+5 𝑉 − Δ𝑉−5 𝑉 vs 𝛾 curve (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2d) shows a cos2𝛾 symmetry (180o period), thus ruling out the DMI origin of the 𝑉g controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' To be more specific,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' the MECD effect can be explicitly expressed as below [4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='6]: 𝐣e T−Pt ∝ 𝜃SH top𝜃SH bottom𝐺S s−m𝐺S m−s𝛔 × (𝐌 × 𝐣e B−Pt) (2) here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 𝐣e T−Pt (𝐣e B−Pt) is the induced (input) charge current density along the T-Pt (B-Pt) electrode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 𝜃SH top(bottom) is the spin Hall angle of the top (bottom) Pt electrode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 𝐺S s−m(𝐺S m−s) is the effective spin-magnon (magnon-spin) convertance at the B-Pt/YIG (YIG/T-Pt) interface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 𝛔 is the spin polarization perpendicular to 𝐣e T−Pt and M is the YIG magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Ruling out the above 3 6 reasons, the MECD voltage can still be potentially manipulated by 𝑉g in the following scenarios: (1) 𝑉g-induced changes in the effective magnetization of YIG, (2) the spin Hall angles (𝜃SH) of Pt or (3) the spin-magnon conversion efficiency across the B-Pt/YIG or YIG/T-Pt interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Hereafter, we experimentally check their possibilities one-by-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Schematics setups for (a) spin pumping measurement where the spin pumping voltage (𝑉SP) was picked up along the B-Pt stripe with H perpendicular to the stripe and 𝑉g applied across the sandwich and for (e) SMR measurement where the resistance change Δ𝑅B of the B-Pt stripe was measured with H rotated in the yoz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (b) The H-dependence of the normalized 𝑉SP(𝐻)/𝑉SP max under different rf frequencies (𝑓) and 𝑉g=-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9, 0 and +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (c) The 𝐻-dependence of 𝑉SP at 𝑓 = 5 GHz and 𝑉g = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9, 0, +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (Error bars from standard deviation by fitting 𝑉SP − 𝐻 curves with the Lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=') (d) The 𝛾-dependences of the Δ𝑅B (open circles) and their ∆𝑅B = Δ𝑅SMR cos 2γ fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (f) The resonance field (𝐻r ) dependence of f under 𝑉g = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9, 0 and + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 V (open circles) and their Kittle fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The 𝑉g-dependence of (g) the peak value of 𝑉SP − 𝐻 curve (𝑉SP peak) under 𝑓 = 5 GHz and (h) the SMR ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (Error bars from standard deviation of the ∆𝑅B = Δ𝑅SMR cos 2γ fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=') Red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3(c&d) are the hyperbolic tangent fitting of the 𝑉SP peak-Vg and SMR ratio- Vg curves, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' To investigate the 𝑉g-dependence of Ms, we conducted spin pumping experiments (experimental details in Method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The spin pumping voltage VSP picked up in the B-Pt electrode at various Vg is exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The H-dependences of a normalized VSP at different f and 𝑉g show no noticeable changes (variation<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3%) in the resonance field (𝐻r) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3(b)) and the overlapped Kittle fittings manifested no changes in the magnetization and anisotropy of YIG under Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Interestingly, the magnitude of 𝑉SP peak changed by Vg (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The tunability defined by (a) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='8 H/ 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 : 12 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='355 rf field +o 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='53 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='0 00 i" 1m 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='101 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 180 360 (e) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 33 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Z TH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3 +6 3 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2 x 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 3 tkx7 𝑉SP peak(𝑉g=+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 𝑉)−𝑉SP peak(𝑉g=−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 𝑉) 𝑉SP peak(𝑉g=−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='9 𝑉) was also ~5 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Moreover, the 𝑉SP peak-Vg tendency seemed similar to the Vdrag-Vg relation, with |𝑏 𝑎 ⁄ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='021 and c=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='54 V-1 extracted from the hyperbolic tangent fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' We also tested VSP along the T-Pt stripe, which had ideally identical Hr but opposite polarity with the B-Pt stripe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S7(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' However, 𝑉SP peak was not changed by Vg for the T-Pt detector (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S7(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Since spin currents were both pumped out from the sandwiched YIG, the different Vg- controllability on VSP for the B-Pt and the T-Pt detectors strongly hinted an interfacial gating origin instead of any bulk YIG reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The following Vg-dependent spin Hall magnetoresistance (SMR) effect also supported this interfacial claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Since SMR originates from spin-transfer at interfaces and shunted by a thick Pt layer, we fabricated another Pt(4)/YIG(80)/Pt(5 nm) sandwich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Its ΔRB-Pt-γ relation at various Vg and the summarized Vg-dependence of the SMR ratio are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3(d,h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The similar coefficients of |𝑏 𝑎 ⁄ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='022 and c=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='51 V-1 were obtained from the hyperbolic tangent fitting (the red line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3(h)), illustrating the Vg-tunability on the SMR ratio also followed the similar trend as the Vg-dependence of 𝛼 and 𝑉SP peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The SMR effect in the T-Pt stripe was independent on Vg (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S7(b,d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (a) The γ-dependence of ΔV(Vg=5V)-ΔV(Vg=-5V) under different T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (b) The T-dependence of the difference in the magnon drag parameter Δα=α(Vg=5 V)-α(Vg=-5 V) between Vg=±5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The solid lines are obtained by fitting data using ∆𝛼 = 𝐴𝑒−∆𝐸 kB𝑇 ⁄ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' (c) The calculated Vg-dependence of Gr by taking redistributed voltage on the contact resistance (Rcontact) into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The red line is the hyperbolic tangent fitting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' After the above analysis we have narrowed possibility for the Vg-controlled MCD effect to (1) a Vg-changeable spin Hall angle in B-Pt or (2) a Vg-controllable spin-magnon conversion efficiency across the B-Pt/YIG interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' If the bulk spin Hall angle was modulated by Vg, we would not expect a substantial difference between the B-Pt and T-Pt stripes since they were both textured in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='38 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='0 A::.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='136V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 240 230 80 2/0 280 290 300 (K8 the (111) orientation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The Vg-independent resistivity of B-Pt (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='S8) did not support a Vg-modulated spin Hall angle of the B-Pt as well [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' We further measured the Vg-controlled MECD effect at different T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The Vg-tunability over the MECD effect was strongly depended on T from 240 K to 300 K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The difference in α under Vg=±5 V increased by a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='6×10-7 at 240 K to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='1×10-7 at 300 K) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' This strong T-dependence cannot favor the possibility of a Vg-controlled intrinsic spin Hall conductivity (𝜎SH int) since the electronic structure of Pt varies little with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Nevertheless, the strong T-dependence can be naturally obtained as following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' According to the spin-mixing conductance model across a NM/MI interface [4,6,26,27,36], the spin-torque-transfer efficiency and the spin- magnon convertance both depend on the s-d exchange coupling strength and thus probability of electrons penetrating into the insulating YIG as evanescent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The probability certainly depends on the interface barrier (thus Vg) and also T since T determines the kinetic energy of electrons in YIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Supposing (1) ±5 V gating leads to the similar band bending at different T and (2) the classic thermal activation theory holds, we would expect an exponential T-dependence (Arrhenius law [37]) for the MECD coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='4(b) shows the fitting well matched the experimental data and the caused difference in the effective tunneling barrier by ±Vg reached 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='13 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Since the spin-mixing conductance depended on the s-d coupling in the same way as the spin- magnon convertance, the SMR shared the same Vg-dependence as the MECD effect naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Band bending at interfaces relies on charged defect density which pins the Fermi level and influences bending degree, which probably accounts for the observation that a smoother and well- crystallized B-Pt/YIG interface (evidenced by a sharper electron diffraction pattern at this region) contributed to the Vg-controllability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Now the above experimental data persuade us to attribute the Vg-controlled MECD effect to the Vg-induced changes in the spin-magnon conductance across the B-Pt/YIG interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' However, the measured Vg-α deviated from the theoretical prediction by the saturation trend at large Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' We attribute this deviation to the redistributed voltage on the contact resistance (Rcontact) since Ileak increases divergently with Vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In practice, we rewrote the Hamiltonian in YIG 𝐻MI = 𝑝2 2𝑚 ⁄ + 𝑉0 + Г𝐒 · 𝛔 − 𝑒𝑧 𝑉g−𝐼leak∙𝑅contact 𝑡 , considering the voltage dropped on Rcontact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The calculated Gr-Vg relation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='4(c)) using parameters for Pt/YIG: Fermi energy 𝜀 = 5 eV, 𝑉0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 eV, Г=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='5 eV [4] and 𝑅contact = 15 MΩ agrees well with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Gr increased (decreased) with positive 9 (negative) Vg and saturated at 𝑉g ≈ ±2 V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The calculated Gr change by Vg saturated at 13%/(MV/cm), also in a quantitative agreement with the experiment value ~10%/(MV/cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The calculated result can also be well fitted with the hyperbolic tangent function with |𝑏 𝑎 ⁄ |=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='041 and c=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='45 V-1, which was the reason why we had used the hyperbolic tangent fitting to mathematically trace the Vg-dependences of α, 𝑉SP peak and SMR ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In summary we have experimentally demonstrated a field-effect magnon transistor based on the MECD effect in the Pt/YIG/Pt sandwich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' With the voltage-induced band bending of YIG, the energy profile of the B-Pt/YIG interfacial barrier and consequently its spin-magnon convertance was modulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' In this sense, the MECD effect was directly modulated by the gate voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Our finding promises direct modulation of spin-magnon conversion by electric fields, which shows a feasible pathway toward electrically controllable magnonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' References: [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Bloch, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 61, 206 (1930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' [2] A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Hillebrands, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 11, 453 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Zhang and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Ryu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Lee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' McMichael, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Stiles, Phys.' 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A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Galwey and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Brown, Thermochim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Acta 386, 91 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Acknowledgements: This work was financial supported by the National Key Research and Development Program of China [MOST Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 2022YFA1402800], the National Natural Science Foundation of China [NSFC, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 51831012, 12134017], and partially supported by the Strategic Priority Research Program (B) of Chinese Academy of Sciences [CAS Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' XDB33000000, Youth Innovation Promotion Association of CAS (2020008)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Contributions: X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' led and was involved in all aspects of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' deposited stacks and fabricated devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' conducted magnetic and transport property measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' contributed to modelling and theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' supervised and designed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' All the authors contributed to data mining and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Conflict of Interests: The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Methods: Sample Preparation: The Si/SiO2//Pt(10)/Y3Fe5O12 (YIG)(80)/Pt(5 nm) heterostructures are deposited by ultrahigh vacuum magnetron sputtering system (ULVAC-MPS-400-HC7) with a base pressure＜5×10-6 Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The bottom Pt Hall bar (B-Pt) with dimensions of 20×200 μm2 was first fabricated on substrates 11 by standard photolithography, followed by deposition of 80 nm YIG film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' After deposition, a high-temperature annealing was carried out in an oxygen atmosphere to improve the crystalline quality of both YIG and Pt/YIG interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Finally, another round of deposition and photolithography was carried out to fabricate top Pt Hall bar (T-Pt) with same dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The terminal of T-Pt and B-Pt Hall bars are designed away from each other allow two Hall bars being input and detected independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' For the spin pumping device, B-Pt and T-Pt are fabricated into two independent stripes with dimensions of 10×360 μm2 by the above mentioned method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' And an 80 nm Au co-planar wave guide (CPW) was deposited afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The two Pt strips are placed in the gap of the CPW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Measurement of Magnetic Property: The M-H hysteresis was measured with a vibrating sample magnetometer (VSM, MicroSense EZ- 9) with field applied parallel to the film plane (IP curve) or perpendicular to film plane (OOP curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Measurement of Transport Property: All the magnon mediate current drag (MCD) and spin magnetoresistance (SMR) test were carried out in a physical property measurement system (PPMS-9 T, Quantum design) with magnetic field up to 9 T and temperatures down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='8 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' During measurements, the input current was supplied by a Keithley 2400 source-meter while a Keithley 2182 nanovoltmeter detected the corresponding voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The gate voltage was provided by another Keithley 2400 across the Hall channel of T-Pt and B-Pt (grounded) Hall bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The magnetic field was fixed at 1 T and sample rotated in xoy, xoz or yoz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' For angular dependent of MCD signal measurements, the input current (Iin) was applied in the long axis of B-Pt Hall bar and the voltage signal (ΔV) was picked up alone the long axis of T-Pt Hall bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Then the magnitude of MCD voltage Vdrag under certain Iin was obtained by fitting the ∆𝑉 − 𝛾 curves measured at different 𝐼in with ∆𝑉 = 𝑉drag cos 2𝛾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The magnon drag parameter was then calculated by 𝛼 ≡ 𝑉drag 𝐼in𝑅T−Pt, where 𝑅T−Pt is the resistance of T-Pt electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' 12 And for SMR test the resistance of the B-Pt (T-Pt) electrode was measured by four-terminal method, and the angular dependence of change in RPt (ΔRPt-γ) was well fitted by ∆𝑅Pt = 𝑅SMR cos 2𝛾 and the SMR ratio was thus obtained by |𝑅SMR 𝑅Pt ⁄ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' The spin pumping test was carried out at room temperature in a home-build electromagnet with magnetic up to ~ 3500 Oe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' A signal generator (ROHDE&SCHWARZ SMB 100A) supplies a microwave signal modulated with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content='172 kHz signal to CPW and the voltage signal was picked up by a lock-in amplifier (Stanford SR830), while external magnetic field H applied perpendicular to the direction that spin pumping voltage was picked up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' To minimize interference, 𝑉g was provided by dry batteries during spin pumping measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} +page_content=' Data availability: The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfbA8f/content/2301.05592v1.pdf'} diff --git a/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf b/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..580802031452b9f81cdc4d732fcfac0fdbb2374a Binary files /dev/null and b/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf differ diff --git a/ZNAyT4oBgHgl3EQf9voF/content/tmp_files/2301.00878v1.pdf.txt b/ZNAyT4oBgHgl3EQf9voF/content/tmp_files/2301.00878v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..46151796d4af79a561ab740c5b8ff4bb1f41b8ed --- /dev/null +++ b/ZNAyT4oBgHgl3EQf9voF/content/tmp_files/2301.00878v1.pdf.txt @@ -0,0 +1,115 @@ +Science Platforms for Heliophysics Data Analysis +Monica G. Bobra (Stanford University), Will T. Barnes (NRC Postdoc residing at the Naval Research +Laboratory), Thomas Y. Chen (Columbia University), Mark C. M. Cheung (Lockheed Martin Solar and +Astrophysics Laboratory), Laura A. Hayes (NASA Goddard Space Flight Center), Jack Ireland (NASA +Goddard Space Flight Center), Miho Janvier (Institut d’Astrophysique Spatiale), Michael S. F. Kirk (NASA +Goddard Space Flight Center and ASTRA llc.), James P. Mason (University of Colorado Boulder), Stuart +J. Mumford (The University of Sheffield and Aperio Software), Paul J. Wright (Stanford University) +Introduction: What is the problem with data analysis in heliophysics today? +In 2020, we live in an era rich with data. NASA instruments designed to study the heliosphere +take more data, at faster rates than ever before. For example, the Solar Dynamics Observatory +(SDO) mission data set currently totals 19 petabytes. The heliophysics community also +conducts coordinated campaigns, where many instruments observe the same target for the +same time, that yield a wide variety of data—image data, spectral data, time series data, and in +situ data on local particle and field conditions. Heliophysics is a naturally multi-messenger +discipline, and many studies rely on using these varied data sets together. +However, scientists struggle to freely explore these data. Network speeds and limited storage +make it difficult to obtain large data sets. Inadequate computing power makes it difficult to +efficiently analyze large data sets. Despite the wide availability of accelerated and scalable +computing resources (e.g. GPUs, cloud computing), most heliophysicists use their laptop or +desktop computers, which are severely limited in both RAM and disk space, for their data +analysis workflows. These hurdles prevent the community from maximizing scientific discovery +and scientific return on investment. +In order to make new scientific breakthroughs with these data, we need to adopt a modern +scientific workflow. In today's prevailing paradigm, heliophysicists download data to analyze on +a local machine. This workflow limits the scope of scientific studies that rely on large data sets. +Scientists only pursue studies feasible within the constraints of their local network speeds, +storage capabilities, software tools, and computing power. +To support much larger studies, we recommend an alternate paradigm. In this workflow, +heliophysicists conduct scientific studies on an external machine equipped with multiple mission +data sets, ample computing power, and software tools. These openly accessible science +platforms, now available in the fields of astrophysics (e.g. Bauer et al. 2019) and earth science +(e.g. Robinson et al. 2019), allow scientists to rapidly analyze petabytes of data. Unburdened by +network speeds, storage, software, and compute, heliophysicists can freely pursue compelling +scientific studies that were previously practically impossible. +Science platforms for other scientific communities +Other scientific communities already recognized the power of this approach. For example, + +astronomers can analyze sky survey data from the Vera Rubin Observatory with a NSF-funded +science platform that co-locates the survey data with high-performance computing at the +National Center for Supercomputing Applications (Dubois-Felsmann et al. 2019). Practically, +this means astronomers can open a terminal window or Jupyter Lab, log onto an external +computing platform, write code, and run it on any of the survey data. Other examples, such as +the NSF-funded science platform called the National Optical Astronomy Observatory Data Lab, +exist as well (Fitzpatrick et al. 2014, Taghizadeh-Popp et al. 2020, Thomas et al. 2020). +The Pangeo data science platform (Odaka et al. 2020), funded by various institutions including +the NSF, NASA, NCAR, Sloan Foundation, and UK Met Office, completely revolutionized the +way many earth scientists think about data science workflows. Earth scientists use Pangeo to +analyze petabytes from a variety of space-based observatories. +The four necessary attributes of a science platform +While the technical design elements of each platform vary, each of them includes four essential +attributes: data storage, computing power, software tools, and open access. The scientific data, +version-controlled and stored in flexible databases, should adhere to a standard and adaptable +format. Computational resources, co-located with the data, should allow parallel processing on +CPUs and GPUs. Science platforms should also provide environments to easily install and use +open-source, openly-developed, and version-controlled scientific software. Finally, the platform +should provide openly accessible collaborative workspaces for the entire community. +Science platforms for heliophysics +The heliophysics community will benefit immensely from a science platform. Heliophysics +observatories produce extremely large data sets that are not used to their full potential. +According to a survey of the solar physics community by the SunPy project (Bobra et al. 2020), +82% of solar physicists work with observational data. +Despite this, most heliophysicists lack access to computational facilities. In the same survey, the +SunPy project found that 14% of the solar physics community uses local or regional clusters, 9% +use GPUs, and 5% use the commercial cloud to do their research. Roughly a third of the +community uses exclusively a laptop or desktop. This also means most of the community does +not take advantage of massively parallel computing, even though it presents the biggest +opportunity to accelerate computing performance (Robinson et al. 2020). +Open-source scientific software already provides powerful, easy-to-use tools that scalabely +accelerate computing performance (e.g. Dask; Rocklin et al. 2015). Furthermore, open-source, +version-controlled instrument calibration software, such as AIAPy (Barnes et al. 2020), +co-located with open-access, version-controlled data can help the community create calibrated, +reproducible, shareable data sets. +Today, science platforms do not exist in the heliophysics community. A solar science platform +prototype (Barnes et al. 2019), which provides interactive access to SDO data and + +high-performance computing with the NASA Pleiades supercomputer, demonstrates how users +can analyze large data sets quickly. Science platforms such as these will usher in a new era of +scalable, interactive supercomputing for data analysis in solar and space physics. +Recommendations +We recommend that NASA maintain and fund science platforms that enable interactive and +scalable data analysis in order to maximize the scientific return of data collected from +space-based instruments1. In support of this vision, we recommend a series of short-term goals +to achieve within the next 10 years: (1) support coordinated and open development of scientific +software that balances interactivity and scalability, (2) provide community education and training +on high-performance and cloud computing for data analysis, and (3) establish a funding model +where grant dollars can buy computing time from commercial cloud vendors. +We also recommend a series of long-term goals to achieve within the next 20 years: (1) +establish dedicated infrastructure at NASA high-performance computing centers (e.g. ADAPT at +Goddard, Pleiades at Ames) for interactive, scalable data analysis, and (2) ensure that the +scientific potential of these data are maximized long past the lifetime of the mission by +co-locating current and final mission data archives with this dedicated infrastructure. Together, +these recommendations allow scientists to accelerate their research workflows, produce +reproducible research, and maximize the scientific return of NASA data sets. +References +Barnes W. T. et al. Zenodo 2020, doi: 10.5281/zenodo.401698. +Barnes W. T. et al. AGU Fall Meeting 2019, bibcode: 2019AGUFMSH41C3317B. +Bauer A. E. et al. 2019, arxiv: 1905.05116. +Bobra M. G. et al. Sol Phys. 295 2020, doi: 10.1007/s11207-020-01622-2. +Dubois-Felsmann G. et al. 2019, url. +Fitzpatrick M. J. et al. SPIE 2014, doi: 10.1117/12.2057445. +Momcheva I. et al. 2015, arxiv: 1507.03989. +National Academies of Sciences, Engineering, and Medicine. 2020, doi: 10.17226/25668. +Odaka T. E. et al. Comm in Comp and Inf Sci. 1190 2020, doi: 10.1007/978-3-030-44728-1_12. +Robinson N. H. et al. 2019, arxiv: 1908.03356. +Rocklin M. Proc. 14th Python in Science Conf. 126 2015, url. +Rollin T. et al. Zenodo 2020, doi: 10.25080/Majora-342d178e-01f. +Taghizadeh-Popp M. et al. 2020, arxiv: 2001.08619. +1 This vision and these short- and long-term goals are consistent with Recommendation 3.2.4 in the 2020 +National Academies report entitled Progress Toward Implementation of the 2013 Decadal Survey for +Solar and Space Physics: A Midterm Assessment, that "NASA and NSF should maximize the scientific +return from large and complex data sets by supporting (1) training opportunities on modern statistical and +computational techniques; (2) science platforms to store, retrieve, and process data using common +standards; (3) funding opportunities for interdisciplinary collaboration; and (4) the development of +open-source software." + diff --git a/ZNAyT4oBgHgl3EQf9voF/content/tmp_files/load_file.txt b/ZNAyT4oBgHgl3EQf9voF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa3850f1b44b9e55e149c6da37d338138310f50e --- /dev/null +++ b/ZNAyT4oBgHgl3EQf9voF/content/tmp_files/load_file.txt @@ -0,0 +1,149 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf,len=148 +page_content='Science Platforms for Heliophysics Data Analysis Monica G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Bobra (Stanford University), Will T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Barnes (NRC Postdoc residing at the Naval Research Laboratory), Thomas Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Chen (Columbia University), Mark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Cheung (Lockheed Martin Solar and Astrophysics Laboratory), Laura A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Hayes (NASA Goddard Space Flight Center), Jack Ireland (NASA Goddard Space Flight Center), Miho Janvier (Institut d’Astrophysique Spatiale), Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Kirk (NASA Goddard Space Flight Center and ASTRA llc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' ), James P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Mason (University of Colorado Boulder), Stuart J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Mumford (The University of Sheffield and Aperio Software), Paul J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Wright (Stanford University) Introduction: What is the problem with data analysis in heliophysics today?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' In 2020, we live in an era rich with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' NASA instruments designed to study the heliosphere take more data, at faster rates than ever before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' For example, the Solar Dynamics Observatory (SDO) mission data set currently totals 19 petabytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' The heliophysics community also conducts coordinated campaigns, where many instruments observe the same target for the same time, that yield a wide variety of data—image data, spectral data, time series data, and in situ data on local particle and field conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Heliophysics is a naturally multi-messenger discipline, and many studies rely on using these varied data sets together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' However, scientists struggle to freely explore these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Network speeds and limited storage make it difficult to obtain large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Inadequate computing power makes it difficult to efficiently analyze large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Despite the wide availability of accelerated and scalable computing resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' GPUs, cloud computing), most heliophysicists use their laptop or desktop computers, which are severely limited in both RAM and disk space, for their data analysis workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' These hurdles prevent the community from maximizing scientific discovery and scientific return on investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' In order to make new scientific breakthroughs with these data, we need to adopt a modern scientific workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=" In today's prevailing paradigm, heliophysicists download data to analyze on a local machine." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' This workflow limits the scope of scientific studies that rely on large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Scientists only pursue studies feasible within the constraints of their local network speeds, storage capabilities, software tools, and computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' To support much larger studies, we recommend an alternate paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' In this workflow, heliophysicists conduct scientific studies on an external machine equipped with multiple mission data sets, ample computing power, and software tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' These openly accessible science platforms, now available in the fields of astrophysics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Bauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2019) and earth science (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2019), allow scientists to rapidly analyze petabytes of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Unburdened by network speeds, storage, software, and compute, heliophysicists can freely pursue compelling scientific studies that were previously practically impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Science platforms for other scientific communities Other scientific communities already recognized the power of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' For example, astronomers can analyze sky survey data from the Vera Rubin Observatory with a NSF-funded science platform that co-locates the survey data with high-performance computing at the National Center for Supercomputing Applications (Dubois-Felsmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Practically, this means astronomers can open a terminal window or Jupyter Lab, log onto an external computing platform, write code, and run it on any of the survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Other examples, such as the NSF-funded science platform called the National Optical Astronomy Observatory Data Lab, exist as well (Fitzpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2014, Taghizadeh-Popp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020, Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' The Pangeo data science platform (Odaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020), funded by various institutions including the NSF, NASA, NCAR, Sloan Foundation, and UK Met Office, completely revolutionized the way many earth scientists think about data science workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Earth scientists use Pangeo to analyze petabytes from a variety of space-based observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' The four necessary attributes of a science platform While the technical design elements of each platform vary, each of them includes four essential attributes: data storage, computing power, software tools, and open access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' The scientific data, version-controlled and stored in flexible databases, should adhere to a standard and adaptable format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Computational resources, co-located with the data, should allow parallel processing on CPUs and GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Science platforms should also provide environments to easily install and use open-source, openly-developed, and version-controlled scientific software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Finally, the platform should provide openly accessible collaborative workspaces for the entire community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Science platforms for heliophysics The heliophysics community will benefit immensely from a science platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Heliophysics observatories produce extremely large data sets that are not used to their full potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' According to a survey of the solar physics community by the SunPy project (Bobra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020), 82% of solar physicists work with observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Despite this, most heliophysicists lack access to computational facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' In the same survey, the SunPy project found that 14% of the solar physics community uses local or regional clusters, 9% use GPUs, and 5% use the commercial cloud to do their research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Roughly a third of the community uses exclusively a laptop or desktop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' This also means most of the community does not take advantage of massively parallel computing, even though it presents the biggest opportunity to accelerate computing performance (Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Open-source scientific software already provides powerful, easy-to-use tools that scalabely accelerate computing performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Dask;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Rocklin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Furthermore, open-source, version-controlled instrument calibration software, such as AIAPy (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020), co-located with open-access, version-controlled data can help the community create calibrated, reproducible, shareable data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Today, science platforms do not exist in the heliophysics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' A solar science platform prototype (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2019), which provides interactive access to SDO data and high-performance computing with the NASA Pleiades supercomputer, demonstrates how users can analyze large data sets quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Science platforms such as these will usher in a new era of scalable, interactive supercomputing for data analysis in solar and space physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Recommendations We recommend that NASA maintain and fund science platforms that enable interactive and scalable data analysis in order to maximize the scientific return of data collected from space-based instruments1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' In support of this vision, we recommend a series of short-term goals to achieve within the next 10 years: (1) support coordinated and open development of scientific software that balances interactivity and scalability, (2) provide community education and training on high-performance and cloud computing for data analysis, and (3) establish a funding model where grant dollars can buy computing time from commercial cloud vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' We also recommend a series of long-term goals to achieve within the next 20 years: (1) establish dedicated infrastructure at NASA high-performance computing centers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' ADAPT at Goddard, Pleiades at Ames) for interactive, scalable data analysis, and (2) ensure that the scientific potential of these data are maximized long past the lifetime of the mission by co-locating current and final mission data archives with this dedicated infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Together, these recommendations allow scientists to accelerate their research workflows, produce reproducible research, and maximize the scientific return of NASA data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' References Barnes W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' Zenodo 2020, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} 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+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 2020, arxiv: 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='08619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' 1 This vision and these short- and long-term goals are consistent with Recommendation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='4 in the 2020 National Academies report entitled Progress Toward Implementation of the 2013 Decadal Survey for Solar and Space Physics: A Midterm Assessment, that "NASA and NSF should maximize the scientific return from large and complex data sets by supporting (1) training opportunities on modern statistical and computational techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' (2) science platforms to store, retrieve, and process data using common standards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' (3) funding opportunities for interdisciplinary collaboration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content=' and (4) the development of open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} +page_content='"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNAyT4oBgHgl3EQf9voF/content/2301.00878v1.pdf'} diff --git a/ZdE4T4oBgHgl3EQfOQzc/content/tmp_files/2301.04964v1.pdf.txt b/ZdE4T4oBgHgl3EQfOQzc/content/tmp_files/2301.04964v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8642c921addbdf283ebde0ab9bdb4b617cb6d89 --- /dev/null +++ b/ZdE4T4oBgHgl3EQfOQzc/content/tmp_files/2301.04964v1.pdf.txt @@ -0,0 +1,2439 @@ +arXiv:2301.04964v1 [math.RT] 12 Jan 2023 +ON GAMMA FACTORS FOR REPRESENTATIONS OF FINITE +GENERAL LINEAR GROUPS +DAVID SOUDRY AND ELAD ZELINGHER +Abstract. We use the Langlands–Shahidi method in order to define the Shahidi gamma +factor for a pair of irreducible generic representations of GLn (Fq) and GLm (Fq). We prove +that the Shahidi gamma factor is multiplicative and show that it is related to the Jacquet– +Piatetski-Shapiro–Shalika gamma factor. As an application, we prove a converse theorem +based on the absolute value of the Shahidi gamma factor, and improve the converse theorem +of Nien. As another application, we give explicit formulas for special values of the Bessel +function of an irreducible generic representation of GLn (Fq). +1. Introduction +In the representation theory of p-adic groups, one method of studying irreducible represen- +tations is by attaching local factors to the representations. These local factors are complex +valued functions of a complex variable. They encode various properties of the representations +in question. These local factors usually arise from global integrals representing L-functions +attached to automorphic representations. Studying these local factors is crucial for under- +standing the global situation. This has been done successfully in many cases, including the +pioneering works of Jacquet–Piatetski-Shapiro–Shalika [9] and Shahidi [19, 20]. +Let F be a finite field with cardinality q. +A local theory of local factors often has a +finite field analog. It allows one to attach “local constants” to irreducible representations of +the F-points version of the group in consideration. We mention the works [18, 16, 25, 13, +14] as examples. These local constants usually encode properties analogous to their local +factors counterparts. Moreover, these local constant theories often allow one to consider “toy +models” for analogous local problems. For instance, shortly after Nien’s proof of the analog +of Jacquet’s conjecture for finite fields [16], Chai proved the conjecture for the p-adic group +case [2], where in his proof he used tools analogous to the ones used by Nien. +In her master’s thesis [18], Roditty-Gershon defined a finite field analog of the gamma fac- +tor of Jacquet–Piatetski-Shapiro–Shalika [9]. This gamma factor represents the tensor prod- +uct representation, attached to two irreducible generic representations π and σ of GLn (F) +and GLm (F), respectively, and is denoted γ(π × σ, ψ). Later, Rongqing Ye showed that +γ(π × σ, ψ) is related to its local field counterpart through level zero supercuspidal represen- +tations [24]. Using this relation and the local Langlands correspondence, Rongqing Ye and +the second author were able to express γ(π × σ, ψ) as a product of Gauss sums [26]. +The theory of the finite field version of the gamma factor associated to the tensor product, +as it currently appears in the literature, is in some sense not complete. The first problem +is that the gamma factor γ(π × σ, ψ) is currently not defined for all irreducible generic +representations π and σ. It is only defined when n ≥ m, and under the assumption that π +is cuspidal (and if n = m, σ is also required to be cuspidal). One can tweak the proofs so +they will work for all irreducible generic representations π and σ, such that π and σ∨ have +disjoint cuspidal support, but that is not enough in order to define γ(π × σ, ψ) for all pairs +1 + +2 +DAVID SOUDRY AND ELAD ZELINGHER +π and σ. One can try to define γ(π × σ, ψ) naively using the expression involving the Bessel +functions of π and σ, but this leads to the second problem. The second problem is that +the current theory lacks the multiplicativity property of the gamma factor. If one naively +extends the definition γ(π × σ, ψ) using the approach suggested above, it is not clear that +the gamma factor would be multiplicative. Both of these difficulties need to be resolved for +applications as in [27]. +The Langlands–Shahidi method provides an alternative approach that solves both of these +problems. In this paper, we use this method to define a finite field version of the Shahidi +gamma factor. We briefly describe the construction now. Let π and σ be representations +of Whittaker type of GLn (F) and GLm (F), respectively. +In Section 3.1, we consider an +intertwining operator Uσ,π : σ◦π → π◦σ, where ◦ denotes parabolic induction. In Section 3.2, +given Whittaker vectors vπ,ψ ∈ π and σ ∈ vσ,ψ, we define Whittaker vectors vπ,σ,ψ ∈ π◦σ and +vσ,π,ψ ∈ σ ◦ π. By uniqueness of the Whittaker vectors, we have that there exists a constant +Γ(π × σ, ψ) ∈ C, such that +Uσ,πvσ,π,ψ = Γ(π × σ, ψ) · vπ,σ,ψ. +We call Γ(π × σ, ψ) the Shahidi gamma factor associated to π and σ. This is a finite analog +of Shahidi’s local coefficient [19]. +We prove properties of Γ(π × σ, ψ), the most important one is that it is multiplicative +(Theorem 3.3). +Theorem 1.1. Let π, σ1 and σ2 be representations of Whittaker type of GLn (F), GLm1 (F) +and GLm2 (F), respectively. Then +Γ(π × (σ1 ◦ σ2), ψ) = Γ(π × σ1, ψ) · Γ(π × σ2, ψ). +We also express Γ(π ×σ, ψ) in terms of the Bessel functions associated with π and σ when +both representations are irreducible. We show that if n ≥ m, then up to some simple factors, +Γ(π × σ, ψ) is given by the naive extension of γ(π × σ∨, ψ) discussed above (Theorem 3.4). +We deduce a relation between the Shahidi gamma factor and the Jacquet–Piatetski-Shapiro– +Shalika gamma factor (Corollary 3.4). +Theorem 1.2. Let π and σ be irreducible generic representations of GLn (F) and GLm (F), +respectively. Suppose that π is cuspidal and n ≥ m. If n = m, suppose that σ is also cuspidal. +Then +Γ(π × σ, ψ) = q +m(2n−m−1) +2 +ωσ (−1) γ(π × σ∨, ψ). +The relation between both gamma factors allows us to give a representation theoretic +interpretation for the absolute value of the Shahidi gamma factor. We show that, in some +sense, the absolute value of the Shahidi gamma factor serves as a good substitute for the +order of the pole of the local L-factor associated with the tensor product representation. Let +us stress that the relation to the Jacquet–Piatetski-Shapiro–Shalika gamma factor is crucial +for these results. The following theorem can be seen as an analog of [9, Section 8.1]. +Theorem 1.3. Let π be an irreducible generic representation of GLn (F) and let σ be an +irreducible cuspidal representation of GLm (F). Then +���q +−nm +2 +· Γ(π × σ, ψ) +��� = q− dπ(σ)m +2 +, +where dπ (σ) is the number of times σ appears in the cuspidal support of π. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +3 +This allows us to deduce a converse theorem based on the absolute value of the normalized +Shahidi gamma factor. Similar theorems in the local setting were given by Gan and his +collaborators in many works (see [7, Lemma 12.3], [6, Lemma A.6] and [1, Lemma A.6]), but +our proof is done on the “group side” rather than on the “Galois side”. +Theorem 1.4. Let π1 and π2 be two irreducible generic representations of GLn1 (F) and +GLn2 (F), respectively. Assume that for every m and every irreducible cuspidal representation +σ of GLm (F) we have +���q− n1m +2 +· Γ(π1 × σ, ψ) +��� = +���q− n2m +2 +· Γ(π2 × σ, ψ) +��� . +Then n1 = n2 and π1 ∼= π2. +Our results combined with Nien’s converse theorem [16] allow us to deduce a converse +theorem that holds under weaker assumptions. This is similar to [10, Section 2.4]. +Theorem 1.5. Let π1 and π2 be irreducible generic representations of GLn (F) with the +same central character. +Suppose that for any 1 ≤ m ≤ +n +2 and any irreducible cuspidal +representation σ of GLm (F) we have +Γ(π1 × σ, ψ) = Γ(π2 × σ, ψ). +Then π1 ∼= π2. +As another application of our results, we find explicit formulas for special values of the +Bessel function of an irreducible generic representation π. The first formula (Theorem 4.5) +expresses Jπ,ψ ( +In−1 +c +) as an exotic Kloosterman sum [11, Page 152]. This formula is already +known in the literature by the work of Curtis–Shinoda [4], but our proof is based on multi- +plicativity of the Shahidi gamma factor, rather than on Deligne–Lusztig theory. The second +formula we find (Theorem 4.6) expresses Jπ,ψ +� +−c′ +In−2 +c +� +as a twisted convolution of values +of the form Jπ,ψ ( +In−1 +c +) and Jπ,ψ +� +c′ +In−1 +� +. Such a formula was given by Chang for n = 3 +[3] and then generalized by Shinoda–Tulunay [21] for n = 4. Chang’s method is based on +the Gelfand–Graev algebra, while our method is based on formulas we found for the Shahidi +gamma factor. +This paper is based on a unpublished note by the first author [23] from 1979. +2. Preliminaries +2.1. Parabolic induction. Given a sequence of positive integers n1, . . . , nr, we denote +by Pn1,...,nr the parabolic subgroup of GLn1+···+nr (F) corresponding to the composition +(n1, . . . , nr). That is, +Pn1,...,nr = Dn1,...,nr ⋊ Nn1,...,nr, +where +Dn1,...,nr = +� +diag (g1, . . . , gr) | ∀1 ≤ j ≤ r, gj ∈ GLnj (F) +� +, +Nn1,...,nr = + + + + + + + + + + + + + + +In1 +∗ +∗ +∗ +In2 +∗ +∗ +... +∗ +Inr + + + + + + + + + + + + + + +. + +4 +DAVID SOUDRY AND ELAD ZELINGHER +Given representations π1, . . . , πr of GLn1 (F) , . . . , GLnr (F), respectively, we denote by +π1⊗ . . . ⊗πr the inflation of π1 ⊗ · · · ⊗ πr to Pn1,...,nr. That is, π1⊗ . . . ⊗πr is a represen- +tation of Pn1,...,nr, acting on the space of π1 ⊗· · ·⊗πr, and its action on pure tensors is given +by +(π1⊗ . . . ⊗πr) (du) v1 ⊗ · · · ⊗ vr = π1 (g1) v1 ⊗ · · · ⊗ πr (gr) vr, +where d = diag (g1, . . . , gr) ∈ Dn1,...,nr and u ∈ Nn1,...,nr, and for every 1 ≤ j ≤ r, vj ∈ πj. +The parabolic induction π1 ◦ . . . ◦ πr is defined as the following representation of +GLn1+···+nr (F): +π1 ◦ . . . ◦ πr = Ind +GLn1+···+nr(F) +Pn1,...,nr +π1⊗ . . . ⊗πr. +By [8, Theorem 2.4], if π is an irreducible representation of GLn (F), then there exist +n1, . . . , nr > 0 with n1 + · · · + nr = n and irreducible cuspidal representations π1, . . . , πr, +of GLn1 (F) , . . . , GLnr (F), such that π is isomorphic to a subrepresentation of the parabolic +induction π1 ◦ . . . ◦ πr. Such π1, . . . , πr are unique up to ordering. We define the cuspidal +support of π to be the multiset {π1, . . . , πr}. +2.2. Generic representations. Let ψ: F → C∗ be a non-trivial additive character. Let +Zn ≤ GLn (F) be the upper triangular unipotent subgroup. We define a character ψ: Zn → +C∗ by the formula +ψ + + + + + + + +1 +a1 +∗ +. . . +∗ +1 +a2 +. . . +∗ +... +... +... +1 +an−1 +1 + + + + + + + += ψ +�n−1 +� +k=1 +ak +� +. +Let π be a finite dimensional representation of GLn (F). π is said to be generic if +HomZn (ResZn π, ψ) ̸= 0. +This condition does not depend on the choice of ψ. See Section 3.2.1. We call a non-zero +element in HomZn (ResZn π, ψ) a ψ-Whittaker functional. The representation π is generic if +and only if there exists 0 ̸= v ∈ π, such that π (u) v = ψ (u) v for every u ∈ Zn. We call such +vector a Whittaker vector with respect to ψ, or a ψ-Whittaker vector. The dimension of the +subspace spanned by the ψ-Whittaker vectors of π is dim HomZn (ResZn π, ψ). +Definition 2.1. We say that π is of Whittaker type if π is generic and the subspace spanned +by its ψ-Whittaker vectors is one-dimensional. +By a well known result of Gelfand and Graev, we have that if π is generic and irreducible, +then it is of Whittaker type [8, Theorem 0.5], [22, Corollary 5.6]. It is well known that +irreducible cuspidal representations of GLn (F) are generic [22, Lemma 5.2]. The following +result is also well known [22, Theorem 5.5]. +Theorem 2.1. Let π1, . . . , πr be representations of Whittaker type of GLn1 (F) , . . . , GLnr (F), +respectively. Then the parabolic induction π1 ◦ . . . ◦ πr is a representation of Whittaker type. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +5 +2.2.1. Whittaker models and Bessel functions. Let π be an irreducible generic representation +of GLn (F). Since π is of Whittaker type, Frobenius reciprocity implies that +dim HomGLn(F) +� +π, IndGLn(F) +Zn +ψ +� += 1. +We denote by W (π, ψ) the unique subspace of IndGLn(F) +Zn +ψ that is isomorphic to π. This is +the Whittaker model of π with respect to ψ. +Recall that for an irreducible representation π of GLn (F), we have that its contragredient +π∨ is isomorphic to πι, where πι is the representation acting on the space of π by πι (g) = +π (gι), where for g ∈ GLn (F), +gι = t� +g−1� +. +(This follows from the fact that for g ∈ GLn (F), the trace characters of π and π∨ are related +by trπ∨ (g) = trπ(g−1), and from the fact that g−1 and t(g−1) are conjugate.) +Using the isomorphism π∨ ∼= πι, we get an isomorphism of vector spaces W (π, ψ) → +W (π∨, ψ−1), given by W �→ ˜W, where +˜W (g) = W (wngι) , +and where wn ∈ GLn (F) is the long Weyl element +wn = + + + + + +1 +1 +... +1 + + + + + . +Under the realization of π by its Whittaker model W (π, ψ), the one-dimensional subspace +spanned by the ψ-Whittaker vectors of π is realized as the one-dimensional subspace of +W (π, ψ) consisting of functions W ∈ W (π, ψ), such that W (gu) = ψ (u) W (g), for every +u ∈ Zn and every g ∈ GLn (F). By [8, Proposition 4.5], there exists a (unique) element W in +this one-dimensional subspace such that W (In) = 1. We call this W the normalized Bessel +function of π with respect to ψ, and denote it by Jπ,ψ. To summarize, the Bessel function +Jπ,ψ is the unique element in W (π, ψ), such that +(1) Jπ,ψ (In) = 1. +(2) Jπ,ψ (gu) = ψ (u) Jπ,ψ (g), for every g ∈ GLn (F) and u ∈ Zn. +The Bessel function enjoys the following identities that relate it to its complex conjugate +and to its contragredient [16, Propositions 2.15 and 3.5]. +Proposition 2.1. For any irreducible generic representation π of GLn (F) and any g ∈ +GLn (F), we have the following identities: +(1) Jπ,ψ (g−1) = Jπ,ψ (g). +(2) Jπ,ψ (g−1) = Jπ∨,ψ−1 (g). +Remark 2.1. Let vπ,ψ be a non-zero ψ-Whittaker vector. If we choose an inner product (·, ·)π +on π which is invariant under the GLn (F)-action, we have that the assignment ℓπ,ψ : π → C +given by vπ �→ (vπ, vπ,ψ)π defines a Whittaker functional. The Whittaker model of π can be +described using Frobenius reciprocity as W (π, ψ) = {Wvπ | vπ ∈ π}, where for g ∈ GLn (F) +and vπ ∈ π, we define Wvπ (g) = (π (g) vπ, vπ,ψ)π. The Bessel function is given by +Jπ,ψ (g) = (π (g) vπ,ψ, vπ,ψ)π +(vπ,ψ, vπ,ψ)π +. + +6 +DAVID SOUDRY AND ELAD ZELINGHER +All of the properties of the Bessel function listed above now follow immediately from the +fact that (·, ·)π is an inner product, and that vπ,ψ is a ψ-Whittaker vector. Moreover, the +projection operator to the one-dimensional subspace spanned by the ψ-Whittaker vectors +prCvπ,ψ can be described in two ways. The first way is by using the inner product, in which +case for vπ ∈ π, +prCvπ,ψ (vπ) = (vπ, vπ,ψ)π +(vπ,ψ, vπ,ψ)π +vπ,ψ. +The second way is by averaging, in which case +prCvπ,ψ (vπ) = +1 +|Zn| +� +u∈Zn +ψ−1 (u) π (u) vπ. +By completing vπ,ψ to an orthogonal basis of π and using the fact that the subspace spanned +by the ψ-Whittaker vectors is one dimensional, we see that +tr +� +prCvπ,ψ ◦π (g) +� += Jπ,ψ (g) . +This is [8, Proposition 4.5]. +2.3. Jacquet–Piatetski-Shapiro–Shalika gamma factors. Let π and σ be irreducible +generic representations of GLn (F) and GLm (F), respectively. For most π and σ, one can +define a constant attached to π and σ called the Jacquet–Piatetski-Shapiro–Shalika gamma +factor of π and σ. It is also known as the Rankin–Selberg gamma factor of π and σ. This +is a finite field analog of the definition given by Jacquet–Piatetski-Shapiro–Shalika [9] for +p-adic groups. These were explained in Piatetski-Shapiro’s lectures in 1976 and studied in +an unpublished note from 1979 by the first author [23]. The case n > m was also studied in +Roddity-Gershon’s master’s thesis under the supervision of the first author. +2.3.1. The case n > m. In her master’s thesis [18], Edva Roditty-Gershon defined the +Jacquet–Piatetski-Shapiro–Shalika gamma factor γ(π × σ, ψ), under the assumption that +π is cuspidal and that n > m. Roddity-Gershon’s thesis is unpublished, but her main results +are presented by Nien in [16]. We briefly review these results now. +The first result is a functional equation that defines the Jacquet–Piatetski-Shapiro–Shalika +gamma factor. Suppose that n > m and that π is cuspidal. For any W ∈ W (π, ψ) and +W ′ ∈ W (σ, ψ−1), and any 0 ≤ j ≤ n − m − 1, we define +Zj (W, W ′; ψ) +� +h∈Zm\GLm(F) +� +x∈M(n−m−j−1)×m(F) +W + + +h +x +In−m−j−1 +Ij+1 + + W ′ (h) . +We are now ready to state the functional equation. +Theorem 2.2 ([16, Theorem 2.10]). There exists a non-zero constant γ(π × σ, ψ) ∈ C, such +that for every 0 ≤ j ≤ m − m − 1, every W ∈ W (π, ψ) and every W ′ ∈ W (σ, ψ−1), we have +qmjγ(π × σ, ψ)Zj (W, W ′; ψ) = Zn−m−j−1 +� +π∨ +� +Im +wn−m +� +˜W, ˜W ′; ψ−1 +� +, +where wn−m ∈ GLn−m (F) is the long Weyl element. +The second result expresses the gamma factor γ(π × σ, ψ) in terms of the Bessel functions +of π and σ. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +7 +Proposition 2.2 ([16, Proposition 2.16]). Under the assumptions above, we have +γ(π × σ, ψ) = +� +h∈Zm\GLm(F) +Jπ,ψ +� +In−m +h +� +Jσ,ψ−1 (h) . +It follows from Proposition 2.2 and Proposition 2.1 that +γ(π × σ, ψ) = γ(π∨ × σ∨, ψ−1). +Moreover, applying Theorem 2.2 twice, we get the following corollary regarding the abso- +lute value of γ(π × σ, ψ). +Corollary 2.1. We have that +γ(π × σ, ψ)γ(π∨ × σ∨, ψ−1) = q−m(n−m−1), +and therefore +|γ(π × σ, ψ)| = q− m(n−m−1) +2 +. +2.3.2. The case n = m. The case n = m was discussed in Piatetski-Shapiro’s lecture and is +explained briefly in Rongqing Ye’s work [24]. +Let S (Fn) be the space of functions φ: Fn → C. For a function φ ∈ S (Fn), we define its +Fourier transform Fψφ: Fn → C by the formula +Fψφ (y) = +� +x∈Fn +φ (x) ψ (⟨x, y⟩) , +where if x = (x1, . . . , xn) ∈ Fn and y = (y1, . . . , yn) ∈ Fn, then ⟨x, y⟩ is the standard pairing +⟨x, y⟩ = +n +� +i=1 +xiyi. +Let π and σ be irreducible cuspidal representations of GLn (F). We define for any W ∈ +W (π, ψ), W ′ ∈ W (σ, ψ−1) and any φ ∈ S (Fn) +Z (W, W ′, φ; ψ) = +� +g∈Zn\GLn(F) +W (g) W ′ (g) φ (eng) , +where en = (0, . . . , 0, 1) ∈ Fn. We are now ready to introduce the functional equation that +defines γ(π × σ, ψ). +Theorem 2.3 ([24, Theorem 2.3]). There exists a non-zero constant γ(π × σ, ψ), such that +for any W ∈ W (π, ψ), W ′ ∈ W (σ, ψ−1), and any φ ∈ S (Fn) with φ (0) = 0, we have +Z( ˜W, ˜W ′, Fψφ; ψ−1) = γ(π × σ, ψ)Z (W, W ′, φ; ψ) . +Similarly to the case n > m, we have an expression of γ(π × σ, ψ) in terms of the Bessel +functions of π and σ. +Proposition 2.3 ([24, Equation (16)]). Let π and σ be irreducible cuspidal representations +of GLn (F). Then +γ(π × σ, ψ) = +� +g∈Zn\GLn(F) +Jπ,ψ (g) Jσ,ψ−1 (g) ψ +�� +eng−1, e1 +�� +, +where e1 = (1, 0, . . . , 0) ∈ Fn. + +8 +DAVID SOUDRY AND ELAD ZELINGHER +It follows from Proposition 2.3 and Proposition 2.1 that +γ(π∨ × σ∨, ψ−1) = γ(π × σ, ψ). +We now move to discuss the absolute value of γ(π × σ, ψ). In order to do that, we first +explain how to extend the functional equation in Theorem 2.3 to all functions in S (Fn) for +most cases. To begin, we notice that for the indicator function of 0 ∈ Fn, which we denote +δ0, we have that Z (W, W ′, δ0; ψ) = 0. We also notice that if π is not isomorphic to σ∨, then +Z (W, W ′, 1; ψ) = 0, where 1 represents the constant function. This is because +Z (W, W ′, 1; ψ) = +� +g∈Zn\GLn(F) +W (g) W ′ (g) +defines a GLn (F)-invariant pairing W (π, ψ) ⊗ W (σ, ψ−1) → C, but such non-trivial pairing +exists only when π is isomorphic to σ∨. These two observations imply the following extension +of the functional equation, in the special case where π is not isomorphic to σ∨. +Proposition 2.4. Suppose that π ≇ σ∨. Then for any φ ∈ S (Fn) we have +Z( ˜W, ˜W ′, Fψφ; ψ−1) = γ(π × σ, ψ)Z (W, W ′, φ; ψ) . +Proof. Write φ = φ0 + φ1, where φ0 = φ − φ(0) and φ1 = φ(0). +Then φ0 (0) = 0 and +Fψφ1 = qnφ (0) δ0. Since Z is linear in φ, we have from the discussion above that +Z (W, W ′, φ; ψ) = Z (W, W ′, φ0; ψ) +and that +Z( ˜W, ˜W ′, Fψφ; ψ−1) = Z( ˜W, ˜W ′, Fψφ0; ψ−1). +The statement now follows from Theorem 2.3. +□ +As a result, we get the following corollary regarding the absolute value of γ(π × σ, ψ). +Corollary 2.2. Let π and σ be irreducible cuspidal representations of GLn (F) such that +π ≇ σ∨. Then +γ(π × σ, ψ)γ(π∨ × σ∨, ψ−1) = qn, +and therefore +|γ(π × σ, ψ)| = q +n +2 . +Proof. This follows by applying Proposition 2.4 twice, and from the fact that the Fourier +transform satisfies +Fψ−1Fψφ = qnφ, +for any φ ∈ S (Fn). +□ +We are left to deal with the case π ∼= σ∨. In this case, the gamma factor γ(π × π∨, ψ) can +be computed explicitly and it equals −1, see Appendix A. +We summarize all cases in the following proposition. +Proposition 2.5. Let π and σ be irreducible cuspidal representations of GLn (F). +• If π ≇ σ∨ then |γ(π × σ, ψ)| = q +n +2 . +• If π ∼= σ∨ then |γ(π × σ, ψ)| = 1. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +9 +3. Shahidi gamma factors (local coefficients) +In this section, we use the Langlands–Shahidi method in order to define a gamma factor for +two representations of Whittaker type of finite general linear groups. This is the finite field +analog of Shahidi’s local coefficient, which uses an intertwining operator. The treatment in +Sections 3.1-3.3 is a finite field analog of Shahidi’s work on local coefficients over local fields +[19]. Unlike the Jacquet–Piatetski-Shapiro–Shalika gamma factors discussed in Section 2.3, +the Shahidi gamma factor can be defined uniformly for all irreducible generic representations +of GLn (F) and GLm (F), regardless of n > m or whether the representations are cuspidal. +We prove properties of the Shahidi gamma factor, where the most important one is the +multiplicativity property, which explains how this gamma factor behaves under parabolic +induction. We end this section by expressing the Shahidi gamma factor in terms of the +Bessel functions associated with the representations, and showing its relation to the Jacquet– +Piatetski-Shapiro–Shalika gamma factor. +3.1. The intertwining operator. Let n and m be positive integers and let π and σ be +representations of GLn (F) and GLm (F), respectively. We define a linear map σ ⊗π → π ⊗σ +acting on pure tensors by component swap: +swσ,π (vσ ⊗ vπ) = vπ ⊗ vσ. +For a function f : GLn+m (F) → σ ⊗ π, we denote by ˜f : GLn+m (F) → π ⊗ σ the function +˜f (g) = swσ,π (f(g)). +We consider the following intertwining operator Tσ,π : σ ◦ π → π ◦ σ, defined for f ∈ σ ◦ π +and g ∈ GLn+m (F) by the formula +Tσ,πf (g) = +� +p∈Pn,m +(π⊗σ) +� +p−1� ˜f ( ˆwn,mpg), +where ˆwn,m is the following Weyl element +ˆwn,m = +� +Im +In +� +. +Writing p ∈ Pn,m as p = du, where d ∈ Dn,m and u ∈ Nn,m, and using the left Dm,n- +equivariance property of f, one checks that +Tσ,πf (g) = |Dn,m| · Uσ,πf (g) , +where +Uσ,πf (g) = +� +u∈Nn,m +˜f ( ˆwn,mug) . +By construction, we have that Tσ,π and Uσ,π are non-zero elements of the space +HomGLn+m(F) (σ ◦ π, π ◦ σ) . +3.2. The Shahidi gamma factor. Suppose now that π and σ are representations of Whit- +taker type of GLn (F) and GLm (F), respectively. By Theorem 2.1 we have that the parabol- +ically induced representations σ ◦ π and π ◦ σ are also of Whittaker type. Let vσ,ψ ∈ σ + +10 +DAVID SOUDRY AND ELAD ZELINGHER +and vπ,ψ ∈ π be non-zero ψ-Whittaker vectors for σ and π, respectively. We may define a +non-zero ψ-Whittaker vector fvσ,ψ,vπ,ψ for σ ◦ π by the formula +fvσ,ψ,vπ,ψ (g) = +�ψ (u) (σ⊗π) (p) vσ,ψ ⊗ vπ,ψ +g = p ˆwn,mu, p ∈ Pm,n, u ∈ Zn+m, +0 +otherwise. +Similarly, we may define fvπ,ψ,vσ,ψ ∈ π ◦ σ. +Since Uσ,π is an intertwining operator, we have that Uσ,πfvσ,ψ,vπ,ψ is a ψ-Whittaker vector +of π ◦ σ. Since fvπ,ψ,vσ,ψ is the unique non-zero ψ-Whittaker vector of π ◦ σ up to scalar, we +must have that +Uσ,πfvσ,ψ,vπ,ψ = γ · fvπ,ψ,vσ,ψ, +where γ ∈ C. It is easy to check that this number γ does not depend on the choice of +ψ-Whittaker vectors vσ,ψ and vπ,ψ. +In order to ease the notation, we denote vσ,π,ψ = fvσ,ψ,vπ,ψ, where we suppress vσ,ψ and vπ,ψ +from the notation. Similarly, we denote vπ,σ,ψ = fvπ,ψ,vσ,ψ. +Definition 3.1. The Shahidi gamma factor of π and σ with respect to ψ is the unique +number Γ(π × σ, ψ) ∈ C, such that +Uσ,πvσ,π,ψ = Γ(π × σ, ψ) · vπ,σ,ψ. +Remark 3.1. If π ◦ σ is irreducible, then so is σ ◦ π, and since Uσ,π is a non-zero intertwining +operator, it is an isomorphism and Γ(π × σ, ψ) must be non-zero. However, in the general +case it is not obvious at this point that Γ(π × σ, ψ) is non-zero. We will show this later. +Remark 3.2. As in Remark 2.1, we may choose invariant inner products (·, ·)π and (·, ·)σ on +π and σ, respectively. We then have a natural inner product (·, ·)σ⊗π on σ ⊗π, which defines +an inner product on σ ◦ π by the formula +(f1, f2)σ◦π = +� +g∈Pm,n\GLn+m(F) +(f1 (g) , f2 (g))σ⊗π . +Using this inner product, the Whittaker functional ℓσ◦π,ψ (f) = (f, vσ,π,ψ)σ◦π is related to the +Whittaker functionals ℓσ,ψ (vσ) = (vσ, vσ,ψ)σ and ℓπ,ψ (vπ) = (vπ, vπ,ψ)π by the formula +ℓσ◦π,ψ (f) = +� +u∈Nn,m +ℓσ,ψ ⊗ ℓπ,ψ (f ( ˆwn,mu)) ψ−1 (u) . +Similarly, by exchanging the roles of π and σ, we have that Whittaker functional ℓπ◦σ,ψ is +given by a similar formula. Using the definitions of the inner products, and the fact that +elements in π ◦ σ are left invariant under Nn,m, we see that Uπ,σ is the adjoint of Uσ,π, with +respect to our choice of inner products. Using the relation between vσ,π,ψ and vπ,σ,ψ, we +obtain the following relation +ℓσ◦π,ψ ◦ Uπ,σ = Γ(π × σ, ψ) · ℓπ◦σ,ψ. +This is how the Shahidi gamma factor is usually defined in the literature. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +11 +3.2.1. Dependence on ψ. For any a ∈ F∗, let ψa : F → C∗ be the additive character +ψa (x) = ψ (ax) . +It is well known that all non-trivial additive characters of F are of the form ψa for some +a ∈ F∗. In this section, we give a relation between Γ(π × σ, ψ) and Γ(π × σ, ψa). +Let a ∈ F∗. Suppose that τ is a generic representation of GLk (F) with a non-zero ψ- +Whittaker vector vτ,ψ. Let +dk = diag +� +1, a, a2, . . . , ak−1� +. +Then we have that τ (dk) vτ,ψ is a non-zero ψa-Whittaker vector of τ. The map v �→ τ (dk) v +is a linear isomorphism from the subspace spanned by the ψ-Whittaker vectors of τ to the +subspace spanned by the ψa-Whittaker vectors of τ. In particular, if vτ,ψ is the unique (up +to scalar multiplication) ψ-Whittaker vector of τ, then τ (dk) vτ,ψ is the unique (up to scalar +multiplication) ψa-Whittaker vector of τ. +Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), respectively. +Let vπ,ψ ∈ π and vσ,ψ ∈ σ be non-zero ψ-Whittaker vectors. Assume that π and σ have +central characters, and denote them by ωπ and ωσ, respectively. Let +vσ,π,ψa = fσ(dm)vσ,ψ,π(dn)vπ,ψ. +Similarly, we define vπ,σ,ψa. +We first express vσ,π,ψa in terms of vσ,π,ψ. We will use this relation later to show a relation +between the gamma factors Γ(π × σ, ψa) and Γ(π × σ, ψ). +Proposition 3.1. We have +vσ,π,ψa = ωσ (a)−n ρ (dn+m) vσ,π,ψ, +where ρ (dn+m) denotes right translation by dn+m. +Proof. Let f = ωσ (a)−n ρ (dn+m) vσ,π,ψ ∈ σ ◦π. By the discussion above, f is a ψa-Whittaker +vector of σ ◦ π. +We have that +f ( ˆwn,m) = ωσ (a)−n vσ,π,ψ ( ˆwn,mdn+m) . +Writing dn+m = diag (dn, andm), we have ˆwn,mdn+m = diag (andm, dn) ˆwn,m, and hence +f ( ˆwn,m) = (σ (dm) ⊗ π (dn)) vσ,π,ψ ( ˆwn,m) = σ (dm) vσ,ψ ⊗ π (dn) vπ,ψ. +This shows that f = vσ,π,ψa, as both are ψa-Whittaker vectors in σ ◦ π, and both agree at +the point ˆwn,m. +□ +Theorem 3.1. We have +Γ(π × σ, ψa) = ωπ (a)m · ωσ (a)−n · Γ(π × σ, ψ). +Proof. By definition, we have that +Γ(π × σ, ψa)vπ,σ,ψa = Uσ,πvσ,π,ψa. +By Proposition 3.1, +Γ(π × σ, ψa)ωπ (a)−m ρ (dn+m) vπ,σ,ψ = ωσ (a)−n Uσ,πρ (dn+m) vσ,π,ψ. +Therefore, we get that +Γ(π × σ, ψa)ωσ (a)n ωπ (a)−m vπ,σ,ψ = Uσ,πvσ,π,ψ, + +12 +DAVID SOUDRY AND ELAD ZELINGHER +which implies that +Γ(π × σ, ψa)ωσ (a)n ωπ (a)−m = Γ(π × σ, ψ), +as required. +□ +3.2.2. Relation between Γ(π × σ, ψ) and Γ(σ∨ × π∨, ψ−1). In this section, we analyze the +relation between Γ(π × σ, ψ) and Γ(σ∨ × π∨, ψ). +Recall that for a finite dimensional representation τ of GLk (F), we have that τ ∨ ∼= τ ι. See +Section 2.2.1. If vτ,ψ is a non-zero ψ-Whittaker vector for τ, then τ (wk) vτ,ψ is a non-zero +ψ−1-Whittaker vector for τ ι. +We have that +πι ◦ σι ∼= (σ ◦ π)ι +by the isomorphism Sσ,π : (σ ◦ π)ι ∼= πι ◦ σι that sends f ∈ (σ ◦ π)ι to the function +(Sσ,πf) (g) = ˜f ( ˆwn,mgι) . +Let +vπι,σι,ψ−1 = fπ(wn)vπ,ψ,σ(wm)vσ,ψ ∈ πι ◦ σι. +Then vπι,σι,ψ−1 is a non-zero ψ−1-Whittaker vector of πι ◦ σι. On the other hand, by the +discussion above, a non-zero ψ−1-Whittaker vector of (σ ◦ π)ι is given by ρ (wm+n) vσ,π,ψ, +where ρ (wm+n) represents right translation by wm+n. Therefore Sσ,πρ (wm+n) vσ,π,ψ is another +non-zero ψ−1-Whittaker vector of πι ◦ σι. +Proposition 3.2. We have +vπι,σι,ψ−1 = Sσ,πρ (wm+n) vσ,π,ψ. +(3.1) +Proof. We have that +Sσ,πρ (wm+n) vσ,π,ψ ( ˆwm,n) = swσ,π vσ,π,ψ (wm+n) = π (wn) vπ,ψ ⊗ σ (wm) vσ,ψ, +where in the last step we used the fact that diag (wm, wn) ˆwn,m = wn+m. +Since Sσ,πρ (wm+n) vσ,π,ψ and vπι,σι,ψ−1 are both ψ−1-Whittaker vectors for the representa- +tion of Whittaker type πι ◦ σι, and they both agree at the point ˆwm,n, they are equal. +□ +Similarly, let +vσι,πι,ψ−1 = fσ(wm)vσ,ψ,π(wn)vπ,ψ ∈ σι ◦ πι. +Using Proposition 3.2 with roles of the representations π and σ exchanged, we have +vσι,πι,ψ−1 = Sπ,σρ (wm+n) vπ,σ,ψ. +(3.2) +Theorem 3.2. Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), +respectively. Then +Γ(π × σ, ψ) = Γ(σ∨ × π∨, ψ−1). +Proof. By definition, +Uπι,σιvπι,σι,ψ−1 = Γ(σι × πι, ψ−1) · vσι,πι,ψ−1. +(3.3) +Substituting (3.1) and (3.2) in (3.3), we get +Uπι,σιSσ,πρ (wm+n) vσ,π,ψ = Γ(σι × πι, ψ−1) · Sπ,σρ (wm+n) vπ,σ,ψ. +A simple computation shows that +Uπι,σι ◦ Sσ,π = Sπ,σ ◦ Uσ,π. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +13 +Hence, we get that +Sπ,σρ (wm+n) Uσ,πvσ,π,ψ = Γ(σι × πι, ψ−1) · Sπ,σρ (wm+n) vπ,σ,ψ, +which implies that +Uσ,πvσ,π,ψ = Γ(σι × πι, ψ−1) · vπ,σ,ψ. +Therefore, we must have +Γ(σι × πι, ψ−1) = Γ(π × σ, ψ), +and the statement in the theorem follows, since σι ∼= σ∨ and πι ∼= π∨. +□ +Combining Theorem 3.2 with Theorem 3.1, we get the following corollary. +Corollary 3.1. Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), +respectively. Assume that both π and σ have central characters, and denote them by ωπ and +ωσ, respectively. Then +Γ(π × σ, ψ) = Γ(σ∨ × π∨, ψ) · ωπ (−1)m ωσ (−1)n . +3.3. Multiplicativity of Gamma factors. In this section, we show that Γ(π × σ, ψ) is +multiplicative. +Let π be a representation of Whittaker type of GLn (F). Let m = m1 + m2, and let σ1 +and σ2 be representations of Whittaker type of GLm1 (F) and GLm2 (F), respectively. By +Theorem 2.1, the parabolic induction σ1 ◦ σ2 is also a representation of Whittaker type. +Hence, the gamma factor Γ(π × (σ1 ◦ σ2), ψ) is well defined. We will show the following +theorem. +Theorem 3.3. +Γ(π × (σ1 ◦ σ2), ψ) = Γ(π1 × σ1, ψ) · Γ(π2 × σ2, ψ). +The proof of this theorem will occupy the remaining subsections of this section. +Remark 3.3. Let σ ⊂ σ1 ◦ σ2 be the unique irreducible generic subrepresentation of σ. We +have that ψ-Whittaker vectors of σ are the same as ψ-Whittaker vectors of σ1 ◦ σ2. Hence, +Theorem 3.3 implies that +Γ(π × σ, ψ) = Γ(π × σ1, ψ) · Γ(π × σ2, ψ). +Before proving the theorem, we mention two other multiplicative properties that follow +immediately from the theorem. +The first property is that the gamma factor is also multiplicative in the first variable. This +follows from Theorem 3.3 combined with Theorem 3.2. +Corollary 3.2. Let π1 and π2 be representations of Whittaker type of GLn1 (F) and GLn2 (F), +respectively, and let σ be a representation of Whittaker type of GLm (F). Then +Γ((π1 ◦ π2) × σ, ψ) = Γ(π1 × σ, ψ) · Γ(π2 × σ, ψ). +The second corollary allows us to express the gamma factor of two parabolically induced +representations as the product of the gamma factors of the components of the parabolic +induction. It follows by repeatedly using multiplicativity in both variables. +Corollary 3.3. Let π1, . . . , πr and σ1, . . . , σt be irreducible generic representations of +GLn1 (F) , . . . , GLnr (F) and GLm1 (F) , . . . , GLmt (F), respectively. Then + +14 +DAVID SOUDRY AND ELAD ZELINGHER +(1) We have +Γ((π1 ◦ . . . ◦ πr) × (σ1 ◦ . . . ◦ σt), ψ) = +r� +i=1 +t� +j=1 +Γ(πi × σj, ψ). +(2) If π is the unique irreducible generic subrepresentation of π1 ◦ . . . ◦ πr and σ is the +unique irreducible generic subrepresentation of σ1 ◦ . . . ◦ σt, then +Γ(π × σ, ψ) = +r� +i=1 +t� +j=1 +Γ(πi × σj, ψ). +In the next subsections we make preparations for the proof of Theorem 3.3. +3.3.1. Transitivity of parabolic induction. Let τ1, τ2 and τ3 be finite dimensional representa- +tions of GLn1 (F), GLn2 (F) and GLn3 (F), respectively. +We realize elements in (τ1 ◦ τ2) ⊗τ3 as functions GLn1+n2 (F) → τ1 ⊗τ2 ⊗τ3 in the obvious +way. Similarly, we realize elements in τ1 ⊗ (τ2 ◦ τ3) as functions GLn2+n3 → τ1 ⊗ τ2 ⊗ τ3 in +the obvious way. +Consider the space (τ1 ◦ τ2) ◦ τ3. We will regard elements of this space as functions +f : GLn1+n2+n3 (F) × GLn1+n2 (F) → τ1 ⊗ τ2 ⊗ τ3, +where f (g; h) means evaluating f at g ∈ GLn1+n2+n3 (F) and then evaluating the resulting +function at h ∈ GLn1+n2 (F). We will similarly regard elements of τ1 ◦ (τ2 ◦ τ3) as functions +f : GLn1+n2+n3 (F) × GLn2+n3 (F) → τ1 ⊗ τ2 ⊗ τ3. +We have an isomorphism of representations +Lτ1,τ2;τ3 : (τ1 ◦ τ2) ◦ τ3 → τ1 ◦ τ2 ◦ τ3, +given by mapping a function f ∈ (τ1 ◦ τ2) ◦ τ3 to +Lτ1,τ2;τ3f (g) = f (g; In1+n2) , +where g ∈ GLn1+n2+n3 (F). +Similarly, we have an isomorphism of representations +Lτ1;τ2,τ3 : τ1 ◦ (τ2 ◦ τ3) → τ1 ◦ τ2 ◦ τ3, +given by mapping a function f ∈ τ1 ◦ (τ2 ◦ τ3) to +Lτ1;τ2,τ3f (g) = f (g; In2+n3) , +where again g ∈ GLn1+n2+n3 (F). +Assume now that τ1, τ2 and τ3 have non-zero ψ-Whittaker vectors, vτ1,ψ, vτ2,ψ and vτ3,ψ, +respectively, and assume that up to scalar multiplication, these Whittaker vectors are unique. +We denote, as before, the following non-zero ψ-Whittaker vectors vτ1,τ2,ψ = fvτ1,ψ,vτ2,ψ ∈ τ1◦τ2 +and vτ2,τ3,ψ = fvτ2,ψ,vτ3,ψ ∈ τ2 ◦ τ3. We also define the following non-zero ψ-Whittaker vectors +vτ1,τ2◦τ3,ψ = fvτ1,ψ,vτ2,τ3,ψ ∈ τ1 ◦ (τ2 ◦ τ3) and vτ1◦τ2,τ3,ψ = fvτ1,τ2,ψ,vτ3,ψ ∈ (τ1 ◦ τ2) ◦ τ3. Finally, +we define +vτ1,τ2,τ3,ψ = Lτ1;τ2,τ3vτ1,τ2◦τ3,ψ = Lτ1,τ2;τ3vτ1◦τ2,τ3,ψ ∈ τ1 ◦ τ2 ◦ τ3. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +15 +Then vτ1,τ2,τ3,ψ is the ψ-Whittaker vector in τ1 ◦ τ2 ◦ τ3 supported on the double coset +Pn1,n2,n3 ˆwn3,n2,n1Zn1+n2+n3, with vτ1,τ2,τ3,ψ ( ˆwn3,n2,n1) = vτ1,ψ ⊗ vτ2,ψ ⊗ vτ3,ψ, where +ˆwn3,n2,n1 = + + +In1 +In2 +In3 + + . +3.3.2. Intertwining operators. We return to the notations of the beginning of this section. +Let π be a representation of Whittaker type of GLn (F). Let m = m1 + m2, and let σ1 and +σ2 be representations of Whittaker type of GLm1 (F) and GLm2 (F), respectively. +Using the isomorphisms from the previous section, we obtain maps such that the following +diagrams are commutative. +σ1 ◦ (σ2 ◦ π) +idσ1 ⊗Uσ2,π � +Lσ1;σ2,π +� +σ1 ◦ (π ◦ σ2) +Lσ1;π,σ2 +� +σ1 ◦ σ2 ◦ π +˜Uσ2,π +� σ1 ◦ π ◦ σ2 +, +(3.4) +(σ1 ◦ π) ◦ σ2 +Uσ1,π⊗idσ2 � +Lσ1,π;σ2 +� +(π ◦ σ1) ◦ σ2 +Lπ,σ1;σ2 +� +σ1 ◦ π ◦ σ2 +˜Uσ1,π +� π ◦ σ1 ◦ σ2 +, +(3.5) +(σ1 ◦ σ2) ◦ π +Lσ1,σ2;π +� +Uσ1◦σ2,π +� π ◦ (σ1 ◦ σ2) +Lπ;σ1,σ2 +� +σ1 ◦ σ2 ◦ π +˜Uσ1◦σ2,π +� π ◦ σ1 ◦ σ2 +. +(3.6) +Let us explain these diagrams. We begin with explaining (3.4). The map idσ1 ⊗Uσ2,π : σ1⊗ +(σ2 ◦ π) → σ1 ⊗ (π ◦ σ2) is a homomorphism of representations. It defines a homomorphism +σ1 ◦ (σ2 ◦ π) → σ1 ◦ (π ◦ σ2), which we keep denoting by idσ1 ⊗Uσ2,π. By unwrapping the +definitions, we see that the map ˜Uσ2,π : σ1 ◦ σ2 ◦ π → σ1 ◦ π ◦ σ2 is given by the formula +˜Uσ2,π (f) (g) = +� +un,m2∈Nn,m2 +swσ2,πf + + + + +Im1 +Im2 +In + + +� +Im1 +un,m2 +� +g + + . +(3.7) +The commutative diagram (3.5) is similar. We get by unwrapping the definitions that the +map ˜Uσ1,π : σ1 ◦ π ◦ σ2 → π ◦ σ1 ◦ σ2 is given by +˜Uσ1,π (f) (g) = +� +un,m1∈Nn,m1 +swσ1,πf + + + + +Im1 +In +Im2 + + +� +un,m1 +Im2 +� +g + + . +(3.8) +Finally, in the diagram (3.6), we have that ˜Uσ1◦σ2,π : σ1 ◦ σ2 ◦ π → π ◦ σ1 ◦ σ2 is given by +˜Uσ1◦σ2,π (f) (g) = +� +un,m∈Nn,m +˜f + + + + +Im1 +Im2 +In + + un,mg + + , +(3.9) +where for g ∈ GLn+m (F), we mean ˜f (g) = swσ1,πswσ2,πf (g). + +16 +DAVID SOUDRY AND ELAD ZELINGHER +Proposition 3.3. We have +˜Uσ1◦σ2,π = ˜Uσ1,π ◦ ˜Uσ2,π. +Proof. Let f ∈ σ1 ◦ σ2 ◦ π and g ∈ GLn+m (F). Then by (3.7) and (3.8), +( ˜Uσ1,π ◦ ˜Uσ2,π) (g) = +� +X∈Mn×m1(F) +� +Y ∈Mn×m2(F) +˜f + + + + +Im1 +Im2 +In + + + + +Im1 +In +Y +Im2 + + +× + + +Im1 +In +Im2 + + + + +In +X +Im1 +Im2 + + g + + . +A simple computation shows that + + +Im1 +Im2 +In + + + + +Im1 +In +Y +Im2 + + + + +Im1 +In +Im2 + + + + +In +X +Im1 +Im2 + + += + + +Im1 +Im2 +In + + + + +In +X +Y +Im1 +Im2 + + . +Hence, we get +( ˜Uσ1,π ◦ ˜Uσ2,π) (g) = +� +X∈Mn×m1(F) +� +Y ∈Mn×m2(F) +˜f + + + + +Im1 +Im2 +In + + + + +In +X +Y +Im1 +Im2 + + g + + , +and the last sum is ˜Uσ1◦σ2,π (f) (g) by (3.9). +□ +3.3.3. Proof of Theorem 3.3. Let vπ,ψ, vσ1,ψ and vσ2,ψ be non-zero ψ-Whittaker vectors of π, +σ1 and σ2, respectively. We keep the notations from the previous section. We are now ready +to prove Theorem 3.3. +Proof. By definition, we have +(idσ1 ⊗Uσ2,π) (vσ1,σ2◦π,ψ) = Γ(π × σ2, ψ)vσ1,π◦σ2,ψ. +Since Lσ1;σ2,πvσ1,σ2◦π,ψ = vσ1,σ2,π,ψ and Lσ1;π,σ2vσ1,π◦σ2,ψ = vσ1,π,σ2,ψ, we get from the commu- +tative diagram (3.4) that +˜Uσ2,πvσ1,σ2,π,ψ = Γ(π × σ2, ψ)vσ1,π,σ2,ψ. +Similarly, we have that +(Uσ1,π ⊗ idσ2) (vσ1◦π,σ2,ψ) = Γ(π × σ1, ψ)vπ◦σ1,σ2,ψ, +and we get from the commutative diagram (3.5) that +˜Uσ1,πvσ1,π,σ2,ψ = Γ(π × σ1, ψ)vπ,σ1,σ2,ψ. +Finally, we have +Uσ1◦σ2,πvσ1◦σ2,π,ψ = Γ(π × (σ1 ◦ σ2), ψ)vπ,σ1◦σ2,ψ. +Since Lσ1,σ2;πvσ1◦σ2,π,ψ = vσ1,σ2,π,ψ and Lπ;σ1,σ2vπ,σ1◦σ2,ψ = vπ,σ1,σ2,ψ, we get from the commu- +tative diagram (3.6) +˜Uσ1◦σ2,πvσ1,σ2,π,ψ = Γ(π × (σ1 ◦ σ2), ψ)vπ,σ1,σ2,ψ. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +17 +Since ˜Uσ1◦σ2,π = ˜Uσ1,π ◦ ˜Uσ2,π, we get that +Γ(π × (σ1 ◦ σ2), ψ)vπ,σ1,σ2,ψ = Γ(π × σ1, ψ)Γ(π × σ2, ψ)vπ,σ1,σ2,ψ, +and the theorem follows. +□ +3.4. Expression in terms of Bessel functions. In this section, we express the Shahidi +gamma factor of two irreducible generic representations in terms of their Bessel functions. +Let π and σ be irreducible generic representations of GLn (F) and GLm (F), respectively. +We assume that π and σ are realized by their Whittaker models W (π, ψ) and W (σ, ψ), +respectively. We choose the Whittaker vectors of π and σ to be their corresponding Bessel +functions, i.e., we choose vπ,ψ = Jπ,ψ and vσ,ψ = Jσ,ψ. We denote Jσ,π,ψ = vσ,π,ψ = fJσ,ψ,Jπ,ψ +and similarly Jπ,σ,ψ = vπ,σ,ψ = fJπ,ψ,Jσ,ψ. +Assume that n ≥ m. By definition, we have that for any g ∈ GLn+m (F), +(Uσ,πJσ,π,ψ) (g) = Γ(π × σ, ψ)Jπ,σ,ψ (g) . +Substituting g = ˆwm,n, we get +Γ(π × σ, ψ)Jπ,σ,ψ ( ˆwm,n) = +� +u∈Nn,m +swσ,π Jσ,π,ψ ( ˆwn,mu ˆwm,n) , +and therefore +Γ(π × σ, ψ)Jπ,ψ ⊗ Jσ,ψ = +� +A∈Mn×m(F) +swσ,π Jσ,π,ψ +� +Im +A +In +� +. +(3.10) +In order for Jσ,π,ψ +� +Im +A +In +� +not to vanish, we must have +� +Im +A +In +� +∈ Pm,n ˆwn,mZn+m, so +there must exist +� +p1 +x +p2 +� +∈ Pm,n and +� +u1 +y +u2 +� +∈ Zn+m, where u1 ∈ Zn and u2 ∈ Zm, such +that +� +p1 +x +p2 +� � +Im +A +In +� += +� +Im +In +� � +u1 +y +u2 +� +, +i.e., +� +p1 + xA +x +p2A +p2 +� += +� +u2 +u1 +y +� +. +Therefore, we have p1 + xA = 0 and x = +� +0m×(n−m), u2 +� +. +In order to proceed, we will separate two cases, the case where n > m and the case where +n = m. +3.4.1. The case n > m. In this case, we write A = +� +A1 +A2 +� +, where A1 ∈ M(n−m)×m (F) and +A2 ∈ Mm×m (F) and x = +� +0m×(n−m), u2 +� +. Then p1 + xA = 0 implies p1 + u2A2 = 0, and +therefore A2 is invertible. + +18 +DAVID SOUDRY AND ELAD ZELINGHER +Write +� +In +A +Im +� += + + +Im +A1 +In−m +A2 +Im + + = + + +Im +−A−1 +2 +A1 +In−m +A2 + + + + +Im +A−1 +2 +In−m +−A1A−1 +2 +Im + + += + + +−A−1 +2 +Im +A1 +In−m +A2 + + ˆwn,m + + +Im +A−1 +2 +In−m +−A1A−1 +2 +Im + + . +Therefore, we have +Jσ,π,ψ +� +Im +A +In +� += ψ +� +In−m +−A1A−1 +2 +Im +� +σ +� +−A−1 +2 +� +⊗ π +� +A1 +In−m +A2 +� +Jσ,ψ ⊗ Jπ,ψ. +(3.11) +Substituting (3.11) back in (3.10), we get +Γ(π × σ, ψ)Jπ,ψ ⊗ Jσ,ψ += +� +A1∈M(n−m)×m(F) +A2∈GLm(F) +ψ +� +In−m +−A1A−1 +2 +Im +� +π +� +A1 +In−m +A2 +� +⊗ σ +� +−A−1 +2 +� +Jπ,ψ ⊗ Jσ,ψ. +(3.12) +We evaluate both sides of (3.12) at (In, Im) to get +Γ(π × σ, ψ) = +� +A1∈M(n−m)×m(F) +A2∈GLm(F) +ψ +� +In−m +−A1A−1 +2 +Im +� +Jπ,ψ +� +A1 +In−m +A2 +� +Jσ,ψ +� +−A−1 +2 +� +. +Writing +� +A1 +In−m +A2 +� += +� +In−m +A1A−1 +2 +Im +� � +In−m +A2 +� +, +we get +Jπ,ψ +� +A1 +In−m +A2 +� += ψ +� +In−m +A1A−1 +2 +Im +� +Jπ,ψ +� +In−m +A2 +� +, +and therefore +Γ(π × σ, ψ) = +� +A1∈M(n−m)×m(F) +A2∈GLm(F) +Jπ,ψ +� +In−m +A2 +� +Jσ,ψ +� +−A−1 +2 +� +. +The summand is independent of A1. Using the equivariance properties of the Bessel function, +we get that the summand is invariant under Zm left translations of A2. Finally, using the +properties of the Bessel function discussed in Section 2.2.1, we get +Γ(π × σ, ψ) = q +m(2n−m−1) +2 +ωσ (−1) +� +x∈Zm\GLm(F) +Jπ,ψ +� +In−m +x +� +Jσ∨,ψ−1 (x) , +where q +m(2n−m−1) +2 += +��M(n−m)×m (F) +�� · |Zm|. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +19 +3.4.2. The case n = m. In this case, we have −p1 = xA, and therefore A is invertible. We +write +� +In +A +In +� += +� +In +−A−1 +A +� � +In +A−1 +In +� += +� +−A−1 +In +A +� +ˆwn,n +� +In +A−1 +In +� +. +Therefore, we have +Jσ,π,ψ +� +Im +A +In +� += ψ +� +In +A−1 +In +� +σ +� +−A−1� +⊗ π (A) Jσ,ψ ⊗ Jπ,ψ. +(3.13) +Substituting (3.13) in (3.10), we get +Γ(π × σ, ψ)Jπ,ψ ⊗ Jσ,ψ = +� +A∈GLn(F) +ψ +� +In +A−1 +In +� +π (A) ⊗ σ +� +−A−1� +Jπ,ψ ⊗ Jσ,ψ. +(3.14) +Evaluating both sides of (3.14) at (In, In), we get +Γ(π × σ, ψ) = +� +A∈GLn(F) +ψ +� +In +A−1 +In +� +Jπ,ψ (A) Jσ,ψ +� +−A−1� +. +The summand is invariant under Zn left translations. Using the properties of the Bessel +function discussed in Section 2.2.1, we get +Γ(π × σ, ψ) = q +n(n−1) +2 +ωσ (−1) +� +x∈Zn\GLn(F) +ψ +� +In +x−1 +In +� +Jπ,ψ (x) Jσ∨,ψ−1 (x) . +3.4.3. Summary of cases. We conclude this section by writing down formulas for the Shahidi +gamma factor for a pair of irreducible generic representations, in terms of their Bessel func- +tions for all cases. In order to do that, we use Theorem 3.2 and the formulas from Sections +3.4.1 and 3.4.2. +Theorem 3.4. Let π and σ be irreducible generic representations of GLn (F) and GLm (F), +respectively. +(1) If n > m, then +Γ(π × σ, ψ) = q +m(2n−m−1) +2 +ωσ (−1) +� +x∈Zm\GLm(F) +Jπ,ψ +� +In−m +x +� +Jσ∨,ψ−1 (x) . +(2) If n = m, then +Γ(π × σ, ψ) = q +n(n−1) +2 +ωσ (−1) +� +x∈Zn\GLn(F) +ψ +� +In +x−1 +In +� +Jπ,ψ (x) Jσ∨,ψ−1 (x) . +(3) If n < m, then +Γ(π × σ, ψ) = q +n(2m−n−1) +2 +ωπ (−1) +� +x∈Zn\GLn(F) +Jπ,ψ (x) Jσ∨,ψ−1 +� +Im−n +x +� +. +Theorem 3.4 allows us to give a relation between the Jacquet–Piatetski-Shapiro–Shalika +gamma factors defined in Section 2.3 and the Shahidi gamma factor. By Proposition 2.2 and +Proposition 2.3, we get the following corollary. + +20 +DAVID SOUDRY AND ELAD ZELINGHER +Corollary 3.4. Let π be an irreducible cuspidal representation of GLn (F) and let σ be an +irreducible generic representation of GLm (F). Then we have the equality +Γ(π × σ, ψ) = q +m(2n−m−1) +2 +ωσ (−1) γ(π × σ∨, ψ) +in either of the following cases: +(1) n > m. +(2) n = m and σ is cuspidal. +4. Applications +4.1. Quantitative interpretation of gamma factors. In this section, we give a repre- +sentation theoretic interpretation of the absolute value of the Shahidi gamma factor. Our +results relate the absolute value of a normalized version of the Shahidi gamma factor with +the cuspidal support of the representations. +For irreducible generic representations π and σ of GLn (F) and GLm (F), respectively, we +define the normalized Shahidi gamma factor by +Γ∗(π × σ, ψ) = q− nm +2 Γ(π × σ, ψ). +It follows from Corollary 3.3 that under this normalization, the gamma factor is still +multiplicative, i.e., the following proposition holds. +Proposition 4.1. Let π1, . . . , πr and σ1, . . . , σt be irreducible generic representations of +GLn1 (F) , . . . , GLnr (F) and GLm1 (F) , . . . , GLmt (F). Suppose that π is the unique irreducible +generic subrepresentation of π1 ◦ . . . ◦ πr and that σ is the unique irreducible generic subrep- +resentation of σ1 ◦ . . . ◦ σt. Then +Γ∗(π × σ, ψ) = +r� +i=1 +t� +j=1 +Γ∗(πi × σj, ψ). +By Corollaries 3.4 and 2.1 and Proposition 2.5, we have the following proposition, which +allows us to express the size of the absolute value of Γ(π × σ, ψ) where π and σ are cuspidal. +Proposition 4.2. Let π and σ be irreducible cuspidal representations of GLn (F) and +GLm (F), respectively. Then +|Γ∗(π × σ, ψ)| = +� +q− n +2 +n = m and π ∼= σ, +1 +otherwise. +Proposition 4.2 tells us that the size of the normalized Shahidi gamma factor serves as a +“Kronecker delta function” for cuspidal representations. It could be thought of an analog +of [9, Section 8.1]. Combining this with the multiplicativity property, we get the following +theorem, that allows us to recover the cuspidal support of a generic irreducible representation +π by computing |Γ∗(π × σ, ψ)| for any irreducible cuspidal σ. +Theorem 4.1. Let π be an irreducible generic representation of GLn (F) and let σ be an +irreducible cuspidal representation of GLm (F). Then +|Γ∗(π × σ, ψ)| = q− dπ(σ)m +2 +, +where dπ (σ) is the number of times that σ appears in the cuspidal support of π. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +21 +Proof. Suppose that the cuspidal support of π is {π1, . . . , πr} . Then π is the unique irre- +ducible generic subrepresentation of π1 ◦ . . . ◦ πr. The result now follows immediately from +Proposition 4.1 and Proposition 4.2. +□ +As a corollary, we get the following converse theorem, which allows us to determine whether +generic representations of GLn (F) and GLm (F) are isomorphic based on the absolute value +of their normalized gamma factors. It is an analog of [1, Lemma A.6], but our proof is on +the “group side” rather than on the “Galois side”. +Theorem 4.2. Let π1 and π2 be irreducible generic representations of GLn1 (F) and GLn2 (F), +respectively. Suppose that for every m > 0 and every irreducible cuspidal representation σ +of GLm (F) we have +|Γ∗(π1 × σ, ψ)| = |Γ∗(π2 × σ, ψ)| . +Then n1 = n2 and π1 ∼= π2. +Proof. By Theorem 4.1, π1 and π2 have the same cuspidal support. By Theorem 2.1, there +exists a unique irreducible generic representation with a given cuspidal support. +□ +As another corollary, we explain that the functional equations in Theorem 2.2 and Propo- +sition 2.4 fail for π and σ, whenever the cuspidal support of π has a non-empty intersection +with the cuspidal support of σ∨. +Corollary 4.1. Suppose that π and σ are irreducible generic representations of GLn (F) and +GLm (F), respectively, and that n > m (respectively, n = m). Suppose that the cuspidal sup- +port of π has a non-empty intersection with the cuspidal support of σ∨. Then the functional +equation in Theorem 2.2 (respectively, Theorem 2.3) does not hold for π and σ. +Proof. If the functional equation holds for π and σ, then it also holds for π∨ and σ∨. This +can be seen by applying complex conjugation to the functional equation, which sends the +ψ-Whittaker functions to ψ−1-Whittaker functions of the contragredient. As in Corollary 2.1 +(respectively, Corollary 2.2), we get that |γ(π × σ, ψ)| = q− m(n−m−1) +2 +. Whenever γ(π × σ, ψ) +is defined, it is given by the formula in Proposition 2.2 (respectively, Proposition 2.3), and +therefore the formula in Corollary 3.4 holds. This implies that |Γ∗(π × σ∨, ψ)| = 1. +On the other hand, because π and σ∨ have common elements in their cuspidal support, +we have that |Γ∗(π × σ∨, ψ)| < 1. +□ +Remark 4.1. In his unpublished manuscript [23], the first author showed that whenever the +cuspidal support of π does not intersect the cuspidal support of σ∨, the relevant functional +equation holds. Due to length considerations, we do not include the proofs here. +4.2. Consequences for the converse theorem. Our results from Section 4.1 allow us to +improve Nien’s results regarding the converse theorem for irreducible generic representations +of finite general linear groups. +Nien showed in [16] the following theorem. +Theorem 4.3. Let π1 and π2 be two irreducible cuspidal representations of GLn (F) with the +same central character. Suppose that for every 1 ≤ m ≤ n +2, and every irreducible generic +representation σ of GLm (F) we have +γ(π1 × σ, ψ) = γ(π2 × σ, ψ). +(4.1) +Then π1 ∼= π2. + +22 +DAVID SOUDRY AND ELAD ZELINGHER +Using our results and Theorem 4.3, we are able to deduce the following converse theorem, +where π1 and π2 can be arbitrary generic representations (rather than just cuspidal repre- +sentations), and (4.1) needs to be verified only for cuspidal representations σ (rather than +for all generic representations). This is similar to [10, Section 2.4]. +Theorem 4.4. Let π1 and π2 be two irreducible generic representations of GLn (F) with the +same central character. Suppose that for every 1 ≤ m ≤ n +2, and every irreducible cuspidal +representation σ of GLm (F) we have +Γ∗(π1 × σ, ψ) = Γ∗(π2 × σ, ψ). +(4.2) +Then π1 ∼= π2. +Proof. Our proof is by induction on the cardinality of the cuspidal support of π1. +We first notice that by Proposition 4.1, we have that for any 1 ≤ m ≤ n +2 and any irreducible +generic representation σ of GLm (F), +Γ∗(π1 × σ, ψ) = Γ∗(π2 × σ, ψ). +Suppose that π1 is cuspidal, then its cuspidal support is of cardinality 1. If π2 is not cusp- +idal, then its cuspidal support contains an irreducible cuspidal representation τ of GLk (F), +where k ≤ +n +2. Since k < n, we have by Theorem 4.1 that |Γ∗(π1 × σ, ψ)| = 1. We also +have by Theorem 4.1 that |Γ∗(π2 × σ, ψ)| < 1, which is a contraction. Therefore, π2 is also +cuspidal, and by Corollary 3.4 and Theorem 4.3, we have that π1 and π2 are isomorphic. +Suppose now that π1 is not cuspidal. Let {τ1, . . . , τr} the cuspidal support of π1 and let +{τ ′ +1, . . . , τ ′ +r′} be the cuspidal support of π2. Without loss of generality, we have that τ1 is +an irreducible cuspidal representation of GLn1 (F), where n1 ≤ n +2. Then by Theorem 4.1 we +have that |Γ∗(π1 × τ1, ψ)| < 1. Since n1 ≤ n +2, we have that Γ∗(π1 × τ1, ψ) = Γ∗(π2 × τ1, ψ), +and therefore |Γ∗(π2 × τ1, ψ)| < 1. By Theorem 4.1, this implies that τ1 is in the cuspidal +support of π2. Without loss of generality, we may assume that τ ′ +1 = τ1. By Proposition 4.1, +we deduce that for any irreducible generic representation σ of GLm (F) where m ≤ n +2, +r� +j=2 +Γ∗(τj × σ, ψ) = +r′ +� +j=2 +Γ∗(τ ′ +j × σ, ψ). +(4.3) +Let π′ +1 be the unique irreducible generic representation of GLn−n1 (F) with cuspidal support +{τ2, . . . , τr}, and let π′ +2 be the unique irreducible generic representation of GLn−n1 (F) with +cuspidal support {τ ′ +2, . . . , τ ′ +r′}. For i = 1, 2, the central characters of πi and π′ +i are related +by ωπi = ωπ′ +i · ωτ1. Therefore, we have that π′ +1 and π′ +2 also have the same central character. +By Proposition 4.1, we have that (4.3) implies that for every m ≤ n +2 and every irreducible +generic representation σ of GLm (F), +Γ∗(π′ +1 × σ, ψ) = Γ∗(π′ +2 × σ, ψ). +By induction π′ +1 ∼= π′ +2, and therefore {τ2, . . . , τr} = {τ ′ +2, . . . , τ ′ +r′}. Hence, π1 ∼= π2, as required. +□ +4.3. Special values of the Bessel function. In this section, we use our results regarding +multiplicativity of the Shahidi gamma factor, and its relation to the Jacquet–Piatetski- +Shapiro–Shalika gamma factor in order to find an explicit formula for special values of the +Bessel function of irreducible generic representations of GLn (F). For two blocks, such a +formula was given by Curtis and Shinoda in [4, Lemma 3.5]. However, their proof uses + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +23 +Deligne–Lusztig theory, while our proof only uses Green’s character values for irreducible +cuspidal representation of GLn (F), see [8, Section 6] and [17, Section 3.1]. We also provide +a formula for a simple value consisting of three blocks. This generalizes a formula of Chang +for irreducible generic representations of GL3 (F) [3]. +4.3.1. Special value formula for two blocks. Fix an algebraic closure F of F. For every pos- +itive integer n, let Fn be the unique extension of degree n in F. Let NFn/F : F∗ +n → F∗ and +TrFn/F : Fn → F be the norm and the trace maps, respectively. Let � +F∗n be the character group +consisting of all multiplicative characters α: F∗ +n → C∗. +It is known that irreducible cuspidal representations of GLn (F) are in a bijection with +Frobenius orbits of size n of � +F∗n, that is, every irreducible cuspidal representation π of GLn (F) +corresponds to a set of size n of the form {α, αq, . . . , αqn−1}, where α ∈ � +F∗n. +We first recall Nien’s result regarding the computation of the Jacquet–Piatetski-Shapiro– +Shalika gamma factor γ(π×χ, ψ) where π is an irreducible cuspidal representation of GLn (F) +and χ is a representation of GL1 (F), that is, χ: F∗ → C∗ is a multiplicative character. Nien’s +result expresses γ(π × χ, ψ) as a Gauss sum. +Proposition 4.3 ([16, Theorem 1.1]). Let π be an irreducible cuspidal representation of +GLn (F) associated with the Frobenius orbit {α, αq, . . . , αqn−1}, where α ∈ � +F∗n. Let χ: F∗ → +C∗ be a multiplicative character. Then +γ(π × χ, ψ) = (−1)n+1 χ(−1)n+1q−n+1 � +ξ∈F∗n +α−1 (ξ) χ−1(NFn/F(ξ))ψ(TrFn/F(ξ)). +Nien’s proof only uses Green’s character formula for irreducible cuspidal representations, +and does not use Deligne–Lusztig theory. +We are ready to state our result regarding special two blocks values of the Bessel function. +Theorem 4.5. Let n > 1, and let π be an irreducible generic representation of GLn (F) +with cuspidal support {π1, . . . , πr}, where for every 1 ≤ j ≤ r, πj is an irreducible cuspidal +representation of GLnj (F) corresponding to the Frobenius orbit {αj, αq +j, . . . , αqnj−1 +j +}, where +αj ∈ � +F∗nj is a multiplicative character. Then for any c ∈ F∗, +Jπ,ψ +� +In−1 +c +� += (−1)n+r q−n+1 +� +ξ1∈F∗ +n1,...,ξr∈F∗ +nr +�r +j=1 NFnj /F(ξj)=(−1)n−1c−1 +r� +j=1 +� +α−1 +j +(ξj) ψ +� +TrFnj /F (ξj) +�� +. +Proof. By Theorem 3.4, we have that +Γ(π × χ, ψ) = qn−1 � +x∈F∗ +Jπ,ψ +� +In−1 +x +� +χ−1 (−x) . +Multiplying by χ (−c) and averaging over all χ ∈ � +F∗, and using the fact that a sum of a +non-trivial character on a group is zero, we get +1 +|F∗| +� +χ∈� +F∗ +Γ(π × χ, ψ)χ (−c) = qn−1Jπ,ψ +� +In−1 +c +� +. +(4.4) + +24 +DAVID SOUDRY AND ELAD ZELINGHER +By Corollary 3.3, we have that +Γ(π × χ, ψ) = +r� +j=1 +Γ(πj × χ, ψ). +By Corollary 3.4 and Proposition 4.3, we have that +Γ(πj × χ, ψ) = (−1)nj+1 χ(−1)nj � +ξ∈F∗nj +α−1 +j +(ξ) χ(NFnj /F(ξ))ψ(TrFnj /F(ξ)). +Therefore, we get that Γ(π × χ, ψ) is given by +(−1)n+r +� +ξ1∈F∗n1,...,ξr∈F∗nr +� r� +j=1 +α−1 +j +(ξj) ψ(TrFnj /F(ξj)) +� +χ +� +(−1)n +r� +j=1 +NFnj /F (ξj) +� +. +(4.5) +Substituting the expression (4.5) for Γ(π × χ, ψ) in (4.4), and using the fact that a sum of +a non-trivial of character over a group is zero, we get the desired result. +□ +Remark 4.2. The expression for Γ(π × χ, ψ) in (4.5) is originally due to Kondo [12]. He +computed it for the Godement–Jacquet gamma factor. +One can show directly that the +Godement–Jacquet gamma factor coincides with the Shahidi gamma factor for representa- +tions for which both factors are defined. Our proof, which is based on Nien’s result and on +multiplicativity of gamma factors, is different than the one given by Kondo. See also another +proof in [15, Chapter IV, Section 6, Example 4]. +Remark 4.3. In [27], a vast generalization of the method in the proof of Theorem 4.5 is used +in order to find formulas for +Jπ,ψ +� +In−m +cIm +� +. +However, [27] relies on the results of [26], which in turn rely on the local Langlands corre- +spondence. The proof given here does not rely on such results. +4.3.2. Special value formula for three blocks. In this subsection, we use our results to prove a +formula for special values of the Bessel function, for a simple value consisting of three blocks. +This generalizes a formula given by Chang [3] for GL3 (F), generalized later by Shinoda and +Tulunay [21] to GL4 (F). Our proof is different from Chang’s proof, which is based on the +Gelfand–Graev algebra. +We start with the following proposition. +Proposition 4.4. Let π be an irreducible generic representation of GLn (F). Then for any +c ∈ F∗, and any g ∈ GLn (F), we have +Jπ,ψ (g) Jπ,ψ +� +In−1 +c +� +=q−(n−1) +� +tx=(x1,...,xn−1)∈Fn−1 +ψ (−xn−1) Jπ,ψ +� +g +� +In−1 +x +1 +� � +In−1 +c +�� +. +Proof. Let m = 1 and let σ = χ : F∗ → C∗ be a multiplicative character. By (3.12), we have +Γ(π × χ, ψ)Jπ,ψ = +� +tx∈Fn−1 +a∈F∗ +χ +� +−a−1� +ψ +� +In−1 +−a−1x +1 +� +π +� +x +In−1 +a +� +Jπ,ψ. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +25 +We multiply by χ (−c) and average over χ ∈ � +F∗. Using the fact that a sum of a non-trivial +character over a group is zero, and using (4.4), we get +qn−1Jπ,ψ +� +In−1 +c +� +Jπ,ψ = +� +tx=(x1,...,xn−1)∈Fn−1 +ψ +� +−c−1xn−1 +� +π +� +x +In−1 +c +� +Jπ,ψ. +Using the decomposition +� +x +In−1 +c +� += +� +In−1 +c−1x +1 +� � +In−1 +c +� +, +and changing the summation variable x to c · x, we get the desired result. +□ +Theorem 4.6. Suppose n ≥ 3. Then for any irreducible generic representation π of GLn (F) +and any c, c′ ∈ F∗, we have +Jπ,ψ + + +−c′ +In−2 +c + + = +� +s∈F∗ +Jπ,ψ +� +In−1 +s−1c +� +Jπ,ψ +� +sc′ +In−1 +� +(ψ (s) − 1) + δcc′,1 +qn−2, +where +δcc′,1 = +�1 +cc′ = 1, +0 +otherwise. +Proof. We substitute g = +� +c′ +In−1 +� +in Proposition 4.4 to get +Jπ,ψ +� +In−1 +c +� +Jπ,ψ +� +c′ +In−1 +� += q−(n−1) +� +tx=(x1,...,xn−1)∈Fn−1 +ψ (−xn−1) Jπ,ψ +� +cc′ +cx +In−1 +� +. +If xn−1 = 0, then +� +cc′ +cx +In−1 +� +lies in the mirabolic subgroup. By [16, Lemma 2.14], we +have that the Bessel function is zero for elements in the mirabolic subgroup that do not lie +in the upper unipotent subgroup Zn. Therefore, we get that if xn−1 = 0, then x = 0 and +ψ (xn−1) Jπ,ψ +� +cc′ +cx +In−1 +� += δcc′,1. +Suppose now that xn−1 = t ̸= 0. Denote tx′ = (x1, . . . , xn−2) ∈ Fn−2. Then we have +� +cc′ +cx +In−1 +� += + + +1 +0 +t−1c′ +In−2 +t−1x′ +1 + + + + +−t−1c′ +In−2 +tc + + + + +1 +0 +(tc)−1 +In−2 +−t−1x′ +1 + + . +Since we have qn−2 elements in Fn−1 with xn−1 = t, we get that +Jπ,ψ +� +In−1 +c +� +Jπ,ψ +� +c′ +In−1 +� += δcc′,1 +qn−1 + q−1 � +t∈F∗ +ψ (−t) Jπ,ψ + + +−t−1c′ +In−2 +tc + + . + +26 +DAVID SOUDRY AND ELAD ZELINGHER +We proceed as in [3, Page 379] and [21, Lemma 4.2]. We replace c with s−1c and c′ with sc′, +where s ∈ F∗, to get +Jπ,ψ +� +In−1 +s−1c +� +Jπ,ψ +� +sc′ +In−1 +� += δcc′,1 +qn−1 + q−1 � +t∈F∗ +ψ (−st) Jπ,ψ + + +−t−1c′ +In−2 +tc + + . +(4.6) +Summing (4.6) over s ∈ F∗, we get +� +s∈F∗ +Jπ,ψ +� +In−1 +s−1c +� +Jπ,ψ +� +sc′ +In−1 +� += q − 1 +qn−1 δcc′,1 − q−1 � +t∈F∗ +Jπ,ψ + + +−t−1c′ +In−2 +tc + + . +(4.7) +Multiplying (4.6) by ψ (s) and summing over s ∈ F∗, we get +� +s∈F∗ +Jπ,ψ +� +In−1 +s−1c +� +Jπ,ψ +� +sc′ +In−1 +� +ψ (s) += − δcc′,1 +qn−1 + q − 1 +q +Jπ,ψ + + +−c′ +In−2 +c + + − q−1 � +1̸=t∈F∗ +Jπ,ψ + + +−t−1c′ +In−2 +tc + + . +(4.8) +Subtracting (4.7) from (4.8), we get the desired result. +□ +Remark 4.4. Using the formulas in Theorem 4.5 and its proof, one can show that if the +cuspidal support of π does not contain any irreducible representation of GL1 (F), then we +have a simpler formula: +Jπ,ψ + + +−c′ +In−2 +c + + = +� +s∈F∗ +Jπ,ψ +� +In−1 +s−1c +� +Jπ,ψ +� +sc′ +In−1 +� +ψ (s) . +(4.9) +However, if the cuspidal support of π contains irreducible representations of GL1 (F), this +simpler formula does not hold. +Remark 4.5. Using the expression in Theorem 4.5, we have that the expression on the right +hand side of (4.9) is an exponential sum that generalizes the Friedlander–Iwaniec character +sum, see [5, Proposition 6]. The Friedlander–Iwaniec character sum played a role in Zhang’s +work on the twin prime conjecture [28]. +Appendix A. Computation of γ(π × π∨, ψ) when π is cuspidal +In this appendix, we compute the Jacquet–Piatetski-Shapiro–Shalika gamma factor +γ(π × σ, ψ) in the special case where π and σ are irreducible cuspidal representations of +GLn (F) and π ∼= σ∨. We will prove the following theorem. +Theorem A.1. Let π be an irreducible cuspidal representation of GLn (F). Then +γ(π × π∨, ψ) = −1. +This was done in [24, Corollary 4.3]. We provide another proof, since the proof in [24] +relies on results of representations of p-adic groups. +For future purposes, we will prove the following general lemma. We will show that Theo- +rem A.1 follows from it. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +27 +We denote by Pn ≤ GLn (F) the mirabolic subgroup. +Lemma A.1. Let G be a finite group and let H ≤ G be a subgroup. Suppose that H is a +semi-direct product of the form H = N ⋊ GLn (F). Let Ψ : H → C∗ be a character which is +trivial on GLn (F). Let τ be an irreducible representation of G, such that +(1) dim HomH (ResH τ, Ψ) = 1. +(2) dim HomN⋊Pn (ResN×Pn τ, ResN⋊Pn Ψ) = 1. +(3) There exists a functional ℓ ∈ HomZn (ResZn τ, C) and a vector v0 ∈ τ, such that +� +p∈Zn\Pn +� +n∈N +ℓ (τ (np) v0) Ψ−1 (n) = 1. +Then +� +g∈Zn\GLn(F) +� +n∈N +ℓ (τ (ng) v0) Ψ−1 (n) ψ (⟨eng, e1⟩) = −1. +Remark A.1. If F∗ ≤ H lies in the center of G, then (1) implies that the restriction of the +central character of τ to F∗ ≤ H is trivial. +Proof. Notice that we have a containment +HomH (ResH τ, Ψ) ⊂ HomN⋊Pn (ResN×Pn τ, ResN⋊Pn Ψ) . +Since both spaces are one dimensional, we have that they are equal. Denote for v ∈ τ, +L (v) = +� +p∈Zn\Pn +� +n∈N +ℓ (τ (np) v) Ψ−1 (n) . +Then L ∈ HomN⋊Pn (ResN×Pn τ, ResN⋊Pn Ψ) and L ̸= 0 because L (v0) = 1. +Therefore, +L ∈ HomH (ResH τ, Ψ), which implies that L (τ (g) v) = L (v) for any v ∈ τ, and any +g ∈ GLn (F). +Denote +S = +� +g∈Zn\GLn(F) +� +n∈N +ℓ (τ (ng) v0) Ψ−1 (n) ψ (⟨eng, e1⟩) . +We have +S = +� +g∈Pn\GLn(F) +L (τ (g) v0) ψ (⟨eng, e1⟩) = +� +g∈Pn\GLn(F) +ψ (⟨eng, e1⟩) . +We decompose this sum through the center of GLn (F) +S = +� +g∈(F∗·Pn)\GLn(F) +� +a∈F∗ +ψ (⟨enag, e1⟩) . +We have that for t ∈ F, +� +a∈F∗ +ψ (at) = +�−1 +t ̸= 0, +q − 1 +t = 0. +Therefore, we get +S = (q − 1) +� +g∈(F∗·Pn)\GLn(F) +⟨eng,e1⟩=0 +1 − +� +g∈(F∗·Pn)\GLn(F) +⟨eng,e1⟩̸=0 +1, + +28 +DAVID SOUDRY AND ELAD ZELINGHER +which we rewrite as +S = +� +g∈Pn\GLn(F) +⟨eng,e1⟩=0 +1 − +1 +|F∗| +� +g∈Pn\GLn(F) +⟨eng,e1⟩̸=0 +1. +Consider the right action of GLn (F) on Fn \ {0}. This action is transitive. The stabilizer +of en is the mirabolic subgroup Pn. Therefore, for x = (x1, . . . , xn) ∈ Fn \ {0}, we have that +Sx = +� +g∈Pn\GLn(F) +eng=x +1 = 1. +This implies that +S = +� +x∈Fn\{0} +x1=0 +Sx − +1 +|F∗| +� +x∈Fn\{0} +x1̸=0 +Sx = +� +qn−1 − 1 +� +− qn−1 = −1, +as required. +□ +We move to prove Theorem A.1. +Proof. We will use Lemma A.1 in the following setup. Let G = GLn (F) × GLn (F), and let +H = GLn (F) embedded diagonally. Let N = {In} and Ψ = 1. +Let π be an irreducible cuspidal representation of GLn (F), then by Schur’s lemma, the +space +HomGLn(F) (π ⊗ π∨, C) +is one-dimensional. Since π is cuspidal, By [8, Theorem 2.2], the restriction of π to the +mirabolic subgroup Pn is irreducible. Therefore, by Schur’s lemma the space +HomPn (ResPn π ⊗ ResPn π∨, C) +is also one-dimensional. +We take τ = W (π, ψ) ⊗ W (π∨, ψ−1), and ℓ : W (π, ψ) ⊗ W (π∨, ψ−1) → C to be the +functional defined on pure tensors by +ℓ (W ⊗ W ′) = W (In) · W ′ (In) . +We have that ℓ ∈ HomZn (ResZn τ, 1). Let v0 = Jπ,ψ ⊗ Jπ∨,ψ−1. +Consider +� +p∈Zn\Pn +� +n∈N +ℓ (τ (np) v0) Ψ−1 (n) = +� +p∈Zn\Pn +Jπ,ψ (p) Jπ∨,ψ−1 (p) . +By [16, Lemma 2.14], we have that if Jπ,ψ (p) ̸= 0 for p ∈ Pn, then p ∈ Zn. Therefore, +� +p∈Zn\Pn +Jπ,ψ (p) Jπ∨,ψ−1 (p) = +� +p∈Zn\Zn +Jπ,ψ (p) Jπ∨,ψ−1 (p) = 1. +Thus, we showed that the required properties for Lemma A.1 are satisfied. +Using Proposition 2.3, we have +γ(π × π∨, ψ) = +� +g∈Zn\GLn(F) +Jπ,ψ (g) Jπ∨,ψ−1 (g) ψ +�� +eng−1, e1 +�� +. + +GAMMA FACTORS FOR REPRESENTATIONS OF GLn +29 +Replacing g with g−1 and using Proposition 2.1, we have +γ(π × π∨, ψ) = +� +g∈Zn\GLn(F) +Jπ,ψ (g) Jπ∨,ψ−1 (g) ψ (⟨eng, e1⟩) , +and therefore by Lemma A.1 +γ(π × π∨, ψ) = +� +g∈Zn\GLn(F) +� +n∈N +ℓ (τ (ng) v) Ψ−1 (n) ψ (⟨eng, e1⟩) = −1, +as required. +□ +References +[1] H. Atobe and W. T. Gan. Local theta correspondence of tempered representations and Langlands +parameters. Invent. Math., 210(2):341–415, 2017. 3, 21 +[2] J. Chai. Bessel functions and local converse conjecture of Jacquet. J. Eur. Math. Soc. (JEMS), +21(6):1703–1728, 2019. 1 +[3] B. Chang. Decomposition of Gelfand-Graev characters of GL3(q). Comm. Algebra, 4(4):375–401, 1976. +3, 23, 24, 26 +[4] C. W. Curtis and K.-i. Shinoda. Zeta functions and functional equations associated with the components +of the Gelfand-Graev representations of a finite reductive group. In Representation theory of algebraic +groups and quantum groups, volume 40 of Adv. Stud. Pure Math., pages 121–139. Math. Soc. Japan, +Tokyo, 2004. 3, 22 +[5] E. Fouvry, E. Kowalski, and P. Michel. The Friedlander-Iwaniec Character Sum, 2013. 26 +[6] W. T. Gan and A. Ichino. The Gross-Prasad conjecture and local theta correspondence. Invent. Math., +206(3):705–799, 2016. 3 +[7] W. T. Gan and G. Savin. Representations of metaplectic groups I: epsilon dichotomy and local Langlands +correspondence. Compos. Math., 148(6):1655–1694, 2012. 3 +[8] S. I. Gel’fand. Representations of the full linear group over a finite field. Math. USSR Sb., 12(13):13–39, +1970. 4, 5, 6, 23, 28 +[9] H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika. Rankin-Selberg convolutions. Amer. J. Math., +105(2):367–464, 1983. 1, 2, 6, 20 +[10] D. Jiang, C. Nien, and S. Stevens. Towards the Jacquet conjecture on the local converse problem for +p-adic GLn. J. Eur. Math. Soc. (JEMS), 17(4):991–1007, 2015. 3, 22 +[11] N. M. Katz. Estimates for Soto-Andrade sums. J. Reine Angew. Math., 438:143–161, 1993. 3 +[12] T. Kondo. On Gaussian sums attached to the general linear groups over finite fields. J. Math. Soc. +Japan, 15:244–255, 1963. 24 +[13] B. Liu and Q. Zhang. Gamma factors and converse theorems for classical groups over finite fields. J. +Number Theory, 234:285–332, 2022. 1 +[14] B. Liu and Q. Zhang. On a converse theorem for G2 over finite fields. Math. Ann., 383(3-4):1217–1283, +2022. 1 +[15] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford university press, 1998. 24 +[16] C. Nien. A proof of the finite field analogue of Jacquet’s conjecture. Amer. J. Math., 136(3):653–674, +2014. 1, 3, 5, 6, 7, 21, 23, 25, 28 +[17] C. Nien. n × 1 local gamma factors and Gauss sums. Finite Fields Appl., 46:255–270, 2017. 23 +[18] E.-A. Roditty. On gamma factors and bessel functions for representations of general linear groups over +finite fields. Master’s thesis, Tel Aviv University, 2010. 1, 6 +[19] F. Shahidi. Fourier transforms of intertwining operators and Plancherel measures for GL(n). Amer. J. +Math., 106(1):67–111, 1984. 1, 2, 9 +[20] F. Shahidi. A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic +groups. Ann. of Math. (2), 132(2):273–330, 1990. 1 +[21] K. Shinoda and I. Tulunay. Representations of the Hecke algebra for GL4(q). J. Algebra Appl., 4(6):631– +644, 2005. 3, 24, 26 + +30 +DAVID SOUDRY AND ELAD ZELINGHER +[22] A. J. Silberger and E.-W. Zink. The characters of the generalized Steinberg representations of finite +general linear groups on the regular elliptic set. Trans. Amer. Math. Soc., 352(7):3339–3356, 2000. 4 +[23] D. Soudry. On gamma functions of pairs over finite fields. 1979. 3, 6, 21 +[24] R. Ye. Rankin-Selberg gamma factors of level zero representations of GLn. Forum Math., 31(2):503–516, +2019. 1, 7, 26 +[25] R. Ye and E. Zelingher. Exterior square gamma factors for cuspidal representations of GLn: finite field +analogs and level-zero representations. Israel J. Math., 240(2):889–934, 2020. 1 +[26] R. Ye and E. Zelingher. Epsilon factors of representations of finite general linear groups. J. Number +Theory, 221:122–142, 2021. 1, 24 +[27] E. Zelingher. On values of the Bessel function for generic representations of finite general linear groups. +2022. arXiv:2211.03678. 2, 24 +[28] Y. Zhang. Bounded gaps between primes. Ann. of Math. (2), 179(3):1121–1174, 2014. 26 +School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv Univer- +sity, Israel 69978 +Email address: soudry@tauex.tau.ac.il +Department of Mathematics, University of Michigan, 1844 East Hall, 530 Church Street, +Ann Arbor, MI 48109-1043 USA +Email address: eladz@umich.edu + diff --git a/ZdE4T4oBgHgl3EQfOQzc/content/tmp_files/load_file.txt b/ZdE4T4oBgHgl3EQfOQzc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7a9372fcf568fee260e43494985d101fe979e1c1 --- /dev/null +++ b/ZdE4T4oBgHgl3EQfOQzc/content/tmp_files/load_file.txt @@ -0,0 +1,1374 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf,len=1373 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='04964v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='RT] 12 Jan 2023 ON GAMMA FACTORS FOR REPRESENTATIONS OF FINITE GENERAL LINEAR GROUPS DAVID SOUDRY AND ELAD ZELINGHER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We use the Langlands–Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of GLn (Fq) and GLm (Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We prove that the Shahidi gamma factor is multiplicative and show that it is related to the Jacquet– Piatetski-Shapiro–Shalika gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' As an application, we prove a converse theorem based on the absolute value of the Shahidi gamma factor, and improve the converse theorem of Nien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' As another application, we give explicit formulas for special values of the Bessel function of an irreducible generic representation of GLn (Fq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Introduction In the representation theory of p-adic groups, one method of studying irreducible represen- tations is by attaching local factors to the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' These local factors are complex valued functions of a complex variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' They encode various properties of the representations in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' These local factors usually arise from global integrals representing L-functions attached to automorphic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Studying these local factors is crucial for under- standing the global situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This has been done successfully in many cases, including the pioneering works of Jacquet–Piatetski-Shapiro–Shalika [9] and Shahidi [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let F be a finite field with cardinality q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' A local theory of local factors often has a finite field analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It allows one to attach “local constants” to irreducible representations of the F-points version of the group in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We mention the works [18, 16, 25, 13, 14] as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' These local constants usually encode properties analogous to their local factors counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Moreover, these local constant theories often allow one to consider “toy models” for analogous local problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For instance, shortly after Nien’s proof of the analog of Jacquet’s conjecture for finite fields [16], Chai proved the conjecture for the p-adic group case [2], where in his proof he used tools analogous to the ones used by Nien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In her master’s thesis [18], Roditty-Gershon defined a finite field analog of the gamma fac- tor of Jacquet–Piatetski-Shapiro–Shalika [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This gamma factor represents the tensor prod- uct representation, attached to two irreducible generic representations π and σ of GLn (F) and GLm (F), respectively, and is denoted γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Later, Rongqing Ye showed that γ(π × σ, ψ) is related to its local field counterpart through level zero supercuspidal represen- tations [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using this relation and the local Langlands correspondence, Rongqing Ye and the second author were able to express γ(π × σ, ψ) as a product of Gauss sums [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The theory of the finite field version of the gamma factor associated to the tensor product, as it currently appears in the literature, is in some sense not complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The first problem is that the gamma factor γ(π × σ, ψ) is currently not defined for all irreducible generic representations π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is only defined when n ≥ m, and under the assumption that π is cuspidal (and if n = m, σ is also required to be cuspidal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' One can tweak the proofs so they will work for all irreducible generic representations π and σ, such that π and σ∨ have disjoint cuspidal support, but that is not enough in order to define γ(π × σ, ψ) for all pairs 1 2 DAVID SOUDRY AND ELAD ZELINGHER π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' One can try to define γ(π × σ, ψ) naively using the expression involving the Bessel functions of π and σ, but this leads to the second problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The second problem is that the current theory lacks the multiplicativity property of the gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If one naively extends the definition γ(π × σ, ψ) using the approach suggested above, it is not clear that the gamma factor would be multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Both of these difficulties need to be resolved for applications as in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Langlands–Shahidi method provides an alternative approach that solves both of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this paper, we use this method to define a finite field version of the Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We briefly describe the construction now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we consider an intertwining operator Uσ,π : σ◦π → π◦σ, where ◦ denotes parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2, given Whittaker vectors vπ,ψ ∈ π and σ ∈ vσ,ψ, we define Whittaker vectors vπ,σ,ψ ∈ π◦σ and vσ,π,ψ ∈ σ ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By uniqueness of the Whittaker vectors, we have that there exists a constant Γ(π × σ, ψ) ∈ C, such that Uσ,πvσ,π,ψ = Γ(π × σ, ψ) · vπ,σ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We call Γ(π × σ, ψ) the Shahidi gamma factor associated to π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is a finite analog of Shahidi’s local coefficient [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We prove properties of Γ(π × σ, ψ), the most important one is that it is multiplicative (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π, σ1 and σ2 be representations of Whittaker type of GLn (F), GLm1 (F) and GLm2 (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then Γ(π × (σ1 ◦ σ2), ψ) = Γ(π × σ1, ψ) · Γ(π × σ2, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We also express Γ(π ×σ, ψ) in terms of the Bessel functions associated with π and σ when both representations are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We show that if n ≥ m, then up to some simple factors, Γ(π × σ, ψ) is given by the naive extension of γ(π × σ∨, ψ) discussed above (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We deduce a relation between the Shahidi gamma factor and the Jacquet–Piatetski-Shapiro– Shalika gamma factor (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible generic representations of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that π is cuspidal and n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If n = m, suppose that σ is also cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then Γ(π × σ, ψ) = q m(2n−m−1) 2 ωσ (−1) γ(π × σ∨, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The relation between both gamma factors allows us to give a representation theoretic interpretation for the absolute value of the Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We show that, in some sense, the absolute value of the Shahidi gamma factor serves as a good substitute for the order of the pole of the local L-factor associated with the tensor product representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let us stress that the relation to the Jacquet–Piatetski-Shapiro–Shalika gamma factor is crucial for these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The following theorem can be seen as an analog of [9, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible generic representation of GLn (F) and let σ be an irreducible cuspidal representation of GLm (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then ���q −nm 2 Γ(π × σ, ψ) ��� = q− dπ(σ)m 2 , where dπ (σ) is the number of times σ appears in the cuspidal support of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 3 This allows us to deduce a converse theorem based on the absolute value of the normalized Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similar theorems in the local setting were given by Gan and his collaborators in many works (see [7, Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3], [6, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6] and [1, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6]), but our proof is done on the “group side” rather than on the “Galois side”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1 and π2 be two irreducible generic representations of GLn1 (F) and GLn2 (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Assume that for every m and every irreducible cuspidal representation σ of GLm (F) we have ���q− n1m 2 Γ(π1 × σ, ψ) ��� = ���q− n2m 2 Γ(π2 × σ, ψ) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then n1 = n2 and π1 ∼= π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Our results combined with Nien’s converse theorem [16] allow us to deduce a converse theorem that holds under weaker assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is similar to [10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1 and π2 be irreducible generic representations of GLn (F) with the same central character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that for any 1 ≤ m ≤ n 2 and any irreducible cuspidal representation σ of GLm (F) we have Γ(π1 × σ, ψ) = Γ(π2 × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then π1 ∼= π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' As another application of our results, we find explicit formulas for special values of the Bessel function of an irreducible generic representation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The first formula (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) expresses Jπ,ψ ( In−1 c ) as an exotic Kloosterman sum [11, Page 152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This formula is already known in the literature by the work of Curtis–Shinoda [4], but our proof is based on multi- plicativity of the Shahidi gamma factor, rather than on Deligne–Lusztig theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The second formula we find (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6) expresses Jπ,ψ � −c′ In−2 c � as a twisted convolution of values of the form Jπ,ψ ( In−1 c ) and Jπ,ψ � c′ In−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Such a formula was given by Chang for n = 3 [3] and then generalized by Shinoda–Tulunay [21] for n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Chang’s method is based on the Gelfand–Graev algebra, while our method is based on formulas we found for the Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This paper is based on a unpublished note by the first author [23] from 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Given a sequence of positive integers n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , nr, we denote by Pn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr the parabolic subgroup of GLn1+···+nr (F) corresponding to the composition (n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' That is, Pn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr = Dn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr ⋊ Nn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr, where Dn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr = � diag (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , gr) | ∀1 ≤ j ≤ r, gj ∈ GLnj (F) � , Nn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed In1 ∗ ∗ ∗ In2 ∗ ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ∗ Inr \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 4 DAVID SOUDRY AND ELAD ZELINGHER Given representations π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr of GLn1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLnr (F), respectively, we denote by π1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ⊗πr the inflation of π1 ⊗ · · · ⊗ πr to Pn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' That is, π1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ⊗πr is a represen- tation of Pn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr, acting on the space of π1 ⊗· · ·⊗πr, and its action on pure tensors is given by (π1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ⊗πr) (du) v1 ⊗ · · · ⊗ vr = π1 (g1) v1 ⊗ · · · ⊗ πr (gr) vr, where d = diag (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , gr) ∈ Dn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr and u ∈ Nn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr, and for every 1 ≤ j ≤ r, vj ∈ πj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The parabolic induction π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr is defined as the following representation of GLn1+···+nr (F): π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr = Ind GLn1+···+nr(F) Pn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',nr π1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ⊗πr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4], if π is an irreducible representation of GLn (F), then there exist n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , nr > 0 with n1 + · · · + nr = n and irreducible cuspidal representations π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr, of GLn1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLnr (F), such that π is isomorphic to a subrepresentation of the parabolic induction π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Such π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr are unique up to ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We define the cuspidal support of π to be the multiset {π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Generic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let ψ: F → C∗ be a non-trivial additive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let Zn ≤ GLn (F) be the upper triangular unipotent subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We define a character ψ: Zn → C∗ by the formula ψ \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 a1 ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ∗ 1 a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 1 an−1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = ψ �n−1 � k=1 ak � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be a finite dimensional representation of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' π is said to be generic if HomZn (ResZn π, ψ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This condition does not depend on the choice of ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We call a non-zero element in HomZn (ResZn π, ψ) a ψ-Whittaker functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The representation π is generic if and only if there exists 0 ̸= v ∈ π, such that π (u) v = ψ (u) v for every u ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We call such vector a Whittaker vector with respect to ψ, or a ψ-Whittaker vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The dimension of the subspace spanned by the ψ-Whittaker vectors of π is dim HomZn (ResZn π, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We say that π is of Whittaker type if π is generic and the subspace spanned by its ψ-Whittaker vectors is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By a well known result of Gelfand and Graev, we have that if π is generic and irreducible, then it is of Whittaker type [8, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5], [22, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is well known that irreducible cuspidal representations of GLn (F) are generic [22, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The following result is also well known [22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr be representations of Whittaker type of GLn1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLnr (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then the parabolic induction π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr is a representation of Whittaker type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Whittaker models and Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible generic representation of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since π is of Whittaker type, Frobenius reciprocity implies that dim HomGLn(F) � π, IndGLn(F) Zn ψ � = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We denote by W (π, ψ) the unique subspace of IndGLn(F) Zn ψ that is isomorphic to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is the Whittaker model of π with respect to ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Recall that for an irreducible representation π of GLn (F), we have that its contragredient π∨ is isomorphic to πι, where πι is the representation acting on the space of π by πι (g) = π (gι), where for g ∈ GLn (F), gι = t� g−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (This follows from the fact that for g ∈ GLn (F), the trace characters of π and π∨ are related by trπ∨ (g) = trπ(g−1), and from the fact that g−1 and t(g−1) are conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=') Using the isomorphism π∨ ∼= πι, we get an isomorphism of vector spaces W (π, ψ) → W (π∨, ψ−1), given by W �→ ˜W, where ˜W (g) = W (wngι) , and where wn ∈ GLn (F) is the long Weyl element wn = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Under the realization of π by its Whittaker model W (π, ψ), the one-dimensional subspace spanned by the ψ-Whittaker vectors of π is realized as the one-dimensional subspace of W (π, ψ) consisting of functions W ∈ W (π, ψ), such that W (gu) = ψ (u) W (g), for every u ∈ Zn and every g ∈ GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By [8, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5], there exists a (unique) element W in this one-dimensional subspace such that W (In) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We call this W the normalized Bessel function of π with respect to ψ, and denote it by Jπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' To summarize, the Bessel function Jπ,ψ is the unique element in W (π, ψ), such that (1) Jπ,ψ (In) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (2) Jπ,ψ (gu) = ψ (u) Jπ,ψ (g), for every g ∈ GLn (F) and u ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Bessel function enjoys the following identities that relate it to its complex conjugate and to its contragredient [16, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='15 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For any irreducible generic representation π of GLn (F) and any g ∈ GLn (F), we have the following identities: (1) Jπ,ψ (g−1) = Jπ,ψ (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (2) Jπ,ψ (g−1) = Jπ∨,ψ−1 (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let vπ,ψ be a non-zero ψ-Whittaker vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If we choose an inner product (·, ·)π on π which is invariant under the GLn (F)-action, we have that the assignment ℓπ,ψ : π → C given by vπ �→ (vπ, vπ,ψ)π defines a Whittaker functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Whittaker model of π can be described using Frobenius reciprocity as W (π, ψ) = {Wvπ | vπ ∈ π}, where for g ∈ GLn (F) and vπ ∈ π, we define Wvπ (g) = (π (g) vπ, vπ,ψ)π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Bessel function is given by Jπ,ψ (g) = (π (g) vπ,ψ, vπ,ψ)π (vπ,ψ, vπ,ψ)π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 6 DAVID SOUDRY AND ELAD ZELINGHER All of the properties of the Bessel function listed above now follow immediately from the fact that (·, ·)π is an inner product, and that vπ,ψ is a ψ-Whittaker vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Moreover, the projection operator to the one-dimensional subspace spanned by the ψ-Whittaker vectors prCvπ,ψ can be described in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The first way is by using the inner product, in which case for vπ ∈ π, prCvπ,ψ (vπ) = (vπ, vπ,ψ)π (vπ,ψ, vπ,ψ)π vπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The second way is by averaging, in which case prCvπ,ψ (vπ) = 1 |Zn| � u∈Zn ψ−1 (u) π (u) vπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By completing vπ,ψ to an orthogonal basis of π and using the fact that the subspace spanned by the ψ-Whittaker vectors is one dimensional, we see that tr � prCvπ,ψ ◦π (g) � = Jπ,ψ (g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is [8, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Jacquet–Piatetski-Shapiro–Shalika gamma factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible generic representations of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For most π and σ, one can define a constant attached to π and σ called the Jacquet–Piatetski-Shapiro–Shalika gamma factor of π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is also known as the Rankin–Selberg gamma factor of π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is a finite field analog of the definition given by Jacquet–Piatetski-Shapiro–Shalika [9] for p-adic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' These were explained in Piatetski-Shapiro’s lectures in 1976 and studied in an unpublished note from 1979 by the first author [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The case n > m was also studied in Roddity-Gershon’s master’s thesis under the supervision of the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The case n > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In her master’s thesis [18], Edva Roditty-Gershon defined the Jacquet–Piatetski-Shapiro–Shalika gamma factor γ(π × σ, ψ), under the assumption that π is cuspidal and that n > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Roddity-Gershon’s thesis is unpublished, but her main results are presented by Nien in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We briefly review these results now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The first result is a functional equation that defines the Jacquet–Piatetski-Shapiro–Shalika gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that n > m and that π is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For any W ∈ W (π, ψ) and W ′ ∈ W (σ, ψ−1), and any 0 ≤ j ≤ n − m − 1, we define Zj (W, W ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) � h∈Zm\\GLm(F) � x∈M(n−m−j−1)×m(F) W \uf8eb \uf8ed h x In−m−j−1 Ij+1 \uf8f6 \uf8f8 W ′ (h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We are now ready to state the functional equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 ([16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' There exists a non-zero constant γ(π × σ, ψ) ∈ C, such that for every 0 ≤ j ≤ m − m − 1, every W ∈ W (π, ψ) and every W ′ ∈ W (σ, ψ−1), we have qmjγ(π × σ, ψ)Zj (W, W ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) = Zn−m−j−1 � π∨ � Im wn−m � ˜W, ˜W ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ−1 � , where wn−m ∈ GLn−m (F) is the long Weyl element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The second result expresses the gamma factor γ(π × σ, ψ) in terms of the Bessel functions of π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 7 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 ([16, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Under the assumptions above, we have γ(π × σ, ψ) = � h∈Zm\\GLm(F) Jπ,ψ � In−m h � Jσ,ψ−1 (h) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 that γ(π × σ, ψ) = γ(π∨ × σ∨, ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Moreover, applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 twice, we get the following corollary regarding the abso- lute value of γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that γ(π × σ, ψ)γ(π∨ × σ∨, ψ−1) = q−m(n−m−1), and therefore |γ(π × σ, ψ)| = q− m(n−m−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The case n = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The case n = m was discussed in Piatetski-Shapiro’s lecture and is explained briefly in Rongqing Ye’s work [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let S (Fn) be the space of functions φ: Fn → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For a function φ ∈ S (Fn), we define its Fourier transform Fψφ: Fn → C by the formula Fψφ (y) = � x∈Fn φ (x) ψ (⟨x, y⟩) , where if x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , xn) ∈ Fn and y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , yn) ∈ Fn, then ⟨x, y⟩ is the standard pairing ⟨x, y⟩ = n � i=1 xiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible cuspidal representations of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We define for any W ∈ W (π, ψ), W ′ ∈ W (σ, ψ−1) and any φ ∈ S (Fn) Z (W, W ′, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) = � g∈Zn\\GLn(F) W (g) W ′ (g) φ (eng) , where en = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , 0, 1) ∈ Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We are now ready to introduce the functional equation that defines γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 ([24, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' There exists a non-zero constant γ(π × σ, ψ), such that for any W ∈ W (π, ψ), W ′ ∈ W (σ, ψ−1), and any φ ∈ S (Fn) with φ (0) = 0, we have Z( ˜W, ˜W ′, Fψφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ−1) = γ(π × σ, ψ)Z (W, W ′, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly to the case n > m, we have an expression of γ(π × σ, ψ) in terms of the Bessel functions of π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 ([24, Equation (16)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible cuspidal representations of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then γ(π × σ, ψ) = � g∈Zn\\GLn(F) Jπ,ψ (g) Jσ,ψ−1 (g) ψ �� eng−1, e1 �� , where e1 = (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , 0) ∈ Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 8 DAVID SOUDRY AND ELAD ZELINGHER It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 that γ(π∨ × σ∨, ψ−1) = γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We now move to discuss the absolute value of γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In order to do that, we first explain how to extend the functional equation in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 to all functions in S (Fn) for most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' To begin, we notice that for the indicator function of 0 ∈ Fn, which we denote δ0, we have that Z (W, W ′, δ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We also notice that if π is not isomorphic to σ∨, then Z (W, W ′, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) = 0, where 1 represents the constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is because Z (W, W ′, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) = � g∈Zn\\GLn(F) W (g) W ′ (g) defines a GLn (F)-invariant pairing W (π, ψ) ⊗ W (σ, ψ−1) → C, but such non-trivial pairing exists only when π is isomorphic to σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' These two observations imply the following extension of the functional equation, in the special case where π is not isomorphic to σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that π ≇ σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then for any φ ∈ S (Fn) we have Z( ˜W, ˜W ′, Fψφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ−1) = γ(π × σ, ψ)Z (W, W ′, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Write φ = φ0 + φ1, where φ0 = φ − φ(0) and φ1 = φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then φ0 (0) = 0 and Fψφ1 = qnφ (0) δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since Z is linear in φ, we have from the discussion above that Z (W, W ′, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) = Z (W, W ′, φ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ) and that Z( ˜W, ˜W ′, Fψφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ−1) = Z( ˜W, ˜W ′, Fψφ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The statement now follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ As a result, we get the following corollary regarding the absolute value of γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible cuspidal representations of GLn (F) such that π ≇ σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then γ(π × σ, ψ)γ(π∨ × σ∨, ψ−1) = qn, and therefore |γ(π × σ, ψ)| = q n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This follows by applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 twice, and from the fact that the Fourier transform satisfies Fψ−1Fψφ = qnφ, for any φ ∈ S (Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ We are left to deal with the case π ∼= σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this case, the gamma factor γ(π × π∨, ψ) can be computed explicitly and it equals −1, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We summarize all cases in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible cuspidal representations of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If π ≇ σ∨ then |γ(π × σ, ψ)| = q n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If π ∼= σ∨ then |γ(π × σ, ψ)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Shahidi gamma factors (local coefficients) In this section, we use the Langlands–Shahidi method in order to define a gamma factor for two representations of Whittaker type of finite general linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is the finite field analog of Shahidi’s local coefficient, which uses an intertwining operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The treatment in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 is a finite field analog of Shahidi’s work on local coefficients over local fields [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Unlike the Jacquet–Piatetski-Shapiro–Shalika gamma factors discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, the Shahidi gamma factor can be defined uniformly for all irreducible generic representations of GLn (F) and GLm (F), regardless of n > m or whether the representations are cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We prove properties of the Shahidi gamma factor, where the most important one is the multiplicativity property, which explains how this gamma factor behaves under parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We end this section by expressing the Shahidi gamma factor in terms of the Bessel functions associated with the representations, and showing its relation to the Jacquet– Piatetski-Shapiro–Shalika gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The intertwining operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let n and m be positive integers and let π and σ be representations of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We define a linear map σ ⊗π → π ⊗σ acting on pure tensors by component swap: swσ,π (vσ ⊗ vπ) = vπ ⊗ vσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For a function f : GLn+m (F) → σ ⊗ π, we denote by ˜f : GLn+m (F) → π ⊗ σ the function ˜f (g) = swσ,π (f(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We consider the following intertwining operator Tσ,π : σ ◦ π → π ◦ σ, defined for f ∈ σ ◦ π and g ∈ GLn+m (F) by the formula Tσ,πf (g) = � p∈Pn,m (π⊗σ) � p−1� ˜f ( ˆwn,mpg), where ˆwn,m is the following Weyl element ˆwn,m = � Im In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Writing p ∈ Pn,m as p = du, where d ∈ Dn,m and u ∈ Nn,m, and using the left Dm,n- equivariance property of f, one checks that Tσ,πf (g) = |Dn,m| · Uσ,πf (g) , where Uσ,πf (g) = � u∈Nn,m ˜f ( ˆwn,mug) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By construction, we have that Tσ,π and Uσ,π are non-zero elements of the space HomGLn+m(F) (σ ◦ π, π ◦ σ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose now that π and σ are representations of Whit- taker type of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 we have that the parabol- ically induced representations σ ◦ π and π ◦ σ are also of Whittaker type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let vσ,ψ ∈ σ 10 DAVID SOUDRY AND ELAD ZELINGHER and vπ,ψ ∈ π be non-zero ψ-Whittaker vectors for σ and π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We may define a non-zero ψ-Whittaker vector fvσ,ψ,vπ,ψ for σ ◦ π by the formula fvσ,ψ,vπ,ψ (g) = �ψ (u) (σ⊗π) (p) vσ,ψ ⊗ vπ,ψ g = p ˆwn,mu, p ∈ Pm,n, u ∈ Zn+m, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, we may define fvπ,ψ,vσ,ψ ∈ π ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since Uσ,π is an intertwining operator, we have that Uσ,πfvσ,ψ,vπ,ψ is a ψ-Whittaker vector of π ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since fvπ,ψ,vσ,ψ is the unique non-zero ψ-Whittaker vector of π ◦ σ up to scalar, we must have that Uσ,πfvσ,ψ,vπ,ψ = γ · fvπ,ψ,vσ,ψ, where γ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is easy to check that this number γ does not depend on the choice of ψ-Whittaker vectors vσ,ψ and vπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In order to ease the notation, we denote vσ,π,ψ = fvσ,ψ,vπ,ψ, where we suppress vσ,ψ and vπ,ψ from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, we denote vπ,σ,ψ = fvπ,ψ,vσ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Shahidi gamma factor of π and σ with respect to ψ is the unique number Γ(π × σ, ψ) ∈ C, such that Uσ,πvσ,π,ψ = Γ(π × σ, ψ) · vπ,σ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If π ◦ σ is irreducible, then so is σ ◦ π, and since Uσ,π is a non-zero intertwining operator, it is an isomorphism and Γ(π × σ, ψ) must be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' However, in the general case it is not obvious at this point that Γ(π × σ, ψ) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will show this later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' As in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we may choose invariant inner products (·, ·)π and (·, ·)σ on π and σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We then have a natural inner product (·, ·)σ⊗π on σ ⊗π, which defines an inner product on σ ◦ π by the formula (f1, f2)σ◦π = � g∈Pm,n\\GLn+m(F) (f1 (g) , f2 (g))σ⊗π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using this inner product, the Whittaker functional ℓσ◦π,ψ (f) = (f, vσ,π,ψ)σ◦π is related to the Whittaker functionals ℓσ,ψ (vσ) = (vσ, vσ,ψ)σ and ℓπ,ψ (vπ) = (vπ, vπ,ψ)π by the formula ℓσ◦π,ψ (f) = � u∈Nn,m ℓσ,ψ ⊗ ℓπ,ψ (f ( ˆwn,mu)) ψ−1 (u) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, by exchanging the roles of π and σ, we have that Whittaker functional ℓπ◦σ,ψ is given by a similar formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the definitions of the inner products, and the fact that elements in π ◦ σ are left invariant under Nn,m, we see that Uπ,σ is the adjoint of Uσ,π, with respect to our choice of inner products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the relation between vσ,π,ψ and vπ,σ,ψ, we obtain the following relation ℓσ◦π,ψ ◦ Uπ,σ = Γ(π × σ, ψ) · ℓπ◦σ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is how the Shahidi gamma factor is usually defined in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Dependence on ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For any a ∈ F∗, let ψa : F → C∗ be the additive character ψa (x) = ψ (ax) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is well known that all non-trivial additive characters of F are of the form ψa for some a ∈ F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this section, we give a relation between Γ(π × σ, ψ) and Γ(π × σ, ψa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let a ∈ F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that τ is a generic representation of GLk (F) with a non-zero ψ- Whittaker vector vτ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let dk = diag � 1, a, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , ak−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then we have that τ (dk) vτ,ψ is a non-zero ψa-Whittaker vector of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The map v �→ τ (dk) v is a linear isomorphism from the subspace spanned by the ψ-Whittaker vectors of τ to the subspace spanned by the ψa-Whittaker vectors of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In particular, if vτ,ψ is the unique (up to scalar multiplication) ψ-Whittaker vector of τ, then τ (dk) vτ,ψ is the unique (up to scalar multiplication) ψa-Whittaker vector of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let vπ,ψ ∈ π and vσ,ψ ∈ σ be non-zero ψ-Whittaker vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Assume that π and σ have central characters, and denote them by ωπ and ωσ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let vσ,π,ψa = fσ(dm)vσ,ψ,π(dn)vπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, we define vπ,σ,ψa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We first express vσ,π,ψa in terms of vσ,π,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will use this relation later to show a relation between the gamma factors Γ(π × σ, ψa) and Γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have vσ,π,ψa = ωσ (a)−n ρ (dn+m) vσ,π,ψ, where ρ (dn+m) denotes right translation by dn+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let f = ωσ (a)−n ρ (dn+m) vσ,π,ψ ∈ σ ◦π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By the discussion above, f is a ψa-Whittaker vector of σ ◦ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that f ( ˆwn,m) = ωσ (a)−n vσ,π,ψ ( ˆwn,mdn+m) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Writing dn+m = diag (dn, andm), we have ˆwn,mdn+m = diag (andm, dn) ˆwn,m, and hence f ( ˆwn,m) = (σ (dm) ⊗ π (dn)) vσ,π,ψ ( ˆwn,m) = σ (dm) vσ,ψ ⊗ π (dn) vπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This shows that f = vσ,π,ψa, as both are ψa-Whittaker vectors in σ ◦ π, and both agree at the point ˆwn,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have Γ(π × σ, ψa) = ωπ (a)m · ωσ (a)−n · Γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By definition, we have that Γ(π × σ, ψa)vπ,σ,ψa = Uσ,πvσ,π,ψa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, Γ(π × σ, ψa)ωπ (a)−m ρ (dn+m) vπ,σ,ψ = ωσ (a)−n Uσ,πρ (dn+m) vσ,π,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we get that Γ(π × σ, ψa)ωσ (a)n ωπ (a)−m vπ,σ,ψ = Uσ,πvσ,π,ψ, 12 DAVID SOUDRY AND ELAD ZELINGHER which implies that Γ(π × σ, ψa)ωσ (a)n ωπ (a)−m = Γ(π × σ, ψ), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Relation between Γ(π × σ, ψ) and Γ(σ∨ × π∨, ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this section, we analyze the relation between Γ(π × σ, ψ) and Γ(σ∨ × π∨, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Recall that for a finite dimensional representation τ of GLk (F), we have that τ ∨ ∼= τ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If vτ,ψ is a non-zero ψ-Whittaker vector for τ, then τ (wk) vτ,ψ is a non-zero ψ−1-Whittaker vector for τ ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that πι ◦ σι ∼= (σ ◦ π)ι by the isomorphism Sσ,π : (σ ◦ π)ι ∼= πι ◦ σι that sends f ∈ (σ ◦ π)ι to the function (Sσ,πf) (g) = ˜f ( ˆwn,mgι) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let vπι,σι,ψ−1 = fπ(wn)vπ,ψ,σ(wm)vσ,ψ ∈ πι ◦ σι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then vπι,σι,ψ−1 is a non-zero ψ−1-Whittaker vector of πι ◦ σι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' On the other hand, by the discussion above, a non-zero ψ−1-Whittaker vector of (σ ◦ π)ι is given by ρ (wm+n) vσ,π,ψ, where ρ (wm+n) represents right translation by wm+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore Sσ,πρ (wm+n) vσ,π,ψ is another non-zero ψ−1-Whittaker vector of πι ◦ σι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have vπι,σι,ψ−1 = Sσ,πρ (wm+n) vσ,π,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that Sσ,πρ (wm+n) vσ,π,ψ ( ˆwm,n) = swσ,π vσ,π,ψ (wm+n) = π (wn) vπ,ψ ⊗ σ (wm) vσ,ψ, where in the last step we used the fact that diag (wm, wn) ˆwn,m = wn+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since Sσ,πρ (wm+n) vσ,π,ψ and vπι,σι,ψ−1 are both ψ−1-Whittaker vectors for the representa- tion of Whittaker type πι ◦ σι, and they both agree at the point ˆwm,n, they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Similarly, let vσι,πι,ψ−1 = fσ(wm)vσ,ψ,π(wn)vπ,ψ ∈ σι ◦ πι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 with roles of the representations π and σ exchanged, we have vσι,πι,ψ−1 = Sπ,σρ (wm+n) vπ,σ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then Γ(π × σ, ψ) = Γ(σ∨ × π∨, ψ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By definition, Uπι,σιvπι,σι,ψ−1 = Γ(σι × πι, ψ−1) · vσι,πι,ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3), we get Uπι,σιSσ,πρ (wm+n) vσ,π,ψ = Γ(σι × πι, ψ−1) · Sπ,σρ (wm+n) vπ,σ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' A simple computation shows that Uπι,σι ◦ Sσ,π = Sπ,σ ◦ Uσ,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 13 Hence, we get that Sπ,σρ (wm+n) Uσ,πvσ,π,ψ = Γ(σι × πι, ψ−1) · Sπ,σρ (wm+n) vπ,σ,ψ, which implies that Uσ,πvσ,π,ψ = Γ(σι × πι, ψ−1) · vπ,σ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we must have Γ(σι × πι, ψ−1) = Γ(π × σ, ψ), and the statement in the theorem follows, since σι ∼= σ∨ and πι ∼= π∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be representations of Whittaker type of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Assume that both π and σ have central characters, and denote them by ωπ and ωσ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then Γ(π × σ, ψ) = Γ(σ∨ × π∨, ψ) · ωπ (−1)m ωσ (−1)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Multiplicativity of Gamma factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this section, we show that Γ(π × σ, ψ) is multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be a representation of Whittaker type of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let m = m1 + m2, and let σ1 and σ2 be representations of Whittaker type of GLm1 (F) and GLm2 (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, the parabolic induction σ1 ◦ σ2 is also a representation of Whittaker type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Hence, the gamma factor Γ(π × (σ1 ◦ σ2), ψ) is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will show the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Γ(π × (σ1 ◦ σ2), ψ) = Γ(π1 × σ1, ψ) · Γ(π2 × σ2, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The proof of this theorem will occupy the remaining subsections of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let σ ⊂ σ1 ◦ σ2 be the unique irreducible generic subrepresentation of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that ψ-Whittaker vectors of σ are the same as ψ-Whittaker vectors of σ1 ◦ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Hence, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 implies that Γ(π × σ, ψ) = Γ(π × σ1, ψ) · Γ(π × σ2, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Before proving the theorem, we mention two other multiplicative properties that follow immediately from the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The first property is that the gamma factor is also multiplicative in the first variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 combined with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1 and π2 be representations of Whittaker type of GLn1 (F) and GLn2 (F), respectively, and let σ be a representation of Whittaker type of GLm (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then Γ((π1 ◦ π2) × σ, ψ) = Γ(π1 × σ, ψ) · Γ(π2 × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The second corollary allows us to express the gamma factor of two parabolically induced representations as the product of the gamma factors of the components of the parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It follows by repeatedly using multiplicativity in both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr and σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , σt be irreducible generic representations of GLn1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLnr (F) and GLm1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLmt (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then 14 DAVID SOUDRY AND ELAD ZELINGHER (1) We have Γ((π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr) × (σ1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ σt), ψ) = r� i=1 t� j=1 Γ(πi × σj, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (2) If π is the unique irreducible generic subrepresentation of π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr and σ is the unique irreducible generic subrepresentation of σ1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ σt, then Γ(π × σ, ψ) = r� i=1 t� j=1 Γ(πi × σj, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In the next subsections we make preparations for the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Transitivity of parabolic induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let τ1, τ2 and τ3 be finite dimensional representa- tions of GLn1 (F), GLn2 (F) and GLn3 (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We realize elements in (τ1 ◦ τ2) ⊗τ3 as functions GLn1+n2 (F) → τ1 ⊗τ2 ⊗τ3 in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, we realize elements in τ1 ⊗ (τ2 ◦ τ3) as functions GLn2+n3 → τ1 ⊗ τ2 ⊗ τ3 in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Consider the space (τ1 ◦ τ2) ◦ τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will regard elements of this space as functions f : GLn1+n2+n3 (F) × GLn1+n2 (F) → τ1 ⊗ τ2 ⊗ τ3, where f (g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' h) means evaluating f at g ∈ GLn1+n2+n3 (F) and then evaluating the resulting function at h ∈ GLn1+n2 (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will similarly regard elements of τ1 ◦ (τ2 ◦ τ3) as functions f : GLn1+n2+n3 (F) × GLn2+n3 (F) → τ1 ⊗ τ2 ⊗ τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have an isomorphism of representations Lτ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='τ3 : (τ1 ◦ τ2) ◦ τ3 → τ1 ◦ τ2 ◦ τ3, given by mapping a function f ∈ (τ1 ◦ τ2) ◦ τ3 to Lτ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='τ3f (g) = f (g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In1+n2) , where g ∈ GLn1+n2+n3 (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, we have an isomorphism of representations Lτ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='τ2,τ3 : τ1 ◦ (τ2 ◦ τ3) → τ1 ◦ τ2 ◦ τ3, given by mapping a function f ∈ τ1 ◦ (τ2 ◦ τ3) to Lτ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='τ2,τ3f (g) = f (g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In2+n3) , where again g ∈ GLn1+n2+n3 (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Assume now that τ1, τ2 and τ3 have non-zero ψ-Whittaker vectors, vτ1,ψ, vτ2,ψ and vτ3,ψ, respectively, and assume that up to scalar multiplication, these Whittaker vectors are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We denote, as before, the following non-zero ψ-Whittaker vectors vτ1,τ2,ψ = fvτ1,ψ,vτ2,ψ ∈ τ1◦τ2 and vτ2,τ3,ψ = fvτ2,ψ,vτ3,ψ ∈ τ2 ◦ τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We also define the following non-zero ψ-Whittaker vectors vτ1,τ2◦τ3,ψ = fvτ1,ψ,vτ2,τ3,ψ ∈ τ1 ◦ (τ2 ◦ τ3) and vτ1◦τ2,τ3,ψ = fvτ1,τ2,ψ,vτ3,ψ ∈ (τ1 ◦ τ2) ◦ τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Finally, we define vτ1,τ2,τ3,ψ = Lτ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='τ2,τ3vτ1,τ2◦τ3,ψ = Lτ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='τ3vτ1◦τ2,τ3,ψ ∈ τ1 ◦ τ2 ◦ τ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 15 Then vτ1,τ2,τ3,ψ is the ψ-Whittaker vector in τ1 ◦ τ2 ◦ τ3 supported on the double coset Pn1,n2,n3 ˆwn3,n2,n1Zn1+n2+n3, with vτ1,τ2,τ3,ψ ( ˆwn3,n2,n1) = vτ1,ψ ⊗ vτ2,ψ ⊗ vτ3,ψ, where ˆwn3,n2,n1 = \uf8eb \uf8ed In1 In2 In3 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Intertwining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We return to the notations of the beginning of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be a representation of Whittaker type of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let m = m1 + m2, and let σ1 and σ2 be representations of Whittaker type of GLm1 (F) and GLm2 (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the isomorphisms from the previous section, we obtain maps such that the following diagrams are commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' σ1 ◦ (σ2 ◦ π) idσ1 ⊗Uσ2,π � Lσ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='σ2,π � σ1 ◦ (π ◦ σ2) Lσ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='π,σ2 � σ1 ◦ σ2 ◦ π ˜Uσ2,π � σ1 ◦ π ◦ σ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4) (σ1 ◦ π) ◦ σ2 Uσ1,π⊗idσ2 � Lσ1,π;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='σ2 � (π ◦ σ1) ◦ σ2 Lπ,σ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='σ2 � σ1 ◦ π ◦ σ2 ˜Uσ1,π � π ◦ σ1 ◦ σ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) (σ1 ◦ σ2) ◦ π Lσ1,σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='π � Uσ1◦σ2,π � π ◦ (σ1 ◦ σ2) Lπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='σ1,σ2 � σ1 ◦ σ2 ◦ π ˜Uσ1◦σ2,π � π ◦ σ1 ◦ σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6) Let us explain these diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We begin with explaining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The map idσ1 ⊗Uσ2,π : σ1⊗ (σ2 ◦ π) → σ1 ⊗ (π ◦ σ2) is a homomorphism of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It defines a homomorphism σ1 ◦ (σ2 ◦ π) → σ1 ◦ (π ◦ σ2), which we keep denoting by idσ1 ⊗Uσ2,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By unwrapping the definitions, we see that the map ˜Uσ2,π : σ1 ◦ σ2 ◦ π → σ1 ◦ π ◦ σ2 is given by the formula ˜Uσ2,π (f) (g) = � un,m2∈Nn,m2 swσ2,πf \uf8eb \uf8ed \uf8eb \uf8ed Im1 Im2 In \uf8f6 \uf8f8 � Im1 un,m2 � g \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='7) The commutative diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We get by unwrapping the definitions that the map ˜Uσ1,π : σ1 ◦ π ◦ σ2 → π ◦ σ1 ◦ σ2 is given by ˜Uσ1,π (f) (g) = � un,m1∈Nn,m1 swσ1,πf \uf8eb \uf8ed \uf8eb \uf8ed Im1 In Im2 \uf8f6 \uf8f8 � un,m1 Im2 � g \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='8) Finally, in the diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6), we have that ˜Uσ1◦σ2,π : σ1 ◦ σ2 ◦ π → π ◦ σ1 ◦ σ2 is given by ˜Uσ1◦σ2,π (f) (g) = � un,m∈Nn,m ˜f \uf8eb \uf8ed \uf8eb \uf8ed Im1 Im2 In \uf8f6 \uf8f8 un,mg \uf8f6 \uf8f8 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='9) where for g ∈ GLn+m (F), we mean ˜f (g) = swσ1,πswσ2,πf (g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 16 DAVID SOUDRY AND ELAD ZELINGHER Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have ˜Uσ1◦σ2,π = ˜Uσ1,π ◦ ˜Uσ2,π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let f ∈ σ1 ◦ σ2 ◦ π and g ∈ GLn+m (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='8), ( ˜Uσ1,π ◦ ˜Uσ2,π) (g) = � X∈Mn×m1(F) � Y ∈Mn×m2(F) ˜f \uf8eb \uf8ed \uf8eb \uf8ed Im1 Im2 In \uf8f6 \uf8f8 \uf8eb \uf8ed Im1 In Y Im2 \uf8f6 \uf8f8 × \uf8eb \uf8ed Im1 In Im2 \uf8f6 \uf8f8 \uf8eb \uf8ed In X Im1 Im2 \uf8f6 \uf8f8 g \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' A simple computation shows that \uf8eb \uf8ed Im1 Im2 In \uf8f6 \uf8f8 \uf8eb \uf8ed Im1 In Y Im2 \uf8f6 \uf8f8 \uf8eb \uf8ed Im1 In Im2 \uf8f6 \uf8f8 \uf8eb \uf8ed In X Im1 Im2 \uf8f6 \uf8f8 = \uf8eb \uf8ed Im1 Im2 In \uf8f6 \uf8f8 \uf8eb \uf8ed In X Y Im1 Im2 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Hence, we get ( ˜Uσ1,π ◦ ˜Uσ2,π) (g) = � X∈Mn×m1(F) � Y ∈Mn×m2(F) ˜f \uf8eb \uf8ed \uf8eb \uf8ed Im1 Im2 In \uf8f6 \uf8f8 \uf8eb \uf8ed In X Y Im1 Im2 \uf8f6 \uf8f8 g \uf8f6 \uf8f8 , and the last sum is ˜Uσ1◦σ2,π (f) (g) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let vπ,ψ, vσ1,ψ and vσ2,ψ be non-zero ψ-Whittaker vectors of π, σ1 and σ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We keep the notations from the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We are now ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By definition, we have (idσ1 ⊗Uσ2,π) (vσ1,σ2◦π,ψ) = Γ(π × σ2, ψ)vσ1,π◦σ2,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since Lσ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='σ2,πvσ1,σ2◦π,ψ = vσ1,σ2,π,ψ and Lσ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='π,σ2vσ1,π◦σ2,ψ = vσ1,π,σ2,ψ, we get from the commu- tative diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4) that ˜Uσ2,πvσ1,σ2,π,ψ = Γ(π × σ2, ψ)vσ1,π,σ2,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Similarly, we have that (Uσ1,π ⊗ idσ2) (vσ1◦π,σ2,ψ) = Γ(π × σ1, ψ)vπ◦σ1,σ2,ψ, and we get from the commutative diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) that ˜Uσ1,πvσ1,π,σ2,ψ = Γ(π × σ1, ψ)vπ,σ1,σ2,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Finally, we have Uσ1◦σ2,πvσ1◦σ2,π,ψ = Γ(π × (σ1 ◦ σ2), ψ)vπ,σ1◦σ2,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since Lσ1,σ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='πvσ1◦σ2,π,ψ = vσ1,σ2,π,ψ and Lπ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='σ1,σ2vπ,σ1◦σ2,ψ = vπ,σ1,σ2,ψ, we get from the commu- tative diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6) ˜Uσ1◦σ2,πvσ1,σ2,π,ψ = Γ(π × (σ1 ◦ σ2), ψ)vπ,σ1,σ2,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 17 Since ˜Uσ1◦σ2,π = ˜Uσ1,π ◦ ˜Uσ2,π, we get that Γ(π × (σ1 ◦ σ2), ψ)vπ,σ1,σ2,ψ = Γ(π × σ1, ψ)Γ(π × σ2, ψ)vπ,σ1,σ2,ψ, and the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Expression in terms of Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this section, we express the Shahidi gamma factor of two irreducible generic representations in terms of their Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible generic representations of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We assume that π and σ are realized by their Whittaker models W (π, ψ) and W (σ, ψ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We choose the Whittaker vectors of π and σ to be their corresponding Bessel functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=', we choose vπ,ψ = Jπ,ψ and vσ,ψ = Jσ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We denote Jσ,π,ψ = vσ,π,ψ = fJσ,ψ,Jπ,ψ and similarly Jπ,σ,ψ = vπ,σ,ψ = fJπ,ψ,Jσ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Assume that n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By definition, we have that for any g ∈ GLn+m (F), (Uσ,πJσ,π,ψ) (g) = Γ(π × σ, ψ)Jπ,σ,ψ (g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Substituting g = ˆwm,n, we get Γ(π × σ, ψ)Jπ,σ,ψ ( ˆwm,n) = � u∈Nn,m swσ,π Jσ,π,ψ ( ˆwn,mu ˆwm,n) , and therefore Γ(π × σ, ψ)Jπ,ψ ⊗ Jσ,ψ = � A∈Mn×m(F) swσ,π Jσ,π,ψ � Im A In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='10) In order for Jσ,π,ψ � Im A In � not to vanish, we must have � Im A In � ∈ Pm,n ˆwn,mZn+m, so there must exist � p1 x p2 � ∈ Pm,n and � u1 y u2 � ∈ Zn+m, where u1 ∈ Zn and u2 ∈ Zm, such that � p1 x p2 � � Im A In � = � Im In � � u1 y u2 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=', � p1 + xA x p2A p2 � = � u2 u1 y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we have p1 + xA = 0 and x = � 0m×(n−m), u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In order to proceed, we will separate two cases, the case where n > m and the case where n = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The case n > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this case, we write A = � A1 A2 � , where A1 ∈ M(n−m)×m (F) and A2 ∈ Mm×m (F) and x = � 0m×(n−m), u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then p1 + xA = 0 implies p1 + u2A2 = 0, and therefore A2 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 18 DAVID SOUDRY AND ELAD ZELINGHER Write � In A Im � = \uf8eb \uf8ed Im A1 In−m A2 Im \uf8f6 \uf8f8 = \uf8eb \uf8ed Im −A−1 2 A1 In−m A2 \uf8f6 \uf8f8 \uf8eb \uf8ed Im A−1 2 In−m −A1A−1 2 Im \uf8f6 \uf8f8 = \uf8eb \uf8ed −A−1 2 Im A1 In−m A2 \uf8f6 \uf8f8 ˆwn,m \uf8eb \uf8ed Im A−1 2 In−m −A1A−1 2 Im \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we have Jσ,π,ψ � Im A In � = ψ � In−m −A1A−1 2 Im � σ � −A−1 2 � ⊗ π � A1 In−m A2 � Jσ,ψ ⊗ Jπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='11) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='11) back in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='10), we get Γ(π × σ, ψ)Jπ,ψ ⊗ Jσ,ψ = � A1∈M(n−m)×m(F) A2∈GLm(F) ψ � In−m −A1A−1 2 Im � π � A1 In−m A2 � ⊗ σ � −A−1 2 � Jπ,ψ ⊗ Jσ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='12) We evaluate both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='12) at (In, Im) to get Γ(π × σ, ψ) = � A1∈M(n−m)×m(F) A2∈GLm(F) ψ � In−m −A1A−1 2 Im � Jπ,ψ � A1 In−m A2 � Jσ,ψ � −A−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Writing � A1 In−m A2 � = � In−m A1A−1 2 Im � � In−m A2 � , we get Jπ,ψ � A1 In−m A2 � = ψ � In−m A1A−1 2 Im � Jπ,ψ � In−m A2 � , and therefore Γ(π × σ, ψ) = � A1∈M(n−m)×m(F) A2∈GLm(F) Jπ,ψ � In−m A2 � Jσ,ψ � −A−1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The summand is independent of A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the equivariance properties of the Bessel function, we get that the summand is invariant under Zm left translations of A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Finally, using the properties of the Bessel function discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we get Γ(π × σ, ψ) = q m(2n−m−1) 2 ωσ (−1) � x∈Zm\\GLm(F) Jπ,ψ � In−m x � Jσ∨,ψ−1 (x) , where q m(2n−m−1) 2 = ��M(n−m)×m (F) �� · |Zm|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The case n = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this case, we have −p1 = xA, and therefore A is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We write � In A In � = � In −A−1 A � � In A−1 In � = � −A−1 In A � ˆwn,n � In A−1 In � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we have Jσ,π,ψ � Im A In � = ψ � In A−1 In � σ � −A−1� ⊗ π (A) Jσ,ψ ⊗ Jπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='13) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='13) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='10), we get Γ(π × σ, ψ)Jπ,ψ ⊗ Jσ,ψ = � A∈GLn(F) ψ � In A−1 In � π (A) ⊗ σ � −A−1� Jπ,ψ ⊗ Jσ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='14) Evaluating both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='14) at (In, In), we get Γ(π × σ, ψ) = � A∈GLn(F) ψ � In A−1 In � Jπ,ψ (A) Jσ,ψ � −A−1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The summand is invariant under Zn left translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the properties of the Bessel function discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we get Γ(π × σ, ψ) = q n(n−1) 2 ωσ (−1) � x∈Zn\\GLn(F) ψ � In x−1 In � Jπ,ψ (x) Jσ∨,ψ−1 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Summary of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We conclude this section by writing down formulas for the Shahidi gamma factor for a pair of irreducible generic representations, in terms of their Bessel func- tions for all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In order to do that, we use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 and the formulas from Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible generic representations of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (1) If n > m, then Γ(π × σ, ψ) = q m(2n−m−1) 2 ωσ (−1) � x∈Zm\\GLm(F) Jπ,ψ � In−m x � Jσ∨,ψ−1 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (2) If n = m, then Γ(π × σ, ψ) = q n(n−1) 2 ωσ (−1) � x∈Zn\\GLn(F) ψ � In x−1 In � Jπ,ψ (x) Jσ∨,ψ−1 (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3) If n < m, then Γ(π × σ, ψ) = q n(2m−n−1) 2 ωπ (−1) � x∈Zn\\GLn(F) Jπ,ψ (x) Jσ∨,ψ−1 � Im−n x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 allows us to give a relation between the Jacquet–Piatetski-Shapiro–Shalika gamma factors defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 and the Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 20 DAVID SOUDRY AND ELAD ZELINGHER Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible cuspidal representation of GLn (F) and let σ be an irreducible generic representation of GLm (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then we have the equality Γ(π × σ, ψ) = q m(2n−m−1) 2 ωσ (−1) γ(π × σ∨, ψ) in either of the following cases: (1) n > m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (2) n = m and σ is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Applications 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Quantitative interpretation of gamma factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this section, we give a repre- sentation theoretic interpretation of the absolute value of the Shahidi gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Our results relate the absolute value of a normalized version of the Shahidi gamma factor with the cuspidal support of the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For irreducible generic representations π and σ of GLn (F) and GLm (F), respectively, we define the normalized Shahidi gamma factor by Γ∗(π × σ, ψ) = q− nm 2 Γ(π × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 that under this normalization, the gamma factor is still multiplicative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=', the following proposition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr and σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , σt be irreducible generic representations of GLn1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLnr (F) and GLm1 (F) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , GLmt (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that π is the unique irreducible generic subrepresentation of π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr and that σ is the unique irreducible generic subrep- resentation of σ1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then Γ∗(π × σ, ψ) = r� i=1 t� j=1 Γ∗(πi × σj, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5, we have the following proposition, which allows us to express the size of the absolute value of Γ(π × σ, ψ) where π and σ are cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π and σ be irreducible cuspidal representations of GLn (F) and GLm (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then |Γ∗(π × σ, ψ)| = � q− n 2 n = m and π ∼= σ, 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 tells us that the size of the normalized Shahidi gamma factor serves as a “Kronecker delta function” for cuspidal representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It could be thought of an analog of [9, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Combining this with the multiplicativity property, we get the following theorem, that allows us to recover the cuspidal support of a generic irreducible representation π by computing |Γ∗(π × σ, ψ)| for any irreducible cuspidal σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible generic representation of GLn (F) and let σ be an irreducible cuspidal representation of GLm (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then |Γ∗(π × σ, ψ)| = q− dπ(σ)m 2 , where dπ (σ) is the number of times that σ appears in the cuspidal support of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that the cuspidal support of π is {π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then π is the unique irre- ducible generic subrepresentation of π1 ◦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' ◦ πr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The result now follows immediately from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ As a corollary, we get the following converse theorem, which allows us to determine whether generic representations of GLn (F) and GLm (F) are isomorphic based on the absolute value of their normalized gamma factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is an analog of [1, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6], but our proof is on the “group side” rather than on the “Galois side”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1 and π2 be irreducible generic representations of GLn1 (F) and GLn2 (F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that for every m > 0 and every irreducible cuspidal representation σ of GLm (F) we have |Γ∗(π1 × σ, ψ)| = |Γ∗(π2 × σ, ψ)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then n1 = n2 and π1 ∼= π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, π1 and π2 have the same cuspidal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, there exists a unique irreducible generic representation with a given cuspidal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ As another corollary, we explain that the functional equations in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 and Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 fail for π and σ, whenever the cuspidal support of π has a non-empty intersection with the cuspidal support of σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that π and σ are irreducible generic representations of GLn (F) and GLm (F), respectively, and that n > m (respectively, n = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that the cuspidal sup- port of π has a non-empty intersection with the cuspidal support of σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then the functional equation in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 (respectively, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3) does not hold for π and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If the functional equation holds for π and σ, then it also holds for π∨ and σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This can be seen by applying complex conjugation to the functional equation, which sends the ψ-Whittaker functions to ψ−1-Whittaker functions of the contragredient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' As in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 (respectively, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2), we get that |γ(π × σ, ψ)| = q− m(n−m−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Whenever γ(π × σ, ψ) is defined, it is given by the formula in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2 (respectively, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3), and therefore the formula in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This implies that |Γ∗(π × σ∨, ψ)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' On the other hand, because π and σ∨ have common elements in their cuspidal support, we have that |Γ∗(π × σ∨, ψ)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In his unpublished manuscript [23], the first author showed that whenever the cuspidal support of π does not intersect the cuspidal support of σ∨, the relevant functional equation holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Due to length considerations, we do not include the proofs here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Consequences for the converse theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Our results from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 allow us to improve Nien’s results regarding the converse theorem for irreducible generic representations of finite general linear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Nien showed in [16] the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1 and π2 be two irreducible cuspidal representations of GLn (F) with the same central character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that for every 1 ≤ m ≤ n 2, and every irreducible generic representation σ of GLm (F) we have γ(π1 × σ, ψ) = γ(π2 × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1) Then π1 ∼= π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 22 DAVID SOUDRY AND ELAD ZELINGHER Using our results and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, we are able to deduce the following converse theorem, where π1 and π2 can be arbitrary generic representations (rather than just cuspidal repre- sentations), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1) needs to be verified only for cuspidal representations σ (rather than for all generic representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This is similar to [10, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π1 and π2 be two irreducible generic representations of GLn (F) with the same central character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that for every 1 ≤ m ≤ n 2, and every irreducible cuspidal representation σ of GLm (F) we have Γ∗(π1 × σ, ψ) = Γ∗(π2 × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2) Then π1 ∼= π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Our proof is by induction on the cardinality of the cuspidal support of π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We first notice that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we have that for any 1 ≤ m ≤ n 2 and any irreducible generic representation σ of GLm (F), Γ∗(π1 × σ, ψ) = Γ∗(π2 × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that π1 is cuspidal, then its cuspidal support is of cardinality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If π2 is not cusp- idal, then its cuspidal support contains an irreducible cuspidal representation τ of GLk (F), where k ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since k < n, we have by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 that |Γ∗(π1 × σ, ψ)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We also have by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 that |Γ∗(π2 × σ, ψ)| < 1, which is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, π2 is also cuspidal, and by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, we have that π1 and π2 are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose now that π1 is not cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let {τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , τr} the cuspidal support of π1 and let {τ ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , τ ′ r′} be the cuspidal support of π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Without loss of generality, we have that τ1 is an irreducible cuspidal representation of GLn1 (F), where n1 ≤ n 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 we have that |Γ∗(π1 × τ1, ψ)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since n1 ≤ n 2, we have that Γ∗(π1 × τ1, ψ) = Γ∗(π2 × τ1, ψ), and therefore |Γ∗(π2 × τ1, ψ)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, this implies that τ1 is in the cuspidal support of π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Without loss of generality, we may assume that τ ′ 1 = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we deduce that for any irreducible generic representation σ of GLm (F) where m ≤ n 2, r� j=2 Γ∗(τj × σ, ψ) = r′ � j=2 Γ∗(τ ′ j × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3) Let π′ 1 be the unique irreducible generic representation of GLn−n1 (F) with cuspidal support {τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , τr}, and let π′ 2 be the unique irreducible generic representation of GLn−n1 (F) with cuspidal support {τ ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , τ ′ r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For i = 1, 2, the central characters of πi and π′ i are related by ωπi = ωπ′ i · ωτ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we have that π′ 1 and π′ 2 also have the same central character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3) implies that for every m ≤ n 2 and every irreducible generic representation σ of GLm (F), Γ∗(π′ 1 × σ, ψ) = Γ∗(π′ 2 × σ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By induction π′ 1 ∼= π′ 2, and therefore {τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , τr} = {τ ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , τ ′ r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Hence, π1 ∼= π2, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Special values of the Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this section, we use our results regarding multiplicativity of the Shahidi gamma factor, and its relation to the Jacquet–Piatetski- Shapiro–Shalika gamma factor in order to find an explicit formula for special values of the Bessel function of irreducible generic representations of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For two blocks, such a formula was given by Curtis and Shinoda in [4, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' However, their proof uses GAMMA FACTORS FOR REPRESENTATIONS OF GLn 23 Deligne–Lusztig theory, while our proof only uses Green’s character values for irreducible cuspidal representation of GLn (F), see [8, Section 6] and [17, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We also provide a formula for a simple value consisting of three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This generalizes a formula of Chang for irreducible generic representations of GL3 (F) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Special value formula for two blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Fix an algebraic closure F of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For every pos- itive integer n, let Fn be the unique extension of degree n in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let NFn/F : F∗ n → F∗ and TrFn/F : Fn → F be the norm and the trace maps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let � F∗n be the character group consisting of all multiplicative characters α: F∗ n → C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' It is known that irreducible cuspidal representations of GLn (F) are in a bijection with Frobenius orbits of size n of � F∗n, that is, every irreducible cuspidal representation π of GLn (F) corresponds to a set of size n of the form {α, αq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , αqn−1}, where α ∈ � F∗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We first recall Nien’s result regarding the computation of the Jacquet–Piatetski-Shapiro– Shalika gamma factor γ(π×χ, ψ) where π is an irreducible cuspidal representation of GLn (F) and χ is a representation of GL1 (F), that is, χ: F∗ → C∗ is a multiplicative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Nien’s result expresses γ(π × χ, ψ) as a Gauss sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3 ([16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible cuspidal representation of GLn (F) associated with the Frobenius orbit {α, αq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , αqn−1}, where α ∈ � F∗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let χ: F∗ → C∗ be a multiplicative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then γ(π × χ, ψ) = (−1)n+1 χ(−1)n+1q−n+1 � ξ∈F∗n α−1 (ξ) χ−1(NFn/F(ξ))ψ(TrFn/F(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Nien’s proof only uses Green’s character formula for irreducible cuspidal representations, and does not use Deligne–Lusztig theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We are ready to state our result regarding special two blocks values of the Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let n > 1, and let π be an irreducible generic representation of GLn (F) with cuspidal support {π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , πr}, where for every 1 ≤ j ≤ r, πj is an irreducible cuspidal representation of GLnj (F) corresponding to the Frobenius orbit {αj, αq j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , αqnj−1 j }, where αj ∈ � F∗nj is a multiplicative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then for any c ∈ F∗, Jπ,ψ � In−1 c � = (−1)n+r q−n+1 � ξ1∈F∗ n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',ξr∈F∗ nr �r j=1 NFnj /F(ξj)=(−1)n−1c−1 r� j=1 � α−1 j (ξj) ψ � TrFnj /F (ξj) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4, we have that Γ(π × χ, ψ) = qn−1 � x∈F∗ Jπ,ψ � In−1 x � χ−1 (−x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Multiplying by χ (−c) and averaging over all χ ∈ � F∗, and using the fact that a sum of a non-trivial character on a group is zero, we get 1 |F∗| � χ∈� F∗ Γ(π × χ, ψ)χ (−c) = qn−1Jπ,ψ � In−1 c � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4) 24 DAVID SOUDRY AND ELAD ZELINGHER By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, we have that Γ(π × χ, ψ) = r� j=1 Γ(πj × χ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, we have that Γ(πj × χ, ψ) = (−1)nj+1 χ(−1)nj � ξ∈F∗nj α−1 j (ξ) χ(NFnj /F(ξ))ψ(TrFnj /F(ξ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we get that Γ(π × χ, ψ) is given by (−1)n+r � ξ1∈F∗n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',ξr∈F∗nr � r� j=1 α−1 j (ξj) ψ(TrFnj /F(ξj)) � χ � (−1)n r� j=1 NFnj /F (ξj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) Substituting the expression (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) for Γ(π × χ, ψ) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4), and using the fact that a sum of a non-trivial of character over a group is zero, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The expression for Γ(π × χ, ψ) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5) is originally due to Kondo [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' He computed it for the Godement–Jacquet gamma factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' One can show directly that the Godement–Jacquet gamma factor coincides with the Shahidi gamma factor for representa- tions for which both factors are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Our proof, which is based on Nien’s result and on multiplicativity of gamma factors, is different than the one given by Kondo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' See also another proof in [15, Chapter IV, Section 6, Example 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In [27], a vast generalization of the method in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5 is used in order to find formulas for Jπ,ψ � In−m cIm � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' However, [27] relies on the results of [26], which in turn rely on the local Langlands corre- spondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The proof given here does not rely on such results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Special value formula for three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' In this subsection, we use our results to prove a formula for special values of the Bessel function, for a simple value consisting of three blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This generalizes a formula given by Chang [3] for GL3 (F), generalized later by Shinoda and Tulunay [21] to GL4 (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Our proof is different from Chang’s proof, which is based on the Gelfand–Graev algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We start with the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible generic representation of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then for any c ∈ F∗, and any g ∈ GLn (F), we have Jπ,ψ (g) Jπ,ψ � In−1 c � =q−(n−1) � tx=(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',xn−1)∈Fn−1 ψ (−xn−1) Jπ,ψ � g � In−1 x 1 � � In−1 c �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let m = 1 and let σ = χ : F∗ → C∗ be a multiplicative character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='12), we have Γ(π × χ, ψ)Jπ,ψ = � tx∈Fn−1 a∈F∗ χ � −a−1� ψ � In−1 −a−1x 1 � π � x In−1 a � Jπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 25 We multiply by χ (−c) and average over χ ∈ � F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the fact that a sum of a non-trivial character over a group is zero, and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4), we get qn−1Jπ,ψ � In−1 c � Jπ,ψ = � tx=(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',xn−1)∈Fn−1 ψ � −c−1xn−1 � π � x In−1 c � Jπ,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the decomposition � x In−1 c � = � In−1 c−1x 1 � � In−1 c � , and changing the summation variable x to c · x, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then for any irreducible generic representation π of GLn (F) and any c, c′ ∈ F∗, we have Jπ,ψ \uf8eb \uf8ed −c′ In−2 c \uf8f6 \uf8f8 = � s∈F∗ Jπ,ψ � In−1 s−1c � Jπ,ψ � sc′ In−1 � (ψ (s) − 1) + δcc′,1 qn−2, where δcc′,1 = �1 cc′ = 1, 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We substitute g = � c′ In−1 � in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4 to get Jπ,ψ � In−1 c � Jπ,ψ � c′ In−1 � = q−(n−1) � tx=(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=',xn−1)∈Fn−1 ψ (−xn−1) Jπ,ψ � cc′ cx In−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If xn−1 = 0, then � cc′ cx In−1 � lies in the mirabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='14], we have that the Bessel function is zero for elements in the mirabolic subgroup that do not lie in the upper unipotent subgroup Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we get that if xn−1 = 0, then x = 0 and ψ (xn−1) Jπ,ψ � cc′ cx In−1 � = δcc′,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose now that xn−1 = t ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Denote tx′ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , xn−2) ∈ Fn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then we have � cc′ cx In−1 � = \uf8eb \uf8ed 1 0 t−1c′ In−2 t−1x′ 1 \uf8f6 \uf8f8 \uf8eb \uf8ed −t−1c′ In−2 tc \uf8f6 \uf8f8 \uf8eb \uf8ed 1 0 (tc)−1 In−2 −t−1x′ 1 \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since we have qn−2 elements in Fn−1 with xn−1 = t, we get that Jπ,ψ � In−1 c � Jπ,ψ � c′ In−1 � = δcc′,1 qn−1 + q−1 � t∈F∗ ψ (−t) Jπ,ψ \uf8eb \uf8ed −t−1c′ In−2 tc \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' 26 DAVID SOUDRY AND ELAD ZELINGHER We proceed as in [3, Page 379] and [21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We replace c with s−1c and c′ with sc′, where s ∈ F∗, to get Jπ,ψ � In−1 s−1c � Jπ,ψ � sc′ In−1 � = δcc′,1 qn−1 + q−1 � t∈F∗ ψ (−st) Jπ,ψ \uf8eb \uf8ed −t−1c′ In−2 tc \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6) Summing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6) over s ∈ F∗, we get � s∈F∗ Jπ,ψ � In−1 s−1c � Jπ,ψ � sc′ In−1 � = q − 1 qn−1 δcc′,1 − q−1 � t∈F∗ Jπ,ψ \uf8eb \uf8ed −t−1c′ In−2 tc \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='7) Multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='6) by ψ (s) and summing over s ∈ F∗, we get � s∈F∗ Jπ,ψ � In−1 s−1c � Jπ,ψ � sc′ In−1 � ψ (s) = − δcc′,1 qn−1 + q − 1 q Jπ,ψ \uf8eb \uf8ed −c′ In−2 c \uf8f6 \uf8f8 − q−1 � 1̸=t∈F∗ Jπ,ψ \uf8eb \uf8ed −t−1c′ In−2 tc \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='8) Subtracting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='7) from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='8), we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the formulas in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5 and its proof, one can show that if the cuspidal support of π does not contain any irreducible representation of GL1 (F), then we have a simpler formula: Jπ,ψ \uf8eb \uf8ed −c′ In−2 c \uf8f6 \uf8f8 = � s∈F∗ Jπ,ψ � In−1 s−1c � Jπ,ψ � sc′ In−1 � ψ (s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='9) However, if the cuspidal support of π contains irreducible representations of GL1 (F), this simpler formula does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using the expression in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='5, we have that the expression on the right hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='9) is an exponential sum that generalizes the Friedlander–Iwaniec character sum, see [5, Proposition 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The Friedlander–Iwaniec character sum played a role in Zhang’s work on the twin prime conjecture [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Computation of γ(π × π∨, ψ) when π is cuspidal In this appendix, we compute the Jacquet–Piatetski-Shapiro–Shalika gamma factor γ(π × σ, ψ) in the special case where π and σ are irreducible cuspidal representations of GLn (F) and π ∼= σ∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible cuspidal representation of GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then γ(π × π∨, ψ) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This was done in [24, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We provide another proof, since the proof in [24] relies on results of representations of p-adic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' For future purposes, we will prove the following general lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will show that Theo- rem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 follows from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 27 We denote by Pn ≤ GLn (F) the mirabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let G be a finite group and let H ≤ G be a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Suppose that H is a semi-direct product of the form H = N ⋊ GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let Ψ : H → C∗ be a character which is trivial on GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let τ be an irreducible representation of G, such that (1) dim HomH (ResH τ, Ψ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (2) dim HomN⋊Pn (ResN×Pn τ, ResN⋊Pn Ψ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' (3) There exists a functional ℓ ∈ HomZn (ResZn τ, C) and a vector v0 ∈ τ, such that � p∈Zn\\Pn � n∈N ℓ (τ (np) v0) Ψ−1 (n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then � g∈Zn\\GLn(F) � n∈N ℓ (τ (ng) v0) Ψ−1 (n) ψ (⟨eng, e1⟩) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' If F∗ ≤ H lies in the center of G, then (1) implies that the restriction of the central character of τ to F∗ ≤ H is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Notice that we have a containment HomH (ResH τ, Ψ) ⊂ HomN⋊Pn (ResN×Pn τ, ResN⋊Pn Ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since both spaces are one dimensional, we have that they are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Denote for v ∈ τ, L (v) = � p∈Zn\\Pn � n∈N ℓ (τ (np) v) Ψ−1 (n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Then L ∈ HomN⋊Pn (ResN×Pn τ, ResN⋊Pn Ψ) and L ̸= 0 because L (v0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, L ∈ HomH (ResH τ, Ψ), which implies that L (τ (g) v) = L (v) for any v ∈ τ, and any g ∈ GLn (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Denote S = � g∈Zn\\GLn(F) � n∈N ℓ (τ (ng) v0) Ψ−1 (n) ψ (⟨eng, e1⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have S = � g∈Pn\\GLn(F) L (τ (g) v0) ψ (⟨eng, e1⟩) = � g∈Pn\\GLn(F) ψ (⟨eng, e1⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We decompose this sum through the center of GLn (F) S = � g∈(F∗·Pn)\\GLn(F) � a∈F∗ ψ (⟨enag, e1⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that for t ∈ F, � a∈F∗ ψ (at) = �−1 t ̸= 0, q − 1 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, we get S = (q − 1) � g∈(F∗·Pn)\\GLn(F) ⟨eng,e1⟩=0 1 − � g∈(F∗·Pn)\\GLn(F) ⟨eng,e1⟩̸=0 1, 28 DAVID SOUDRY AND ELAD ZELINGHER which we rewrite as S = � g∈Pn\\GLn(F) ⟨eng,e1⟩=0 1 − 1 |F∗| � g∈Pn\\GLn(F) ⟨eng,e1⟩̸=0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Consider the right action of GLn (F) on Fn \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This action is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' The stabilizer of en is the mirabolic subgroup Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, for x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' , xn) ∈ Fn \\ {0}, we have that Sx = � g∈Pn\\GLn(F) eng=x 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' This implies that S = � x∈Fn\\{0} x1=0 Sx − 1 |F∗| � x∈Fn\\{0} x1̸=0 Sx = � qn−1 − 1 � − qn−1 = −1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ We move to prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We will use Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 in the following setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let G = GLn (F) × GLn (F), and let H = GLn (F) embedded diagonally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let N = {In} and Ψ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let π be an irreducible cuspidal representation of GLn (F), then by Schur’s lemma, the space HomGLn(F) (π ⊗ π∨, C) is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Since π is cuspidal, By [8, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='2], the restriction of π to the mirabolic subgroup Pn is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, by Schur’s lemma the space HomPn (ResPn π ⊗ ResPn π∨, C) is also one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We take τ = W (π, ψ) ⊗ W (π∨, ψ−1), and ℓ : W (π, ψ) ⊗ W (π∨, ψ−1) → C to be the functional defined on pure tensors by ℓ (W ⊗ W ′) = W (In) · W ′ (In) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' We have that ℓ ∈ HomZn (ResZn τ, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Let v0 = Jπ,ψ ⊗ Jπ∨,ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Consider � p∈Zn\\Pn � n∈N ℓ (τ (np) v0) Ψ−1 (n) = � p∈Zn\\Pn Jπ,ψ (p) Jπ∨,ψ−1 (p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' By [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='14], we have that if Jπ,ψ (p) ̸= 0 for p ∈ Pn, then p ∈ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Therefore, � p∈Zn\\Pn Jπ,ψ (p) Jπ∨,ψ−1 (p) = � p∈Zn\\Zn Jπ,ψ (p) Jπ∨,ψ−1 (p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Thus, we showed that the required properties for Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='3, we have γ(π × π∨, ψ) = � g∈Zn\\GLn(F) Jπ,ψ (g) Jπ∨,ψ−1 (g) ψ �� eng−1, e1 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' GAMMA FACTORS FOR REPRESENTATIONS OF GLn 29 Replacing g with g−1 and using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1, we have γ(π × π∨, ψ) = � g∈Zn\\GLn(F) Jπ,ψ (g) Jπ∨,ψ−1 (g) ψ (⟨eng, e1⟩) , and therefore by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='1 γ(π × π∨, ψ) = � g∈Zn\\GLn(F) � n∈N ℓ (τ (ng) v) Ψ−1 (n) ψ (⟨eng, e1⟩) = −1, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' □ References [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' Atobe and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='il Department of Mathematics, University of Michigan, 1844 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043 USA Email address: eladz@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdE4T4oBgHgl3EQfOQzc/content/2301.04964v1.pdf'} diff --git a/ZtE1T4oBgHgl3EQfKAPm/content/tmp_files/2301.02960v1.pdf.txt b/ZtE1T4oBgHgl3EQfKAPm/content/tmp_files/2301.02960v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0844a59249998ee7fe7d696eb5e88b495d91783 --- /dev/null +++ b/ZtE1T4oBgHgl3EQfKAPm/content/tmp_files/2301.02960v1.pdf.txt @@ -0,0 +1,2121 @@ +Directed flow and global polarization in Au+Au collisions across BES energies at RHIC +Ze-Fang Jiang,1, 2, ∗ Xiang-Yu Wu,2 Shanshan Cao,3, † and Ben-Wei Zhang2, 4 +1Department of Physics and Electronic-Information Engineering, +Hubei Engineering University, Xiaogan, Hubei, 432000, China +2Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), +Central China Normal University, Wuhan, Hubei, 430079, China +3Institute of Frontier and Interdisciplinary Science, Shandong University, Qingdao, Shandong, 266237, China +4Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter, +South China Normal University, Guangzhou, Guangdong, 510006, China +We study the directed flow of identified particles in Au+Au collisions at √sNN = 7.7 to 62.4 GeV. The +Glauber model is extended to include both a tilted deformation of the QGP fireball with respect to the longitudi- +nal direction and a non-zero longitudinal flow velocity gradient in the initial state. By combining this improved +initial condition with a (3+1)-dimensional viscous hydrodynamic model calculation, we obtain a satisfactory +description of the transverse momentum spectra and the rapidity dependent directed flow coefficient of different +hadron species. Our calculation indicates the sensitivity of the hadron directed flow, especially its splitting be- +tween protons and antiprotons, to both the initial geometry and the initial longitudinal flow velocity. Therefore, +the combination of directed flow of different hadrons can provide a tight constraint on the initial condition of +nuclear matter created in heavy-ion collisions. The initial condition extracted from the directed flow is fur- +ther tested with the global polarization of Λ and ¯Λ within the same theoretical framework, where we obtain a +reasonable description of these hyperon polarization observed at different collision energies at RHIC. +I. +INTRODUCTION +A new state of strongly coupled nuclear matter, known +as the Quark-Gluon Plasma (QGP), is created in relativistic +heavy-ion collisions at the BNL Relativistic Heavy-Ion Col- +lider (RHIC) and the CERN Large Hadron Collider (LHC) [1– +8]. Lattice QCD calculations suggest that the transition from +hadronic matter to the QGP is a smooth crossover at zero +baryon density [9]. +Whether one may obtain a first-order +phase transition at finite baryon density by creating a com- +pressed baryonic matter (CBM) is still an open question [10]. +A first-order transition indicates the existence of a “softest +point” in the equation of state (EoS), which may leave sig- +nature in the final state observables such as the transverse +collective flow of hadrons [11]. +To search for the first- +order phase transition, various CBM experiments have been +constructed to investigate the QCD phase diagram at high +baryon density, such as the Beam Energy Scan (BES) ex- +periment at RHIC [12], Nuclotron-based Ion Collider fAcil- +ity (NICA) [13], Japan Proton Accelerator Research Complex +for Heavy-Ion (JPARC-HI) [14], Alternating Gradient Syn- +chrontron (AGS) [15, 16] at BNL and Facility for Antiproton +and Ion Research (FAIR) [17]. Various observables have been +proposed to seek signals of the first order phase transition and +locate the critical endpoint (CEP) in the QCD phase diagram, +such as higher-order cumulants of conserved charges [18], +collective flow of emitted particles [19–23], amplification of +the light nuclei multiplicity ratio [24], and even jet quenching +in low energy collisions [25, 26]. +The first-order Fourier coefficient of the azimuthal distribu- +tion of particles, known as the rapidity-odd directed flow (v1) +∗Electronic address: jiangzf@mails.ccnu.edu.cn +†Electronic address: shanshan.cao@sdu.edu.cn +[19, 20, 27–32], is among the most popular observables in +analyzing the QGP properties, considering that they are sen- +sitive to the initial size and geometry of the nuclear matter +produced in energetic collisions [1, 27, 33–44]. Within hydro- +dynamic models, the directed flow observed in heavy-ion ex- +periments can be understood with an expanding fireball from +an initial energy density that is asymmetric (tilted or shifted) +with respect to the beam axis. The related phenomenologi- +cal model calculations provide a reasonable description of the +charged particle v1 measured in Au+Au, Zr+Zr, Ru+Ru and +Pb+Pb collisions [45–50]. However, with zero baryon den- +sity, the splitting of v1 between baryon and anti-baryon cannot +be explained by the deformed initial energy density distribu- +tion alone. It is a great challenge to quantitatively describe +the different directed flow coefficients between protons and +antiprotons measured at different collision energies at RHIC- +BES [19], NA49 [51] and E895 [16] using current hydrody- +namic and transport models [39, 52–56]. Recently, this prob- +lem is resolved for Au+Au collisions at √sNN = 200 GeV in +Ref. [45] by assuming that the baryon density distribution is +also counterclockwise tilted in the reaction plane with respect +to the longitudinal direction, similar to the deformation of the +energy density profile. Therefore, it is of great interest to fur- +ther explore whether this proposal can also be verified at other +colliding energies, and whether the corresponding initial ge- +ometry of nuclear matter consists with other observables such +as the global polarization of Λ and Λ hyperons in heavy-ion +collisions. +In this work, we investigate the splitting of directed flow +between protons and antiprotons in Au+Au collisions across +the BES energies (√sNN = 7.7 - 62.4 GeV) using the (3+1)- +dimensional viscous hydrodynamic model CLVisc [57–60]. +The 3-D initial condition of the QGP is developed from our +earlier study [48] to further include the tilted deformation of +the baryon density distribution [45] and the longitudinal flow +velocity gradient of the QGP beyond the Bjorken approxima- +arXiv:2301.02960v1 [nucl-th] 8 Jan 2023 + +2 +tion [49, 50]. By combining this improved initial condition +and the CLVisc model, we are able to provide a satisfactory +description of the transverse momentum (pT) spectra of iden- +tified hadrons (π+, K+, p and ¯p) in different centrality classes +of Au-Au collisions at the BES energies. We further show +that a simultaneous description of v1 of mesons, baryons and +anti-baryons rely on the initial geometry of both the medium +energy density and the baryon number density, and the ini- +tial longitudinal flow velocity profile. In the end, the medium +geometry and longitudinal flow constrained from the rapidity +(y) dependence of v1 is further tested by the global polariza- +tion of hyperons, from which the correlation between directed +flow and global polarization can be inferred. +The rest of this paper is organized as follows. In Sec. II, we +will present our modified Glauber model for initializing the +QGP, and the CLVisc hydrodynamic model simulation of its +further evolution. In Sec. III, we will calculate the transverse +momentum spectra, directed flow and global polarization of +identified particles measured in relativistic heavy-ion colli- +sions at the BES energies, and investigate their dependence +on the initial geometry and longitudinal flow of the QGP. In +the end, we will summarize and discuss necessary future de- +velopments in Sec. IV. +II. +MODEL FRAMEWORK +A. +Initial condition +We use a modified Glauber model to generate the initial +condition of the QGP fireball, which is tilted in the reaction +plane of nuclear collisions [47, 48, 61]. The nuclear thickness +function of an incoming nucleus is obtained using the Woods- +Saxon (WS) distribution of nucleons as +T(x, y) = +� ∞ +−∞ +dz +n0 +1 + exp +� +r−R0(1+β2Y 0 +2 (θ)) +d +�, +(1) +where n0 is the average nucleon density, r = +� +x2 + y2 + z2 +is the radial position with x, y, z being the space coordinates, +θ is the polar angle, d is the surface diffusiveness parame- +ter, β2 is the quadruple deformity, R0 is the radius parame- +ter of the nucleus, and the spherical harmonic function reads +Y 0 +2 (θ) = 1 +4 +� +5 +π(3 cos2 θ − 1). For Au+Au collision systems +at the BES energies (7.7 - 62.4 GeV), the parameters are listed +in Table I. +Nucleus n0 [1/fm3] R0 [fm] d [fm] β2 +197 +79 Au +0.17 +6.38 +0.535 0.0 +TABLE I: Parameters of the Woods-Saxon distribution for the Au +nucleus [62, 63]. +For two nuclei moving along the beam direction (±ˆz) and +collide with an impact parameter b, their corresponding thick- +ness functions can be expressed as +T+(xT) = T(xT − b/2), +T−(xT) = T(xT + b/2), +(2) +where xT = (x, y) is the transverse plane coordinate. Accord- +ing to the Glauber model, the density distributions of partici- +pant nucleons from the two nuclei are then +T1(xT) = T+(xT) +� +1 − +� +1 − σNNT−(xT) +A +�A� +, +(3) +T2(xT) = T−(xT) +� +1 − +� +1 − σNNT+(xT) +A +�A� +, +(4) +in which A is the mass number and σNN is the inelastic +nucleon-nucleon scattering cross section [62]. +Non-central collisions deposit energy into the QGP asym- +metrically along the longitudinal direction. As illustrated in +Fig. 1, a counterclockwise tilt of the medium profile is ex- +pected in the reaction plane [46]. This deformation can be +introduced into the initial condition of the QGP via a rapid- +ity dependent wounded (or participant) nucleon distribution +function as [47, 48, 64] +WN(x, y, ηs) = T1(x, y) + T2(x, y) ++ Ht[T1(x, y) − T2(x, y)] tan +�ηs +ηt +� +, +(5) +where the parameter Ht reflects the overall strength of imbal- +ance between the forward and backward spacetime rapidities +(ηs), and the function tan(ηs/ηt) models the rapidity depen- +dence of this imbalance. A fixed parameter ηt = 8.0 will +be used in the present study, which provides a reasonable de- +scription of the directed flow (v1) of charged particles in our +earlier work [48]. +The energy density ε(x, y, ηs) and the local baryon density +at the initial time then read [58, 60] +ε(x, y, ηs) = K · W(x, y, ηs) · H(ηs) , +(6) +n(x, y, ηs) = 1 +N · W(x, y, ηs) · H(ηs) · HB(ηs) , +(7) +in which the overall factor K is determined by the multiplic- +ity distribution (dNch/dη or dNch/dy) of soft hadrons, N is a +normalization factor constrained by the number of participant +nucleons Npart, and W(x, y, ηs) is the total weight function +that combines contributions from wounded nucleons and bi- +nary collisions as +W(x, y, ηs) = +(1 − α)WN(x, y, ηs) + αnBC(x, y) +[(1 − α)WN(0, 0, 0) + αnBC(0, 0)] |b=0 +, +(8) +with nBC(x, y) = σNNT+(x, y)T−(x, y) being the number of +binary (hard) collisions, and α = 0.05 being the collision +hardness parameter determined by the centrality (or b) depen- +dence of the soft hadron yield [58, 62]. +In Eqs. (6) and (7), a function +H(ηs) = exp +� +−(|ηs| − ηw)2 +2σ2η +θ(|ηs| − ηw) +� +(9) + +3 +4 +2 +0 +2 +4 +s +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +x [fm] +[GeV/fm3] +Energy density + 10 - 40% Au+Au @ 62.4 GeV +0 +2 +4 +6 +8 +10 +12 +14 +16 +4 +2 +0 +2 +4 +s +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +x [fm] +[GeV/fm3] +Energy density + 10 - 40% Au+Au @ 27 GeV +0.0 +1.2 +2.4 +3.6 +4.8 +6.0 +7.2 +8.4 +4 +2 +0 +2 +4 +s +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +x [fm] +[GeV/fm3] +Energy density + 10 - 40% Au+Au @ 7.7 GeV +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4 +2 +0 +2 +4 +s +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +x [fm] +[1/fm3] +Local baryon density + 10 - 40% Au+Au @ 62.4 GeV +0.0 +0.3 +0.6 +0.9 +1.2 +1.5 +1.8 +4 +2 +0 +2 +4 +s +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +x [fm] +[1/fm3] +Local baryon density + 10 - 40% Au+Au @ 27 GeV +0.00 +0.16 +0.32 +0.48 +0.64 +0.80 +0.96 +1.12 +1.28 +4 +2 +0 +2 +4 +s +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +x [fm] +[1/fm3] +Local baryon density (1.2*Ht) + 10 - 40% Au+Au @ 7.7 GeV +0.00 +0.16 +0.32 +0.48 +0.64 +0.80 +0.96 +1.12 +FIG. 1: (Color online) Distributions of the initial energy density (upper row) and net baryon number density (lower row) on the ηs-x plane for +10-40% Au+Au collisions at √sNN = 62.4, 27 and 7.7 GeV. The arrows (lime color) sketch propagation towards the forward and backward +rapidity directions. +is introduced to describe the plateau structure of the longi- +tudinal distribution of emitted hadrons, in which ηw controls +the width of the central rapidity plateau and ση determines +the width (speed) of the Gaussian decay outside the plateau +region [58]. Following the recent study Ref. [45], the longitu- +dinal dependence of the baryon density is also introduced into +the initial condition via +HB(ηs) = exp +� +−(ηs − ηn)2 +2σ2n +� ++ exp +� +−(ηs + ηn)2 +2σ2n +� +, +(10) +where parameters ηn and σn are calibrated by the pT spectra of +protons and antiprotons. This phenomenological ansatz [45] +can be qualitatively justified in the string models of the ini- +tial state [60, 65, 66], considering that the titled energy and +baryon density profiles originate from strings that connect va- +lence and sea quarks inside nuclei. +Since we aim at understanding the directed flow of soft +hadrons and the hyperon polarization within the same frame- +work, the latter of which is sensitive to the gradient of fluid +velocity in the longitudinal direction [67], it is necessary to +extend the initialization of fluid velocity beyond the Bjorken +approximation. Following Refs. [49, 50, 68], the longitudi- +nal fluid velocity at the initial proper time τ0 can be given +by vηs = T τηs/T ττ, in which the energy-momentum tensor +components read +T ττ = ε(x, y, ηs) cosh(yL) , +(11) +T τηs = 1 +τ0 +ε(x, y, ηs) sinh(yL) , +(12) +where the rapidity variable is parameterized with +yL ≡ fvyCM. +(13) +Here, the center of mass rapidity yCM is related to the partici- +pant thickness function imbalance as +yCM = arctanh +�T1 − T2 +T1 + T2 +tanh(ybeam) +� +, +(14) +in which ybeam ≡ arccosh[√sNN/(2mN)] is the beam rapid- +ity with mN being the nucleon mass; and fv ∈ [0, 1] models +the fractional longitudinal momentum attributed to the corre- +sponding flow velocity. This fv parameter allows us to vary +the magnitude of the longitudinal flow velocity, which fur- +ther affects the slope of the directed flow with respect to ra- +pidity (dv1/dy) and the global polarization of hyperons. For +fv = 0, one has yL = 0, and the velocity field is reduced to +the Bjorken flow scenario [49]. The initial fluid velocity in +transverse directions are still set as 0 via T τx = T τy = 0, +considering that they have little impact on the observables we +investigate in this work. +In Table II, we summarize the parameters used for initial- +izing the QGP produced at the BES energies. The first four +parameters (K, τ0, ση and ηw) are adjusted according to rapid- +ity dependence of the charged particle yields (dNch/dy) in the +most central collisions at each colliding energy, and the next +two parameters (σn and ηn) are from the pT spectra of protons +and antiprotons. The last two parameters (fv and Ht) are de- +termined by the directed flow of hadrons. Note that since we +include both the geometric tilt and the longitudinal velocity +in the initial QGP profile, values extracted for fv in this work +can be different from those in Ref. [50] where only the latter +effect is taken into account. +Using these parameterizations, we first present in Fig. 1 +the distributions of the energy density (upper row) and net +baryon number density (lower row) at τ0 on the ηs-x plane + +4 +√sNN [GeV] +K +τ0 [fm] ση [fm] ηw +σn +ηn +fv +Ht +62.4 +11.8 +1.0 +0.3 +2.25 1.34 2.7 0.18 10.0 +39 +8.25 +1.3 +0.3 +1.9 1.13 2.1 0.22 13.0 +27 +7.40 +1.4 +0.3 +1.6 1.06 1.8 0.23 13.5 +19.6 +5.60 +1.8 +0.3 +1.3 0.85 1.5 0.24 15.5 +14.5 +3.90 +2.2 +0.3 +1.15 0.81 1.4 0.24 18.0 +11.5 +3.05 +2.4 +0.3 +1.16 0.79 1.22 0.25 22.0 +7.7 +2.50 +2.6 +0.3 +0.9 0.70 1.05 0.26 28.0 +TABLE II: Parameters for the 3-dimensional optical Glauber model +of the initial condition of the QGP [49, 60]. +for 10-40% Au+Au collisions at three different colliding en- +ergies (√sNN = 62.4, 27, 7.7 GeV). From the figure, one +observes that the energy and baryon densities are not only +shifted asymmetrically along the forward and backward ra- +pidity directions, but also tilted counterclockwise in the ηs- +x plane. Due to different parametrizations between Eq. (6) +and Eq. (7), the initial baryon density tends to be shifted to- +wards larger forward and backward rapidity regions than the +energy distribution. The asymmetric distribution of baryon +density will in the end affect the different abundance between +protons and antiprotons at different locations of the QGP fire- +ball [45]. Since a stronger drag on the participant nucleons +from spectators is expected at lower collisional energies, we +obtain a stronger tilt of the density profile and thus a larger +Ht parameter at lower energies. Meanwhile, a larger frac- +tional longitudinal momentum is deposited from the colliding +beams into the QGP at lower energies, as reflected by the in- +creasing value of fv as √sNN decreases. At very low energy +(for √sNN = 11.5 and 7.7 GeV), we also need to assume the +baryon density has a stronger tilt than the energy density by +applying 1.2Ht for the former, in order to improve our phe- +nomenological description of the proton v1. This might result +from effects of the phase transition [39, 53, 54, 56], emission +of the spectator matter [42] and the electromagnetic field [69] +that have not been taken into account in our present work. +B. +Hydrodynamic evolution +Starting with the initial condition constructed in the previ- +ous subsection, we utilize a (3+1)-D viscous hydrodynamic +model CLVisc [57–60] to simulate the subsequent evolution +of the QGP medium. The hydrodynamic equations read [70– +74] +∇µT µν = 0 , +(15) +∇µJµ = 0 , +(16) +where the energy-momentum tensor T µν and the net baryon +current Jµ are defined as +T µν = εU µU ν − P∆µν + πµν , +(17) +Jµ = nU µ + V µ , +(18) +with ε, P, n, uµ, πµν, V µ being the local energy den- +sity, pressure, net baryon density, flow velocity field, shear +stress tensor and baryon diffusion current respectively. The +projection tensor is given by ∆µν += gµν − uµuν, with +gµν = diag(1, −1, −1, −1) being the metric tensor. +Ef- +fects of the bulk viscosity is not included in the present study +yet [49, 60, 65, 75, 76]. +Based on the Israel-Stewart second order hydrodynamic ex- +pansion, the dissipative currents πµν and V µ are expressed as +follows [76]: +∆µν +αβ(u · ∂)παβ = − 1 +τπ +(πµν − ηvσµν) − 4 +3πµνθ +− 5 +7πα<µσν> +α ++ 9 +70 +4 +e + P π<µ +α πν>α , +∆µν(u · ∂)Vν = − 1 +τV +� +V µ − κB ▽µ µB +T +� +− V µθ +− 3 +10Vνσµν , +(19) +where θ = ∂ · u is the expansion rate, σµν = ∂<µuν> is +the shear tensor, ηv and κB are the shear viscosity and baryon +diffusion coefficient. For an arbitrary tensor Aµν, its trace- +less symmetric part is given by A<µν> = +1 +2[(∆µα∆νβ + +∆να∆µβ) − 2 +3∆µν∆αβ]Aαβ [60]. +In hydrodynamic simulation, the specific shear viscosity +Cηv and the baryon diffusion coefficient κB are treated as +model parameters, which are related to ηv and CB as: +Cηv = +ηvT +e + P , +(20) +κB = CB +T n +�1 +3 cot +�µB +T +� +− +nT +e + P +� +, +(21) +where µB stands for the baryon chemical potential. They con- +nect to the relaxation times as +τπ = 5Cηv +T +, +τV = CB +T . +(22) +In this work, we set Cηv = 0.08 and CB = 0.4 for all collision +energies. +The hydrodynamic equations are then solved together with +the NEOS-BQS equation of state (EOS) [77, 78], which is +based on the lattice QCD calculation at high temperature and +vanishing net baryon density and then extended to finite net +baryon density according to the Taylor expansion method [77, +78]. It connects the QGP and hadron phases with a smooth +crossover under the strangeness neutrality (nS = 0) and the +electric charge density nQ = 0.4nB conditions. +C. +Particlization +The isothermal freeze-out condition [58] is applied in our +study, with the freeze-out hypersurface determined by a con- +stant freeze-out energy density (efrz= 0.4 GeV/fm3) [60]. On +this hypersurface, the Cooper-Frye formalism is implemented +to obtain the momentum distribution of hadrons: +dN +pTdpTdφdy = +gi +(2π)3 +� +Σ +pµdΣµfeq(1 + δfπ + δfV ) . (23) + +5 +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT ++ +0-5% ×10 +0 +5-10% ×10 +1 +K+ +10-20% ×10 +2 +20-40% ×10 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT +p +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +Au+Au @ sNN = 62.4 GeV +p +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT ++ +0-5% ×10 +0 +5-10% ×10 +1 +K+ +10-20% ×10 +2 +20-40% ×10 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT +p +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +Au+Au @ sNN = 39 GeV +p +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT ++ +0-5% ×10 +0 +5-10% ×10 +1 +K+ +10-20% ×10 +2 +20-40% ×10 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT +p +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +Au+Au @ sNN = 27 GeV +p +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT ++ +0-5% ×10 +0 +5-10% ×10 +1 +K+ +10-20% ×10 +2 +20-40% ×10 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT +p +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +Au+Au @ sNN = 19.6 GeV +p +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT ++ +0-5% ×10 +0 +5-10% ×10 +1 +K+ +10-20% ×10 +2 +20-30% ×10 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT +p +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +Au+Au @ sNN = 14.5 GeV +p +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT ++ +0-5% ×10 +0 +5-10% ×10 +1 +K+ +10-20% ×10 +2 +20-40% ×10 +3 +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +10 +5 +10 +3 +10 +1 +101 +103 +dNch/2 dY PT dPT +p +0.0 +0.5 +1.0 +1.5 +2.0 +PT (GeV) +Au+Au @ sNN = 7.7 GeV +p +FIG. 2: (Color online) The transverse momentum spectra of identified charged hadrons (π+, K+, p and ¯p) at mid-rapidity (|y| < 0.25) in +different centrality classes of Au+Au collisions at √sNN = 7.7, 14.5, 19.6, 27, 39 and 62.4 GeV, compared to the STAR data [25]. +In the above equation, gi is the spin-color degeneracy factor +for identified hadrons, and dΣµ is the hypersurface element +determined by the projection method [58]. The thermal dis- +tribution (feq) and its out-of-equilibrium corrections (δfπ and +δfV ) are given by +feq = +1 +exp [(pµU µ − BµB) /Tf] ∓ 1 , +(24) +δfπ(x, p) = (1 ± f eq(x, p)) pµpνπµν +2T 2 +f (e + P), +(25) +δfV (x, p) = (1 ± f eq(x, p)) +� nB +e + P − +B +U µpµ +� pµVµ +κB/τV +, +(26) +where Tf is the chemical freeze-out temperature, and B repre- +sents the baryon number of an identified hadron. The out-of- +equilibrium corrections above are derived from the Boltzmann +equation via the relaxation time approximation [79]. Contri- +butions from resonance decay have been taken into account +in this work based on Ref. [58], although hadronic scatterings +after the QGP phase has not been included yet. +III. +NUMERICAL RESULTS +In this section, we present our numerical results on light +hadrons in Au+Au collisions at the BES energies using the +(3+1)-D CLVisc hydrodynamics model with finite net baryon +density and the tilted initial condition. We first present the +transverse momentum spectra of identified particles π+, K+, +p and ¯p in Sec. III A. Then, we study the rapidity dependence +of the directed flow of π+, p and ¯p in Sec. III B. The rela- +tion between the slope of v1 vs. y and two competing effects +– the tilt of the fireball (Ht) and the fractional longitudinal + +6 +momentum transferred into the initial flow velocity (fv) – is +investigated in Sec. III C. In the end, we verify the initial con- +dition constrained from the directed flow by reproducing the +global polarization of Λ and ¯Λ hyperons in Sec. III D within +the same theoretical framework. +A. +Identified particle spectra +We start with validating our model setup by comparing the +transverse momentum spectra of the identified light hadrons +between our calculation and the STAR data [25] in Fig. 2. As +discussed in Sec. II A, the first six model parameters summa- +rized in Tab. II are adjusted to describe the rapidity depen- +dence of charged particle yields (dNch/dy) and the pT spectra +of proton (antiproton) yield in the most central collisions at +each colliding energy. With these parameters, the combination +of our initial condition and hydrodynamic evolution is able to +provide a reasonable description of the pT spectra of different +species of identified particles (π+, K+, p and ¯p) across dif- +ferent centrality bins at these colliding energies. This implies +the success of this model setup in describing the bulk evolu- +tion of the nuclear matter produced by heavy-ion collisions at +the BES energies. More detailed discussions on pT spectra +of identified particles, such as their mean pT, can be found in +Ref. [60], which provides an insight into the radial flow of the +QGP medium. This offers a reliable baseline for our further +study of the directed flow coefficient and global polarization. +The last two parameters in Tab. II (fv and Ht) cannot be +determined yet, because they control the initial velocity and +deformed geometry of the QGP medium, and are insensitive +to the integrated particle spectra over the azimuthal angle [45– +50, 60]. In other words, varying these two parameters accord- +ing to the directed flow coefficient later will have little impact +on the hadron spectra presented in this subsection. +B. +Directed flow coefficients of π+, p and ¯p +In this subsection, we study the directed flow coefficients of +π+, p and ¯p in 10-40% Au+Au collisions at the BES energies +(7.7 - 62.4 GeV). As the first order Fourier component of the +azimuthal angle distribution, the directed flow coefficient at a +given rapidity can be calculated as +v1(y) = ⟨cos(φ − Ψ1)⟩ = +� +cos(φ − Ψ1) dN +dydφdφ +� +dN +dydφdφ +, +(27) +where Ψ1 is the first order event plane angle of a nucleus- +nucleus collision. Since we use a smooth initial condition for +the energy density and baryon number density distributions, +event-by-event fluctuations are ignored in the present work. +Therefore, the event plane here is the same as the spectator +plane characterized by the deflected neutrons in experimental +measurements. Effects of the initial-state fluctuations on the +final-state hadron v1 will be left for our future exploration. +Shown in Fig. 3 is the directed flow of π+ as a function +of rapidity in 10-40% Au+Au collisions at the BES ener- +gies. Here, v1 is analyzed using soft hadrons within 0 < +pT < 3.0 GeV. One can see that our calculation provides a +reasonable description of the pion v1(y) around mid-rapidity +(y ∈ [−1, 1]) observed at STAR [19]. +In this work, the +directed flow is contributed by two mechanisms, the longi- +tudinally tilted geometry of the QGP medium [Eq. (5)], as +discussed in Ref. [47], and the non-zero initial gradient of +the longitudinal flow velocity along the direction of the im- +pact parameter (∂vz/∂x) [Eqs. (11) - (14)], as proposed in +Ref. [49, 50, 68]. Detailed discussions on how these two ef- +fects contribute to the hadron v1 and how their corresponding +model parameters (Ht and fv) are adjusted will be presented +later in Sec. III C. In the last two panels for √sNN = 11.5 and +7.7 GeV, we also present calculations using a larger tilt param- +eter (1.2Ht) for the initial baryon number density distribution. +As expected, the pion v1 is not sensitive to this baryon distri- +bution. +In Fig. 4, we further study the rapidity dependence of v1 +for protons and antiprotons in 10-40% Au+Au collisions at the +BES energies. And for a more clear display of this rapidity de- +pendence, its slope is extracted in Fig. 5 around mid rapidity +(y ∈ [−0.5, 0.5]) as a function of the colliding energy. Consis- +tent with the findings of Ref. [45], extending the tilted initial +geometry from energy density to baryon number density leads +to a separation of v1 between protons and antiprotons. Within +our model, using the same Ht parameter is able to provide a +good description of the experiment data when the colliding en- +ergy is not very low. However, at √sNN = 11.5 and 7.7 GeV, +this minimal assumption seems insufficient. A slightly larger +value of Ht for baryon number density than for energy density +is required to generate the increasing trend of proton v1 with +respect to y, as well as the splitting between protons and an- +tiprotons observed at very low energies. This larger value of +Ht for the baryon density than the overall QGP medium might +result from the possible phase transition in the pre-equilibrium +stage [56], the strong electromagnetic field [69] and hadronic +scatterings [60]. +As shown in Fig. 5, with the increase of colliding energy, +the splitting of v1 between protons and antiprotons becomes +smaller. This can be understood with the weaker tilt of the +baryon number density distribution and the slower longitu- +dinal flow velocity in higher energy collisions, and can be +confirmed by the increasing values of Ht and fv extracted in +Tab. II as √sNN becomes smaller. Our calculation indicates +the essential role of the geometric distribution of the baryon +number density in producing the baryon and anti-baryon v1. +And it also suggests the necessity of utilizing pions and pro- +tons (antiprotons) together to simultaneously constrain both +the medium geometry and its longitudinal flow profile in the +initial state. +C. +Dependence of dv1/dy on the medium geometry and the +longitudinal flow velocity +Both the tilted QGP profile and the longitudinal flow ve- +locity gradient contribute to the directed flow of hadrons in +heavy-ion collisions. In this subsection, we investigate how +these two effects compete with each other and illustrate how + +7 +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 + v1 +10-40% Au+Au @ 62.4 GeV ++ +STAR, ++ +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +v1 +10-40% Au+Au @ 39 GeV ++ +STAR, ++ +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +v1 +10-40% Au+Au @ 27 GeV ++ +STAR, ++ +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +v1 +10-40% Au+Au @ 19.6 GeV ++ +STAR, ++ +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +v1 +10-40% Au+Au @ 11.5 GeV ++, 1.2*Ht for baryon ++, 1.0*Ht for baryon +STAR, ++ +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +v1 +10-40% Au+Au @ 7.7 GeV ++, 1.2*Ht for baryon ++, 1.0*Ht for baryon +STAR, ++ +FIG. 3: (Color online) Rapidity dependence of the directed flow coefficient of π+ in 10-40% Au+Au collisions at the BES energies, compared +between our model calculation and the STAR data [19]. +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 + v1 +10-40% Au+Au @ 62.4 GeV +proton +anti-proton +STAR, proton +STAR, anti-proton +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +v1 +10-40% Au+Au @ 39 GeV +proton +anti-proton +STAR, proton +STAR, anti-proton +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 + v1 +10-40% Au+Au @ 27 GeV +proton +anti-proton +STAR, proton +STAR, anti-proton +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 + v1 +10-40% Au+Au @ 19.6 GeV +proton +anti-proton +STAR, proton +STAR, anti-proton +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.2 +0.1 +0.0 +0.1 +0.2 +v1 +10-40% Au+Au @ 11.5 GeV +proton, 1.2*Ht for baryon +anti-proton, 1.2*Ht for baryon +proton, 1.0*Ht for baryon +anti-proton, 1.0*Ht for baryon +STAR, proton +STAR, anti-proton +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.2 +0.1 +0.0 +0.1 +0.2 +v1 +10-40% Au+Au @ 7.7 GeV +proton, 1.2*Ht for baryon +anti-proton, 1.2*Ht for baryon +proton, 1.0*Ht for baryon +anti-proton, 1.0*Ht for baryon +STAR, proton +STAR, anti-proton +FIG. 4: (Color online) Rapidity dependence of the directed flow coefficient of protons and antiprotons in 10-40% Au+Au collisions at the BES +energies, compared between our model calculation and the STAR data [19]. +the two parameters Ht and fv are determined in our model +calculation. +In the upper panel of Fig. 6, we show the slope of the di- +rected flow coefficient dv1/dy around mid-rapidity for π+, +proton and antiproton as a function of the Ht parameter in 10- +40% Au+Au collisions at √sNN = 27 GeV, while fv is fixed +at 0.23. When the longitudinal velocity profile is fixed, we +observe the increase of Ht from 0 to 23 leads to a decrease +of the slope from positive to negative values for pions and an- +tiprotons. In other words, with finite initial longitudinal flow +velocity, the v1 of pions and antiprotons are positive (negative) +in the forward (backward) rapidity region when Ht is small, +but flip when Ht is large. Since pions and antiprotons are +mainly composed of partons newly produced by nuclear col- + +8 +10 +1 +10 +2 +sNN [GeV] +0.15 +0.10 +0.05 +0.00 +0.05 +dv1/dy ++ +proton +anti-proton +STAR, ++ +STAR, protron +STAR, antiprotron +NA49, protron +FIG. 5: (Color online) The slope of the rapidity dependence of the di- +rected flow coefficients of pions, protons and antiprotons as functions +of the colliding energy in 10-40% Au+Au collisions. Results from +our model calculation are compared to the STAR [19] and NA49 [51] +data. Here, 1.2Ht is applied to the baryon number density distribu- +tion at √sNN = 11.5 and 7.7 GeV. +lisions, their v1 both follow the energy density distribution of +the QGP. On the other hand, protons carry baryons deposited +by the beam nuclei and therefore their v1 is significantly af- +fected by the distribution of the initial baryon number density +and show different features compared to pions and antipro- +tons. +In the lower panel of Fig. 6, we show the slope of v1(y) as a +function fv when Ht is fixed at 13.5. Compared to Fig. 6, we +observe different dependences of dv1/dy on the medium de- +formation and its longitudinal flow velocity. While the slopes +of pions and antiprotons decreases as the medium becomes +more tilted (or Ht increases), the opposite is seen when the +longitudinal flow velocity (or fv) increases. +Figure 6 suggests the sensitivity of the slope of v1(y) to +both the medium geometry and its longitudinal flow profile in +the initial state. Here, Ht = 13.5 and fv = 0.23 provide the +best simultaneous description of the v1 of pions, protons and +antiprotons at √sNN = 27 GeV. Values of these two parame- +ters at other collision energies are constrained in the same way +in our work. +D. +Global polarization +Apart from the directed flow, the longitudinal flow veloc- +ity gradient can also induce polarization of constituent par- +tons inside the QGP via the spin-orbit coupling [50, 68, 80– +86]. It has also been found in Ref. [67] that the global po- +larization can be affected by the initial geometry of the QGP +medium when the same initial longitudinal velocity field is ap- +plied. Therefore, due to their similar origins, directed flow and +global polarization should be correlated with each other. This +correlation has recently been investigated in Refs. [50, 87]. In +this subsection, using the same model setup as previously im- +plemented for studying the hadron v1, we further explore the +global polarization of Λ and Λ in heavy-ion collisions. +0 +5 +10 +15 +20 +Ht +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +0.025 +0.050 + dv1/dy +10-40% Au+Au @ 27 GeV, fv=0.23 ++ +protron +antiprotron +STAR, ++ +STAR, protron +STAR, antiprotron +0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 +fv +0.08 +0.06 +0.04 +0.02 +0.00 +0.02 + dv1/dy +10-40% Au+Au @ 27 GeV, Ht = 13.5 ++ +protron +antiprotron +STAR, ++ +STAR, protron +STAR, antiprotron +FIG. 6: (Color online) Upper panel: the slope of the directed flow +coefficient dv1/dy as a function of the tilt parameter Ht when fv +is fixed, in 10-40% Au+Au collisions at √sNN = 27 GeV. Lower +panel: the slope of the directed flow coefficient as a function of the +longitudinal rapidity fraction parameter fv when Ht is fixed. The +experimental data are from the STAR Collaboration [19]. +We assume quarks have reached local thermal equilib- +rium on their freezeout hypersurface. Meanwhile, the spin +of quarks or hadrons are not modified during particlization +and resonance decay [60, 86, 88, 89]. Then, the polarization +pseudo vector for spin 1/2 fermions can be obtained using the +modified Cooper-Frye formalism as [90, 91], +Sµ(p) = +� +dΣ · pJ µ +5 (p, X) +2m +� +dΣ · N(p, X), +(28) +where J µ +5 is the axial charge current density and N µ(p, X) +is the number density of fermions in the phase space. Fol- +lowing the quantum kinetic theory [88, 89, 92], Sµ(p) can be +decomposed into different sources, +Sµ(p) = Sµ +thermal(p) + Sµ +shear(p) + Sµ +accT(p) ++Sµ +chemical(p) + Sµ +EB(p), +(29) +where +Sµ +thermal(p) = +� +dΣσFσϵµναβpν∂α +uβ +T , +Sµ +shear(p) = +� +dΣσFσ +ϵµναβpνuβ +(u · p)T +×pρ(∂ρuα + ∂αuρ − uρDuα), +Sµ +accT(p) = − +� +dΣσFσ +ϵµναβpνuα +T +� +Duβ − ∂βT +T +� +, + +9 +Sµ +chemical(p) = 2 +� +dΣσFσ +1 +(u · p)ϵµναβpαuβ∂ν +µ +T , +Sµ +EB(p) = 2 +� +dΣσFσ +�ϵµναβpαuβEν +(u · p)T ++ Bµ +T +� +, +(30) +with +F µ = +ℏ +8mΛΦ(p)pµfeq(1 − feq), +Φ(p) = +� +dΣµpµfeq. +(31) +The five terms in Eq. (30) represent polarization induced by +the thermal vorticity (Sµ +thermal), the shear tensor (Sµ +shear), the +fluid acceleration minus temperature gradient (Sµ +accT), the gra- +dient of chemical potential over temperature (Sµ +chemical), and +the external electromagnetic field (Sµ +EB), respectively. +De- +tailed expressions of these terms can be found in Refs. [60, +88, 89, 92] or derived from the statistic model [93, 94] and +the Kubo formula [95–98]. Here, Sµ +shear and Sµ +chemical are also +named as the shear-induced polarization (SIP) and the bary- +onic spin Hall effect (SHE) in literature. Since the electro- +magnetic field decay rapidly in heavy-ion collisions, the Sµ +EB +term is neglected in our current work. +The average polarization vector in the rest frame of Λ (or +¯Λ) is then given by +⃗P ∗(p) = ⃗P(p) − +⃗P(p) · ⃗p +p0(p0 + m)⃗p, +(32) +where +P µ(p) ≡ 1 +sSµ(p), +(33) +with s = 1/2 being the spin of the particle. After averaging +over the transverse momentum, one obtains the local polariza- +tion as +⟨⃗P(φp)⟩ = +� ymax +ymin dy +� pTmax +pTmin pTdpT[Φ(p)⃗P ∗(p)] +� ymax +ymin dy +� pTmax +pTmin pTdpTΦ(p) +, +(34) +in which φp is the azimuthal angle. +In the present study, +the mass of Λ (or Λ) is set as m = 1.116 GeV, and the +kinematic regions for analyzing the hyperon polarization are +pT ∈ [0.5 GeV, 3.0 GeV] and y ∈ [−1, 1]. Finally, the +global polarization of Λ and Λ is obtained by further averag- +ing ⃗P ∗(p) over φp in Eq. (34). +Shown in Fig. 7 is the global polarization of Λ and ¯Λ hyper- +ons along the out-of-plane direction as a function of the col- +lision energy in 20-50% Au+Au collisions. Using the model +parameters Ht and fv extracted from the directed flow of pi- +ons, protons and antiprotons as discussed earlier, our model +calculation also provides a reasonable description of the hy- +peron polarization here. This helps further validate our mod- +eling of the tilted geometry of the QGP together with its lon- +gitudinal flow gradient, and also confirms the correlation be- +tween the directed flow and the global polarization. We have +10 +1 +10 +2 +sNN [GeV] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +-Py (%) + (th+shear+accT+chem) + (th+shear+accT+chem) + (th+accT+chem) + (th+accT+chem) +STAR, +STAR, +10 +1 +10 +2 +sNN [GeV] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +-Py (%) + (total 1.0*Ht for baryon) + (total 1.0*Ht for baryon) + (total 1.2*Ht for baryon) + (total 1.2*Ht for baryon) +STAR, +STAR, +FIG. 7: (Color online) The global polarization of Λ and Λ as a func- +tion of the collision energy in 20-50% Au+Au collisions, analyzed +within pT ∈ [0.5 GeV, 3.0 GeV], y ∈ [−1, 1] and compared to the +STAR data [99] (rescaled by 0.877 due to the updated hyperon decay +parameter [50, 100]). In the lower panel, “total” denotes “thermal + +shear + accT + chemical” for short. +confirmed that the total amount of polarization presented here +is mainly contributed by the thermal vorticity term. The shear +tensor relies on the tilted geometry of the medium: a coun- +terclockwise tilt in the reaction plane induces a negative shear +tensor, while a clockwise tilt induces a positive shear tensor. +Therefore, as shown in the upper panel of Fig. 7, introduc- +ing the shear contribution increases the magnitude of −Py. +In the lower panel of Fig. 7, we study the effects of the pos- +sible larger value of Ht for the net baryon number density +than for the energy density at low energies (√sNN = 11.5 and +7.7 GeV), as previously suggested by the proton v1. After +introducing this stronger tilt of baryon number density distri- +bution, the global polarization becomes smaller because of a +positive contribution from the chemical term to Py. We note +that the difference between Λ and ¯Λ polarization relies on +both the geometry of baryon distribution and the longitudinal +flow velocity. This is why our current result appears different +from the earlier study [86] using the same hydrodynamic cal- +culation but different initial condition. The electromagnetic +field may cause further difference between their polarization, +which has not been included in our current study. + +10 +IV. +CONCLUSIONS +We have developed an initial condition model for studying +the directed flow and global polarization of identified hadrons +in heavy-ion collisions across the BES energies. The Glauber +model has been extended such that effects of the tilted geome- +try of both the energy density and the net baryon number den- +sity distribution has been included. The initial longitudinal +flow velocity profile has also been introduced. By combin- +ing this initial condition with a hydrodynamic simulation of +the QGP evolution, we have provided a satisfactory descrip- +tion of the transverse momentum spectra of identified particles +(π+, K+, p and ¯p) from √sNN = 7.7 to 62.4 GeV. The di- +rected flow coefficient of these identified hadrons, especially +the splitting of v1 between protons and antiprotons have been +investigated in detail. +Our calculation shows the essential role of the tilted +medium geometry and the early-time longitudinal flow veloc- +ity in generating the directed flow of hadrons. While both +the strength of tilt (or the Ht parameter) and the fractional +longitudinal momentum deposition (or fv) become larger at +lower collisional energies, they have different impacts on the +slope of v1(y) of different hadron species. An increasing Ht +results in a decrease of dv1/dy around mid-rapidity for both +pions and antiprotons, while an increasing fv causes their in- +crease. The opposite trend could be seen for protons, which +is strongly affected by the baryon number density distribution +deposited by the colliding beam. Therefore, a simultaneous +comparison of the pion, proton and antiproton v1 to their ex- +perimental data sets a tight constraint on the inhomogeneous +distribution of the initial energy and baryon number density, +and the longitudinal flow velocity profile of the nuclear matter +created in non-central heavy-ion collisions. The initial geom- +etry and flow velocity extracted from the hadron v1 is further +tested with the global polarization of Λ and ¯Λ hyperons that is +also affected by the medium geometry and the fluid velocity +gradient in the longitudinal direction. Using the same hydro- +dynamic framework, we obtain a reasonable description of the +hyperon polarization across the BES energies, and find that +the separation of global polarization between Λ and ¯Λ could +also be sensitive to the tilted geometry of the net baryon num- +ber density distribution. +Our study constitutes a step forward in understanding the +origin of the splitting of v1 between different particle species +produced in Au+Au collisions at √sNN = 7.7 - 62.4 GeV. +Nevertheless, in addition to the deformed medium geome- +try and the initial longitudinal flow velocity gradient, other +sources exist for splitting the directed flow between protons +and antiprotons. For example, the electromagnetic field pro- +duced in non-central heavy-ion collisions results in directional +drift of charged quarks (u, d, s) and therefore different values +of v1 between different final state charged particles. Addi- +tionally, the light hadron v1 can also be affected by the de- +celerated baryon flow and hadronic cascade after the QGP +expansion, especially at lower collision energy [45, 49, 50]. +Therefore, a combination of hydrodynamic model and after- +burner hadronic transport is necessary for a better constraint +on the initial state. Furthermore, due to the limit of our cur- +rent smooth initial condition, our calculation is restricted to +the rapidity odd component of v1. The rapidity even com- +ponent, especially its non-trivial pT dependence even at mid- +rapidity [101–103] is another interesting topic to investigate +after our initialization method is extended to include event- +by-event fluctuations. +Last but not least, the initial condi- +tion and the QGP profile we propose here can be straightfor- +wardly coupled to studies of hard probes in low energy colli- +sions, such as the collective flow of high pT hadrons and heavy +quarks [104] and their correlations [105, 106]. These will be +addressed in our upcoming efforts. +Acknowledgments +This work was supported by the National Natural Science +Foundation of China (NSFC) under Grant Nos. 11935007, +12175122 and 2021-867, Guangdong Major Project of Basic +and Applied Basic Research No. 2020B0301030008, the Nat- +ural Science Foundation of Hubei Province No. 2021CFB272, +the Education Department of Hubei Province of China with +Young Talents Project No. Q20212703, the Open Founda- +tion of Key Laboratory of Quark and Lepton Physics (MOE) +No. QLPL202104 and the Xiaogan Natural Science Founda- +tion under Grant No. XGKJ2021010016. +[1] Jean-Yves Ollitrault. Anisotropy as a signature of transverse +collective flow. Phys. Rev. D, 46:229–245, 1992. +[2] Dirk H. Rischke, S. Bernard, and J. A. Maruhn. Relativis- +tic hydrodynamics for heavy ion collisions. 1. General aspects +and expansion into vacuum. +Nucl. Phys. A, 595:346–382, +1995. +[3] H. Sorge. 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C, 80(10):979, 2020. + diff --git a/ZtE1T4oBgHgl3EQfKAPm/content/tmp_files/load_file.txt b/ZtE1T4oBgHgl3EQfKAPm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b4c9512fbaf28dd1399cd908c9192afdde8ba35 --- /dev/null +++ b/ZtE1T4oBgHgl3EQfKAPm/content/tmp_files/load_file.txt @@ -0,0 +1,1543 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf,len=1542 +page_content='Directed flow and global polarization in Au+Au collisions across BES energies at RHIC Ze-Fang Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ∗ Xiang-Yu Wu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2 Shanshan Cao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' † and Ben-Wei Zhang2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 4 1Department of Physics and Electronic-Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Hubei Engineering University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Xiaogan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Hubei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 432000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' China 2Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Central China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Wuhan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Hubei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 430079,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' China 3Institute of Frontier and Interdisciplinary Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Shandong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Qingdao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Shandong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 266237,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' China 4Guangdong Provincial Key Laboratory of Nuclear Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Institute of Quantum Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' South China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Guangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Guangdong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 510006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' China We study the directed flow of identified particles in Au+Au collisions at √sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The Glauber model is extended to include both a tilted deformation of the QGP fireball with respect to the longitudi- nal direction and a non-zero longitudinal flow velocity gradient in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' By combining this improved initial condition with a (3+1)-dimensional viscous hydrodynamic model calculation, we obtain a satisfactory description of the transverse momentum spectra and the rapidity dependent directed flow coefficient of different hadron species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Our calculation indicates the sensitivity of the hadron directed flow, especially its splitting be- tween protons and antiprotons, to both the initial geometry and the initial longitudinal flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, the combination of directed flow of different hadrons can provide a tight constraint on the initial condition of nuclear matter created in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The initial condition extracted from the directed flow is fur- ther tested with the global polarization of Λ and ¯Λ within the same theoretical framework, where we obtain a reasonable description of these hyperon polarization observed at different collision energies at RHIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' INTRODUCTION A new state of strongly coupled nuclear matter, known as the Quark-Gluon Plasma (QGP), is created in relativistic heavy-ion collisions at the BNL Relativistic Heavy-Ion Col- lider (RHIC) and the CERN Large Hadron Collider (LHC) [1– 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Lattice QCD calculations suggest that the transition from hadronic matter to the QGP is a smooth crossover at zero baryon density [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Whether one may obtain a first-order phase transition at finite baryon density by creating a com- pressed baryonic matter (CBM) is still an open question [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' A first-order transition indicates the existence of a “softest point” in the equation of state (EoS), which may leave sig- nature in the final state observables such as the transverse collective flow of hadrons [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' To search for the first- order phase transition, various CBM experiments have been constructed to investigate the QCD phase diagram at high baryon density, such as the Beam Energy Scan (BES) ex- periment at RHIC [12], Nuclotron-based Ion Collider fAcil- ity (NICA) [13], Japan Proton Accelerator Research Complex for Heavy-Ion (JPARC-HI) [14], Alternating Gradient Syn- chrontron (AGS) [15, 16] at BNL and Facility for Antiproton and Ion Research (FAIR) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Various observables have been proposed to seek signals of the first order phase transition and locate the critical endpoint (CEP) in the QCD phase diagram, such as higher-order cumulants of conserved charges [18], collective flow of emitted particles [19–23], amplification of the light nuclei multiplicity ratio [24], and even jet quenching in low energy collisions [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The first-order Fourier coefficient of the azimuthal distribu- tion of particles, known as the rapidity-odd directed flow (v1) ∗Electronic address: jiangzf@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='ccnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='cn †Electronic address: shanshan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='cao@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='cn [19, 20, 27–32], is among the most popular observables in analyzing the QGP properties, considering that they are sen- sitive to the initial size and geometry of the nuclear matter produced in energetic collisions [1, 27, 33–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Within hydro- dynamic models, the directed flow observed in heavy-ion ex- periments can be understood with an expanding fireball from an initial energy density that is asymmetric (tilted or shifted) with respect to the beam axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The related phenomenologi- cal model calculations provide a reasonable description of the charged particle v1 measured in Au+Au, Zr+Zr, Ru+Ru and Pb+Pb collisions [45–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' However, with zero baryon den- sity, the splitting of v1 between baryon and anti-baryon cannot be explained by the deformed initial energy density distribu- tion alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' It is a great challenge to quantitatively describe the different directed flow coefficients between protons and antiprotons measured at different collision energies at RHIC- BES [19], NA49 [51] and E895 [16] using current hydrody- namic and transport models [39, 52–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Recently, this prob- lem is resolved for Au+Au collisions at √sNN = 200 GeV in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [45] by assuming that the baryon density distribution is also counterclockwise tilted in the reaction plane with respect to the longitudinal direction, similar to the deformation of the energy density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, it is of great interest to fur- ther explore whether this proposal can also be verified at other colliding energies, and whether the corresponding initial ge- ometry of nuclear matter consists with other observables such as the global polarization of Λ and Λ hyperons in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In this work, we investigate the splitting of directed flow between protons and antiprotons in Au+Au collisions across the BES energies (√sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 - 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV) using the (3+1)- dimensional viscous hydrodynamic model CLVisc [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The 3-D initial condition of the QGP is developed from our earlier study [48] to further include the tilted deformation of the baryon density distribution [45] and the longitudinal flow velocity gradient of the QGP beyond the Bjorken approxima- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02960v1 [nucl-th] 8 Jan 2023 2 tion [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' By combining this improved initial condition and the CLVisc model, we are able to provide a satisfactory description of the transverse momentum (pT) spectra of iden- tified hadrons (π+, K+, p and ¯p) in different centrality classes of Au-Au collisions at the BES energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' We further show that a simultaneous description of v1 of mesons, baryons and anti-baryons rely on the initial geometry of both the medium energy density and the baryon number density, and the ini- tial longitudinal flow velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the end, the medium geometry and longitudinal flow constrained from the rapidity (y) dependence of v1 is further tested by the global polariza- tion of hyperons, from which the correlation between directed flow and global polarization can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' II, we will present our modified Glauber model for initializing the QGP, and the CLVisc hydrodynamic model simulation of its further evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III, we will calculate the transverse momentum spectra, directed flow and global polarization of identified particles measured in relativistic heavy-ion colli- sions at the BES energies, and investigate their dependence on the initial geometry and longitudinal flow of the QGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the end, we will summarize and discuss necessary future de- velopments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' MODEL FRAMEWORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Initial condition We use a modified Glauber model to generate the initial condition of the QGP fireball, which is tilted in the reaction plane of nuclear collisions [47, 48, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The nuclear thickness function of an incoming nucleus is obtained using the Woods- Saxon (WS) distribution of nucleons as T(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y) = � ∞ −∞ dz n0 1 + exp � r−R0(1+β2Y 0 2 (θ)) d �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (1) where n0 is the average nucleon density,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' r = � x2 + y2 + z2 is the radial position with x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' z being the space coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' θ is the polar angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' d is the surface diffusiveness parame- ter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' β2 is the quadruple deformity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' R0 is the radius parame- ter of the nucleus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' and the spherical harmonic function reads Y 0 2 (θ) = 1 4 � 5 π(3 cos2 θ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' For Au+Au collision systems at the BES energies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 - 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV), the parameters are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Nucleus n0 [1/fm3] R0 [fm] d [fm] β2 197 79 Au 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 TABLE I: Parameters of the Woods-Saxon distribution for the Au nucleus [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' For two nuclei moving along the beam direction (±ˆz) and collide with an impact parameter b, their corresponding thick- ness functions can be expressed as T+(xT) = T(xT − b/2), T−(xT) = T(xT + b/2), (2) where xT = (x, y) is the transverse plane coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Accord- ing to the Glauber model, the density distributions of partici- pant nucleons from the two nuclei are then T1(xT) = T+(xT) � 1 − � 1 − σNNT−(xT) A �A� , (3) T2(xT) = T−(xT) � 1 − � 1 − σNNT+(xT) A �A� , (4) in which A is the mass number and σNN is the inelastic nucleon-nucleon scattering cross section [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Non-central collisions deposit energy into the QGP asym- metrically along the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 1, a counterclockwise tilt of the medium profile is ex- pected in the reaction plane [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This deformation can be introduced into the initial condition of the QGP via a rapid- ity dependent wounded (or participant) nucleon distribution function as [47, 48, 64] WN(x, y, ηs) = T1(x, y) + T2(x, y) + Ht[T1(x, y) − T2(x, y)] tan �ηs ηt � , (5) where the parameter Ht reflects the overall strength of imbal- ance between the forward and backward spacetime rapidities (ηs), and the function tan(ηs/ηt) models the rapidity depen- dence of this imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' A fixed parameter ηt = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 will be used in the present study, which provides a reasonable de- scription of the directed flow (v1) of charged particles in our earlier work [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The energy density ε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) and the local baryon density at the initial time then read [58,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 60] ε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) = K · W(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) · H(ηs) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (6) n(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) = 1 N · W(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) · H(ηs) · HB(ηs) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (7) in which the overall factor K is determined by the multiplic- ity distribution (dNch/dη or dNch/dy) of soft hadrons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' N is a normalization factor constrained by the number of participant nucleons Npart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' and W(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) is the total weight function that combines contributions from wounded nucleons and bi- nary collisions as W(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) = (1 − α)WN(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' ηs) + αnBC(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y) [(1 − α)WN(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 0) + αnBC(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 0)] |b=0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (8) with nBC(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y) = σNNT+(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y)T−(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y) being the number of binary (hard) collisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='05 being the collision hardness parameter determined by the centrality (or b) depen- dence of the soft hadron yield [58, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (6) and (7), a function H(ηs) = exp � −(|ηs| − ηw)2 2σ2η θ(|ηs| − ηw) � (9) 3 4 2 0 2 4 s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 x [fm] [GeV/fm3] Energy density 10 - 40% Au+Au @ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV 0 2 4 6 8 10 12 14 16 4 2 0 2 4 s 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 x [fm] [GeV/fm3] Energy density 10 - 40% Au+Au @ 27 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='8 6.' metadata={'source': 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+page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 1: (Color online) Distributions of the initial energy density (upper row) and net baryon number density (lower row) on the ηs-x plane for 10-40% Au+Au collisions at √sNN = 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4, 27 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The arrows (lime color) sketch propagation towards the forward and backward rapidity directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' is introduced to describe the plateau structure of the longi- tudinal distribution of emitted hadrons, in which ηw controls the width of the central rapidity plateau and ση determines the width (speed) of the Gaussian decay outside the plateau region [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Following the recent study Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [45], the longitu- dinal dependence of the baryon density is also introduced into the initial condition via HB(ηs) = exp � −(ηs − ηn)2 2σ2n � + exp � −(ηs + ηn)2 2σ2n � , (10) where parameters ηn and σn are calibrated by the pT spectra of protons and antiprotons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This phenomenological ansatz [45] can be qualitatively justified in the string models of the ini- tial state [60, 65, 66], considering that the titled energy and baryon density profiles originate from strings that connect va- lence and sea quarks inside nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Since we aim at understanding the directed flow of soft hadrons and the hyperon polarization within the same frame- work, the latter of which is sensitive to the gradient of fluid velocity in the longitudinal direction [67], it is necessary to extend the initialization of fluid velocity beyond the Bjorken approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [49, 50, 68], the longitudi- nal fluid velocity at the initial proper time τ0 can be given by vηs = T τηs/T ττ, in which the energy-momentum tensor components read T ττ = ε(x, y, ηs) cosh(yL) , (11) T τηs = 1 τ0 ε(x, y, ηs) sinh(yL) , (12) where the rapidity variable is parameterized with yL ≡ fvyCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (13) Here, the center of mass rapidity yCM is related to the partici- pant thickness function imbalance as yCM = arctanh �T1 − T2 T1 + T2 tanh(ybeam) � , (14) in which ybeam ≡ arccosh[√sNN/(2mN)] is the beam rapid- ity with mN being the nucleon mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' and fv ∈ [0, 1] models the fractional longitudinal momentum attributed to the corre- sponding flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This fv parameter allows us to vary the magnitude of the longitudinal flow velocity, which fur- ther affects the slope of the directed flow with respect to ra- pidity (dv1/dy) and the global polarization of hyperons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' For fv = 0, one has yL = 0, and the velocity field is reduced to the Bjorken flow scenario [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The initial fluid velocity in transverse directions are still set as 0 via T τx = T τy = 0, considering that they have little impact on the observables we investigate in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In Table II, we summarize the parameters used for initial- izing the QGP produced at the BES energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The first four parameters (K, τ0, ση and ηw) are adjusted according to rapid- ity dependence of the charged particle yields (dNch/dy) in the most central collisions at each colliding energy, and the next two parameters (σn and ηn) are from the pT spectra of protons and antiprotons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The last two parameters (fv and Ht) are de- termined by the directed flow of hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Note that since we include both the geometric tilt and the longitudinal velocity in the initial QGP profile, values extracted for fv in this work can be different from those in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [50] where only the latter effect is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Using these parameterizations, we first present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 1 the distributions of the energy density (upper row) and net baryon number density (lower row) at τ0 on the ηs-x plane 4 √sNN [GeV] K τ0 [fm] ση [fm] ηw σn ηn fv Ht 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='26 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 TABLE II: Parameters for the 3-dimensional optical Glauber model of the initial condition of the QGP [49, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' for 10-40% Au+Au collisions at three different colliding en- ergies (√sNN = 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4, 27, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' From the figure, one observes that the energy and baryon densities are not only shifted asymmetrically along the forward and backward ra- pidity directions, but also tilted counterclockwise in the ηs- x plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Due to different parametrizations between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (6) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (7), the initial baryon density tends to be shifted to- wards larger forward and backward rapidity regions than the energy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The asymmetric distribution of baryon density will in the end affect the different abundance between protons and antiprotons at different locations of the QGP fire- ball [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Since a stronger drag on the participant nucleons from spectators is expected at lower collisional energies, we obtain a stronger tilt of the density profile and thus a larger Ht parameter at lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Meanwhile, a larger frac- tional longitudinal momentum is deposited from the colliding beams into the QGP at lower energies, as reflected by the in- creasing value of fv as √sNN decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' At very low energy (for √sNN = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV), we also need to assume the baryon density has a stronger tilt than the energy density by applying 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2Ht for the former, in order to improve our phe- nomenological description of the proton v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This might result from effects of the phase transition [39, 53, 54, 56], emission of the spectator matter [42] and the electromagnetic field [69] that have not been taken into account in our present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Hydrodynamic evolution Starting with the initial condition constructed in the previ- ous subsection, we utilize a (3+1)-D viscous hydrodynamic model CLVisc [57–60] to simulate the subsequent evolution of the QGP medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The hydrodynamic equations read [70– 74] ∇µT µν = 0 , (15) ∇µJµ = 0 , (16) where the energy-momentum tensor T µν and the net baryon current Jµ are defined as T µν = εU µU ν − P∆µν + πµν , (17) Jµ = nU µ + V µ , (18) with ε, P, n, uµ, πµν, V µ being the local energy den- sity, pressure, net baryon density, flow velocity field, shear stress tensor and baryon diffusion current respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The projection tensor is given by ∆µν = gµν − uµuν, with gµν = diag(1, −1, −1, −1) being the metric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Ef- fects of the bulk viscosity is not included in the present study yet [49, 60, 65, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Based on the Israel-Stewart second order hydrodynamic ex- pansion, the dissipative currents πµν and V µ are expressed as follows [76]: ∆µν αβ(u · ∂)παβ = − 1 τπ (πµν − ηvσµν) − 4 3πµνθ − 5 7πα<µσν> α + 9 70 4 e + P π<µ α πν>α , ∆µν(u · ∂)Vν = − 1 τV � V µ − κB ▽µ µB T � − V µθ − 3 10Vνσµν , (19) where θ = ∂ · u is the expansion rate, σµν = ∂<µuν> is the shear tensor, ηv and κB are the shear viscosity and baryon diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' For an arbitrary tensor Aµν, its trace- less symmetric part is given by A<µν> = 1 2[(∆µα∆νβ + ∆να∆µβ) − 2 3∆µν∆αβ]Aαβ [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In hydrodynamic simulation, the specific shear viscosity Cηv and the baryon diffusion coefficient κB are treated as model parameters, which are related to ηv and CB as: Cηv = ηvT e + P , (20) κB = CB T n �1 3 cot �µB T � − nT e + P � , (21) where µB stands for the baryon chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' They con- nect to the relaxation times as τπ = 5Cηv T , τV = CB T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (22) In this work, we set Cηv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='08 and CB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 for all collision energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The hydrodynamic equations are then solved together with the NEOS-BQS equation of state (EOS) [77, 78], which is based on the lattice QCD calculation at high temperature and vanishing net baryon density and then extended to finite net baryon density according to the Taylor expansion method [77, 78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' It connects the QGP and hadron phases with a smooth crossover under the strangeness neutrality (nS = 0) and the electric charge density nQ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4nB conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Particlization The isothermal freeze-out condition [58] is applied in our study, with the freeze-out hypersurface determined by a con- stant freeze-out energy density (efrz= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV/fm3) [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' On this hypersurface, the Cooper-Frye formalism is implemented to obtain the momentum distribution of hadrons: dN pTdpTdφdy = gi (2π)3 � Σ pµdΣµfeq(1 + δfπ + δfV ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (23) 5 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT + 0-5% ×10 0 5-10% ×10 1 K+ 10-20% ×10 2 20-40% ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) Au+Au @ sNN = 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV p 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT + 0-5% ×10 0 5-10% ×10 1 K+ 10-20% ×10 2 20-40% ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) Au+Au @ sNN = 39 GeV p 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT + 0-5% ×10 0 5-10% ×10 1 K+ 10-20% ×10 2 20-40% ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) Au+Au @ sNN = 27 GeV p 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT + 0-5% ×10 0 5-10% ×10 1 K+ 10-20% ×10 2 20-40% ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) Au+Au @ sNN = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='6 GeV p 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT + 0-5% ×10 0 5-10% ×10 1 K+ 10-20% ×10 2 20-30% ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) Au+Au @ sNN = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 GeV p 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT + 0-5% ×10 0 5-10% ×10 1 K+ 10-20% ×10 2 20-40% ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) 10 5 10 3 10 1 101 103 dNch/2 dY PT dPT p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 PT (GeV) Au+Au @ sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV p FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 2: (Color online) The transverse momentum spectra of identified charged hadrons (π+, K+, p and ¯p) at mid-rapidity (|y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='25) in different centrality classes of Au+Au collisions at √sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='6, 27, 39 and 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV, compared to the STAR data [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the above equation, gi is the spin-color degeneracy factor for identified hadrons, and dΣµ is the hypersurface element determined by the projection method [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The thermal dis- tribution (feq) and its out-of-equilibrium corrections (δfπ and δfV ) are given by feq = 1 exp [(pµU µ − BµB) /Tf] ∓ 1 , (24) δfπ(x, p) = (1 ± f eq(x, p)) pµpνπµν 2T 2 f (e + P), (25) δfV (x, p) = (1 ± f eq(x, p)) � nB e + P − B U µpµ � pµVµ κB/τV , (26) where Tf is the chemical freeze-out temperature, and B repre- sents the baryon number of an identified hadron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The out-of- equilibrium corrections above are derived from the Boltzmann equation via the relaxation time approximation [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Contri- butions from resonance decay have been taken into account in this work based on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [58], although hadronic scatterings after the QGP phase has not been included yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, we present our numerical results on light hadrons in Au+Au collisions at the BES energies using the (3+1)-D CLVisc hydrodynamics model with finite net baryon density and the tilted initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' We first present the transverse momentum spectra of identified particles π+, K+, p and ¯p in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Then, we study the rapidity dependence of the directed flow of π+, p and ¯p in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The rela- tion between the slope of v1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' y and two competing effects – the tilt of the fireball (Ht) and the fractional longitudinal 6 momentum transferred into the initial flow velocity (fv) – is investigated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the end, we verify the initial con- dition constrained from the directed flow by reproducing the global polarization of Λ and ¯Λ hyperons in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III D within the same theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Identified particle spectra We start with validating our model setup by comparing the transverse momentum spectra of the identified light hadrons between our calculation and the STAR data [25] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' II A, the first six model parameters summa- rized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' II are adjusted to describe the rapidity depen- dence of charged particle yields (dNch/dy) and the pT spectra of proton (antiproton) yield in the most central collisions at each colliding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' With these parameters, the combination of our initial condition and hydrodynamic evolution is able to provide a reasonable description of the pT spectra of different species of identified particles (π+, K+, p and ¯p) across dif- ferent centrality bins at these colliding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This implies the success of this model setup in describing the bulk evolu- tion of the nuclear matter produced by heavy-ion collisions at the BES energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' More detailed discussions on pT spectra of identified particles, such as their mean pT, can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [60], which provides an insight into the radial flow of the QGP medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This offers a reliable baseline for our further study of the directed flow coefficient and global polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The last two parameters in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' II (fv and Ht) cannot be determined yet, because they control the initial velocity and deformed geometry of the QGP medium, and are insensitive to the integrated particle spectra over the azimuthal angle [45– 50, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In other words, varying these two parameters accord- ing to the directed flow coefficient later will have little impact on the hadron spectra presented in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Directed flow coefficients of π+, p and ¯p In this subsection, we study the directed flow coefficients of π+, p and ¯p in 10-40% Au+Au collisions at the BES energies (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 - 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' As the first order Fourier component of the azimuthal angle distribution, the directed flow coefficient at a given rapidity can be calculated as v1(y) = ⟨cos(φ − Ψ1)⟩ = � cos(φ − Ψ1) dN dydφdφ � dN dydφdφ , (27) where Ψ1 is the first order event plane angle of a nucleus- nucleus collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Since we use a smooth initial condition for the energy density and baryon number density distributions, event-by-event fluctuations are ignored in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, the event plane here is the same as the spectator plane characterized by the deflected neutrons in experimental measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Effects of the initial-state fluctuations on the final-state hadron v1 will be left for our future exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 3 is the directed flow of π+ as a function of rapidity in 10-40% Au+Au collisions at the BES ener- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Here, v1 is analyzed using soft hadrons within 0 < pT < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' One can see that our calculation provides a reasonable description of the pion v1(y) around mid-rapidity (y ∈ [−1, 1]) observed at STAR [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In this work, the directed flow is contributed by two mechanisms, the longi- tudinally tilted geometry of the QGP medium [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (5)], as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [47], and the non-zero initial gradient of the longitudinal flow velocity along the direction of the im- pact parameter (∂vz/∂x) [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (11) - (14)], as proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [49, 50, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Detailed discussions on how these two ef- fects contribute to the hadron v1 and how their corresponding model parameters (Ht and fv) are adjusted will be presented later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the last two panels for √sNN = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV, we also present calculations using a larger tilt param- eter (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2Ht) for the initial baryon number density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' As expected, the pion v1 is not sensitive to this baryon distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 4, we further study the rapidity dependence of v1 for protons and antiprotons in 10-40% Au+Au collisions at the BES energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' And for a more clear display of this rapidity de- pendence, its slope is extracted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 5 around mid rapidity (y ∈ [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5]) as a function of the colliding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Consis- tent with the findings of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [45], extending the tilted initial geometry from energy density to baryon number density leads to a separation of v1 between protons and antiprotons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Within our model, using the same Ht parameter is able to provide a good description of the experiment data when the colliding en- ergy is not very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' However, at √sNN = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV, this minimal assumption seems insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' A slightly larger value of Ht for baryon number density than for energy density is required to generate the increasing trend of proton v1 with respect to y, as well as the splitting between protons and an- tiprotons observed at very low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This larger value of Ht for the baryon density than the overall QGP medium might result from the possible phase transition in the pre-equilibrium stage [56], the strong electromagnetic field [69] and hadronic scatterings [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 5, with the increase of colliding energy, the splitting of v1 between protons and antiprotons becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This can be understood with the weaker tilt of the baryon number density distribution and the slower longitu- dinal flow velocity in higher energy collisions, and can be confirmed by the increasing values of Ht and fv extracted in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' II as √sNN becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Our calculation indicates the essential role of the geometric distribution of the baryon number density in producing the baryon and anti-baryon v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' And it also suggests the necessity of utilizing pions and pro- tons (antiprotons) together to simultaneously constrain both the medium geometry and its longitudinal flow profile in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Dependence of dv1/dy on the medium geometry and the longitudinal flow velocity Both the tilted QGP profile and the longitudinal flow ve- locity gradient contribute to the directed flow of hadrons in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In this subsection, we investigate how these two effects compete with each other and illustrate how 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 v1 10-40% Au+Au @ 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV + STAR, + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 v1 10-40% Au+Au @ 39 GeV + STAR, + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 v1 10-40% Au+Au @ 27 GeV + STAR, + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': 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@ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 GeV proton, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2*Ht for baryon anti-proton, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2*Ht for baryon proton, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0*Ht for baryon anti-proton, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0*Ht for baryon STAR, proton STAR, anti-proton 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0*Ht for baryon STAR, proton STAR, anti-proton FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 4: (Color online) Rapidity dependence of the directed flow coefficient of protons and antiprotons in 10-40% Au+Au collisions at the BES energies, compared between our model calculation and the STAR data [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' the two parameters Ht and fv are determined in our model calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 6, we show the slope of the di- rected flow coefficient dv1/dy around mid-rapidity for π+, proton and antiproton as a function of the Ht parameter in 10- 40% Au+Au collisions at √sNN = 27 GeV, while fv is fixed at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' When the longitudinal velocity profile is fixed, we observe the increase of Ht from 0 to 23 leads to a decrease of the slope from positive to negative values for pions and an- tiprotons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In other words, with finite initial longitudinal flow velocity, the v1 of pions and antiprotons are positive (negative) in the forward (backward) rapidity region when Ht is small, but flip when Ht is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Since pions and antiprotons are mainly composed of partons newly produced by nuclear col- 8 10 1 10 2 sNN [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='05 dv1/dy + proton anti-proton STAR, + STAR, protron STAR, antiprotron NA49, protron FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 5: (Color online) The slope of the rapidity dependence of the di- rected flow coefficients of pions, protons and antiprotons as functions of the colliding energy in 10-40% Au+Au collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Results from our model calculation are compared to the STAR [19] and NA49 [51] data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Here, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2Ht is applied to the baryon number density distribu- tion at √sNN = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' lisions, their v1 both follow the energy density distribution of the QGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' On the other hand, protons carry baryons deposited by the beam nuclei and therefore their v1 is significantly af- fected by the distribution of the initial baryon number density and show different features compared to pions and antipro- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 6, we show the slope of v1(y) as a function fv when Ht is fixed at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 6, we observe different dependences of dv1/dy on the medium de- formation and its longitudinal flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' While the slopes of pions and antiprotons decreases as the medium becomes more tilted (or Ht increases), the opposite is seen when the longitudinal flow velocity (or fv) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Figure 6 suggests the sensitivity of the slope of v1(y) to both the medium geometry and its longitudinal flow profile in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Here, Ht = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 and fv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='23 provide the best simultaneous description of the v1 of pions, protons and antiprotons at √sNN = 27 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Values of these two parame- ters at other collision energies are constrained in the same way in our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Global polarization Apart from the directed flow, the longitudinal flow veloc- ity gradient can also induce polarization of constituent par- tons inside the QGP via the spin-orbit coupling [50, 68, 80– 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' It has also been found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [67] that the global po- larization can be affected by the initial geometry of the QGP medium when the same initial longitudinal velocity field is ap- plied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, due to their similar origins, directed flow and global polarization should be correlated with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This correlation has recently been investigated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [50, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In this subsection, using the same model setup as previously im- plemented for studying the hadron v1, we further explore the global polarization of Λ and Λ in heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 0 5 10 15 20 Ht 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='050 dv1/dy 10-40% Au+Au @ 27 GeV, fv=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='23 + protron antiprotron STAR, + STAR, protron STAR, antiprotron 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='35 fv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='02 dv1/dy 10-40% Au+Au @ 27 GeV, Ht = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 + protron antiprotron STAR, + STAR, protron STAR, antiprotron FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 6: (Color online) Upper panel: the slope of the directed flow coefficient dv1/dy as a function of the tilt parameter Ht when fv is fixed, in 10-40% Au+Au collisions at √sNN = 27 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Lower panel: the slope of the directed flow coefficient as a function of the longitudinal rapidity fraction parameter fv when Ht is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The experimental data are from the STAR Collaboration [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' We assume quarks have reached local thermal equilib- rium on their freezeout hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Meanwhile, the spin of quarks or hadrons are not modified during particlization and resonance decay [60, 86, 88, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Then, the polarization pseudo vector for spin 1/2 fermions can be obtained using the modified Cooper-Frye formalism as [90, 91], Sµ(p) = � dΣ · pJ µ 5 (p, X) 2m � dΣ · N(p, X), (28) where J µ 5 is the axial charge current density and N µ(p, X) is the number density of fermions in the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Fol- lowing the quantum kinetic theory [88,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 92],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Sµ(p) can be decomposed into different sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Sµ(p) = Sµ thermal(p) + Sµ shear(p) + Sµ accT(p) +Sµ chemical(p) + Sµ EB(p),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (29) where Sµ thermal(p) = � dΣσFσϵµναβpν∂α uβ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Sµ shear(p) = � dΣσFσ ϵµναβpνuβ (u · p)T ×pρ(∂ρuα + ∂αuρ − uρDuα),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Sµ accT(p) = − � dΣσFσ ϵµναβpνuα T � Duβ − ∂βT T � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 9 Sµ chemical(p) = 2 � dΣσFσ 1 (u · p)ϵµναβpαuβ∂ν µ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Sµ EB(p) = 2 � dΣσFσ �ϵµναβpαuβEν (u · p)T + Bµ T � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (30) with F µ = ℏ 8mΛΦ(p)pµfeq(1 − feq),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Φ(p) = � dΣµpµfeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (31) The five terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (30) represent polarization induced by the thermal vorticity (Sµ thermal), the shear tensor (Sµ shear), the fluid acceleration minus temperature gradient (Sµ accT), the gra- dient of chemical potential over temperature (Sµ chemical), and the external electromagnetic field (Sµ EB), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' De- tailed expressions of these terms can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [60, 88, 89, 92] or derived from the statistic model [93, 94] and the Kubo formula [95–98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Here, Sµ shear and Sµ chemical are also named as the shear-induced polarization (SIP) and the bary- onic spin Hall effect (SHE) in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Since the electro- magnetic field decay rapidly in heavy-ion collisions, the Sµ EB term is neglected in our current work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The average polarization vector in the rest frame of Λ (or ¯Λ) is then given by ⃗P ∗(p) = ⃗P(p) − ⃗P(p) · ⃗p p0(p0 + m)⃗p, (32) where P µ(p) ≡ 1 sSµ(p), (33) with s = 1/2 being the spin of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' After averaging over the transverse momentum, one obtains the local polariza- tion as ⟨⃗P(φp)⟩ = � ymax ymin dy � pTmax pTmin pTdpT[Φ(p)⃗P ∗(p)] � ymax ymin dy � pTmax pTmin pTdpTΦ(p) , (34) in which φp is the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the present study, the mass of Λ (or Λ) is set as m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='116 GeV, and the kinematic regions for analyzing the hyperon polarization are pT ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 GeV, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 GeV] and y ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Finally, the global polarization of Λ and Λ is obtained by further averag- ing ⃗P ∗(p) over φp in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 7 is the global polarization of Λ and ¯Λ hyper- ons along the out-of-plane direction as a function of the col- lision energy in 20-50% Au+Au collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Using the model parameters Ht and fv extracted from the directed flow of pi- ons, protons and antiprotons as discussed earlier, our model calculation also provides a reasonable description of the hy- peron polarization here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This helps further validate our mod- eling of the tilted geometry of the QGP together with its lon- gitudinal flow gradient, and also confirms the correlation be- tween the directed flow and the global polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' We have 10 1 10 2 sNN [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 Py (%) (th+shear+accT+chem) (th+shear+accT+chem) (th+accT+chem) (th+accT+chem) STAR, STAR, 10 1 10 2 sNN [GeV] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 Py (%) (total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0*Ht for baryon) (total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0*Ht for baryon) (total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2*Ht for baryon) (total 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='2*Ht for baryon) STAR, STAR, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 7: (Color online) The global polarization of Λ and Λ as a func- tion of the collision energy in 20-50% Au+Au collisions, analyzed within pT ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 GeV, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='0 GeV], y ∈ [−1, 1] and compared to the STAR data [99] (rescaled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='877 due to the updated hyperon decay parameter [50, 100]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the lower panel, “total” denotes “thermal + shear + accT + chemical” for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' confirmed that the total amount of polarization presented here is mainly contributed by the thermal vorticity term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The shear tensor relies on the tilted geometry of the medium: a coun- terclockwise tilt in the reaction plane induces a negative shear tensor, while a clockwise tilt induces a positive shear tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, as shown in the upper panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 7, introduc- ing the shear contribution increases the magnitude of −Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' In the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 7, we study the effects of the pos- sible larger value of Ht for the net baryon number density than for the energy density at low energies (√sNN = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='5 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 GeV), as previously suggested by the proton v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' After introducing this stronger tilt of baryon number density distri- bution, the global polarization becomes smaller because of a positive contribution from the chemical term to Py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' We note that the difference between Λ and ¯Λ polarization relies on both the geometry of baryon distribution and the longitudinal flow velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' This is why our current result appears different from the earlier study [86] using the same hydrodynamic cal- culation but different initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The electromagnetic field may cause further difference between their polarization, which has not been included in our current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 10 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' CONCLUSIONS We have developed an initial condition model for studying the directed flow and global polarization of identified hadrons in heavy-ion collisions across the BES energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The Glauber model has been extended such that effects of the tilted geome- try of both the energy density and the net baryon number den- sity distribution has been included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The initial longitudinal flow velocity profile has also been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' By combin- ing this initial condition with a hydrodynamic simulation of the QGP evolution, we have provided a satisfactory descrip- tion of the transverse momentum spectra of identified particles (π+, K+, p and ¯p) from √sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 to 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The di- rected flow coefficient of these identified hadrons, especially the splitting of v1 between protons and antiprotons have been investigated in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Our calculation shows the essential role of the tilted medium geometry and the early-time longitudinal flow veloc- ity in generating the directed flow of hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' While both the strength of tilt (or the Ht parameter) and the fractional longitudinal momentum deposition (or fv) become larger at lower collisional energies, they have different impacts on the slope of v1(y) of different hadron species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' An increasing Ht results in a decrease of dv1/dy around mid-rapidity for both pions and antiprotons, while an increasing fv causes their in- crease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The opposite trend could be seen for protons, which is strongly affected by the baryon number density distribution deposited by the colliding beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, a simultaneous comparison of the pion, proton and antiproton v1 to their ex- perimental data sets a tight constraint on the inhomogeneous distribution of the initial energy and baryon number density, and the longitudinal flow velocity profile of the nuclear matter created in non-central heavy-ion collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The initial geom- etry and flow velocity extracted from the hadron v1 is further tested with the global polarization of Λ and ¯Λ hyperons that is also affected by the medium geometry and the fluid velocity gradient in the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Using the same hydro- dynamic framework, we obtain a reasonable description of the hyperon polarization across the BES energies, and find that the separation of global polarization between Λ and ¯Λ could also be sensitive to the tilted geometry of the net baryon num- ber density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Our study constitutes a step forward in understanding the origin of the splitting of v1 between different particle species produced in Au+Au collisions at √sNN = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='7 - 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content='4 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Nevertheless, in addition to the deformed medium geome- try and the initial longitudinal flow velocity gradient, other sources exist for splitting the directed flow between protons and antiprotons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' For example, the electromagnetic field pro- duced in non-central heavy-ion collisions results in directional drift of charged quarks (u, d, s) and therefore different values of v1 between different final state charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Addi- tionally, the light hadron v1 can also be affected by the de- celerated baryon flow and hadronic cascade after the QGP expansion, especially at lower collision energy [45, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Therefore, a combination of hydrodynamic model and after- burner hadronic transport is necessary for a better constraint on the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Furthermore, due to the limit of our cur- rent smooth initial condition, our calculation is restricted to the rapidity odd component of v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' The rapidity even com- ponent, especially its non-trivial pT dependence even at mid- rapidity [101–103] is another interesting topic to investigate after our initialization method is extended to include event- by-event fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Last but not least, the initial condi- tion and the QGP profile we propose here can be straightfor- wardly coupled to studies of hard probes in low energy colli- sions, such as the collective flow of high pT hadrons and heavy quarks [104] and their correlations [105, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' These will be addressed in our upcoming efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 11935007, 12175122 and 2021-867, Guangdong Major Project of Basic and Applied Basic Research No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 2020B0301030008, the Nat- ural Science Foundation of Hubei Province No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' 2021CFB272, the Education Department of Hubei Province of China with Young Talents Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' Q20212703, the Open Founda- tion of Key Laboratory of Quark and Lepton Physics (MOE) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' QLPL202104 and the Xiaogan Natural Science Founda- tion under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' XGKJ2021010016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZtE1T4oBgHgl3EQfKAPm/content/2301.02960v1.pdf'} +page_content=' [1] Jean-Yves Ollitrault.' metadata={'source': 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b/b9AzT4oBgHgl3EQfLftx/content/tmp_files/2301.01115v1.pdf.txt @@ -0,0 +1,1316 @@ +Supercurrent in Bi4Te3 Topological Material-Based Three-Terminal Junctions +Jonas K¨olzer,1, 2 Abdur Rehman Jalil,1, 2 Daniel Rosenbach,1, 2 Lisa Arndt,3 Gregor Mussler,1, 2 +Peter Sch¨uffelgen,1, 2 Detlev Gr¨utzmacher,1, 2 Hans L¨uth,1, 2 and Thomas Sch¨apers1, 2, ∗ +1Peter Gr¨unberg Institut (PGI-9), Forschungszentrum J¨ulich, Wilhelm-Johnen-Straße, 52425 J¨ulich, Germany +2JARA-Fundamentals of Future Information Technology, J¨ulich-Aachen Research Alliance, +Forschungszentrum J¨ulich and RWTH Aachen University, 52425 J¨ulich, Germany +3JARA Institute for Quantum Information, RWTH Aachen University, Germany +(Dated: January 4, 2023) +In an in-situ prepared three-terminal Josephson junction based on the topological insulator Bi4Te3 +and the superconductor Nb the transport properties are studied. The differential resistance maps as +a function of two bias currents reveal extended areas of Josephson supercurrent including coupling +effects between adjacent superconducting electrodes. The observed dynamics for the coupling of the +junctions is interpreted using a numerical simulation of a similar geometry based on a resistively +and capacitively shunted Josephson junction model. The temperature dependency indicates that the +device behaves similar to prior experiments with single Josephson junctions comprising topological +insulators weak links. Irradiating radio frequencies to the junction we find a spectrum of integer +Shapiro steps and an additional fractional step, which is interpreted by a skewed current-phase +relationship. In a perpendicular magnetic field we observe Fraunhofer-like interference patterns of +the switching currents. +I. +INTRODUCTION +Hybrid structures comprising three-dimensional topo- +logical insulator nanoribbons combined with supercon- +ductors are a very promising platform for realizing cir- +cuits for fault-tolerant topological quantum computing +[1–4]. For its operation Majorana bound states are em- +ployed, which are formed by aligning an external mag- +netic field with a nanoribbon proximitized with an s-type +superconductor [5–7]. For the braiding of different pairs +of Majorana states for qubit operation multi-terminal +structures are required [2, 8, 9]. Braiding can be per- +formed by adjusting the superconducting phase of the +superconducting electrodes to each other. +Multi-terminal Josephson junctions are the backbone +of Majorana braiding mechanism in a topological qubit; +where a three-terminal Josephson junction acts as a basic +building block [2]. Understanding the superconducting +transport in such a device holds a key importance for the +realization of a topological quantum system. Generally, +the use of hybrid devices with multiple connections leads +to rich physics in terms of transport properties. Indeed, +theoretical studies have investigated singularities, such +as Weyl nodes, in the Andreev spectra of multi-terminal +Josephson junctions [10–12]. +Moreover, multi-terminal +Josephson junctions with topologically trivial supercon- +ducting leads may lead to realizations where the junction +itself can be regarded as an artificial topological material +[13]. +Furthermore, three-terminal junctions also allow +transport via the quartet mechanism and non-local An- +dreev processes [14–17]. +On the experimental side, multi-terminal Josephson +junctions were fabricated with different materials for the +∗ th.schaepers@fz-juelich.de +weak link. In three-terminal Josephson junctions with +a Cu or InAs nanowire subgap states [18, 19] and half- +integer Shapiro steps [20] were observed, indicating trans- +port via quartets of entangled Cooper pairs. Supercur- +rent flow affected by dissipative currents in an adjacent +junction was studied on graphene-based junctions [21]. +Moreover, the higher-dimensional phase space was found +to lead to fractional Shapiro steps in this type of junctions +due to the inverse AC Josephson effect [4]. By combin- +ing a multi-terminal junction with a top gate, the effect +of gate voltage and magnetic field on the critical current +contour has been studied [3, 24, 25]. Recently, flakes of +the topological insulator Bi2Se3 were also used as a weak +link in an interferometer structure, and evidence for a +non-sinusoidal current-phase relationship was observed +[26]. In flux-controlled three-terminal junctions based on +Bi2Te3, the opening and closing of a minigap was studied +using normal probes [27]. +Here, we report on the transport properties of a three- +terminal Josephson junction based on the Bi4Te3 mate- +rial system as the weak link and Nb as the supercon- +ductor. To fabricate the samples, we used selective-area +growth for the Bi4Te3 layer in combination with an in- +situ bridge technology to define the superconducting elec- +trodes [2]. Bi4Te3 is a natural superlattice of alternating +Bi2 bilayers and Bi2Te3 quintuple layers. Initially, Bi4Te3 +has been reported to be a semimetal with zero band gap +and a Dirac cone at the Γ point [29]. However, recent +band structure calculations in conjunction with scanning +tunneling spectroscopy and angular photoemission spec- +troscopy measurements suggest that the material is a +semimetal with topological surface states [30–32]. In par- +ticular, advanced GW-band structure calculations have +shown that a band gap of about 0.2 eV opens at the Γ +point, which significantly reduces the density of the bulk +state in this energy range [32]. Bi4Te3 is classified as a +dual topological insulator, a strong topological insulator +arXiv:2301.01115v1 [cond-mat.supr-con] 3 Jan 2023 + +2 +with a non-zero mirror Chern number, i.e. a topological +crystalline insulator phase. Though Bi4Te3 does not ex- +hibit the proposed Dirac semimetal phase, it is still a very +interesting material as it resides in close proximity to the +critical point of band crossing in the topological phase +diagram of BixTey alloys [33]. Such a transition is pro- +posed by Yang et al. [34] where a topological crystalline +insulator (Bi2Te3) [35] can be topologically transformed +into a topological Dirac semimetal through alloying it +with other materials. On our multi-terminal junctions, +we first investigated the DC properties and related the +results to simulations based on the resistively and capac- +itively shunted Josephson junction (RCSJ) model. We +then measured the radio frequency (rf) response, finding +evidence for coupling of adjacent junctions. Finally, the +behavior of our three-terminal junctions when an out-of- +plane magnetic field is applied is investigated. +II. +EXPERIMENTAL +Using the previously introduced technologies of topo- +logical insulator selective-area growth and in-situ bridge +technology we fabricated three-terminal Josephson junc- +tions, as illustrated in Fig. 1(a) [2, 36]. +The geome- +try of the nanoribbon T-shaped junction for selective- +area growth is defined by trenches in a SiO2/Si3N4 +(5 nm/15 nm) layer on a highly-resistive Si (111) sub- +strate [37]. +First, the 600-nm-wide nanotrenches are +etched into the top Si3N4 layer using a combination of +electron beam lithography and reactive ion etching. Sub- +sequently, a second set of layers, i.e. a 100-nm-thick SiO2 +layer and a 300-nm-thick Si3N4 layer, is deposited on top +to define the stencil mask for the in-situ Nb deposition +[2]. After patterning the structures for the stencil mask +into Si3N4, SiO2 is etched in hydrofluoric acid (HF) form- +ing the free-hanging bridge structures. Simultaneously, +the Si(111) surface in the selective-area growth trenches +is released in the bottom SiO2 layer defined by the Si3N4 +layer on top. The Bi4Te3 layer is selectively grown within +these trenches, while the Si3N4 bridge structures are em- +ployed to define the geometry of the in situ deposited su- +perconducting electrodes [2]. The Bi4Te3 layer is grown +at a temperature of 310◦C using molecular beam epitaxy. +Subsequently, the 50-nm-thick superconducting Nb elec- +trodes are deposited by electron beam evaporation fol- +lowed by covering the whole structure with a 5-nm-thick +Al2O3 dielectric capping layer. Our processing scheme +ensured a high-quality crystalline topological insulator +material with clean superconductor interfaces [2, 38], as +reported in previous transmission electron microscopy +studies. +An electron microscopy image of the investi- +gated device is presented in Fig. 1(b). +The measurements of the three-terminal Josephson +junction were carried out in a dilution refrigerator with +base temperature of T = 25 mK. containing a 1 - 1 - 6 T +vector magnet. As indicated in Fig. 1(b), the left, right, +and bottom junction electrodes are labeled as ”L”, ”R”, +and ”B”, respectively. Two current sources supply cur- +rents ILB and IRB from L and R to the bottom electrode, +respectively, with the according voltages VLB and VRB +measured. The differential resistances are measured by +adding an ac current of 10 nA to the DC current bias us- +ing a lock-in amplifier. The rf-irradiation for the Shapiro +step measurements was provided by an antenna placed +in close vicinity to the sample. +III. +RESULTS AND DISCUSSION +DC characteristics +Information about the basic junction characteristics is +obtained by measuring the differential resistances RLB = +∆VLB/∆ILB and RRB = ∆VRB/∆IRB as a function of the +bias currents ILB and IRB, respectively. Starting with the +left junction we find that RLB shown in Figs 2(a) and (b) +contains a superconducting region in the center when ILB +and IRB are varied. The observed critical current contour +is similar to what has been observed in induced supercon- +ducting nano junctions made of high mobility materials +such as InAs/Al [3, 24] or graphene [21]. The supercon- +ducting region extends along an inclined line indicated by +the dashed line in Fig. 2(a). The switching to the super- +conducting state can be seen in the line cuts at fix values +IRB = 0 and ±0.7µA provided in Fig. 2(b). The exten- +sion of the superconducting state originates from a part of +IRB which flows via R to L through the junction between +L and B compensating the current ILR partly and by that +reducing the total current. For our three-terminal device +no reduced differential resistance is observed along the +line ILB = IRB, which would indicate the presence of a +Josephson supercurrent between the junction formed be- +tween electrodes L and R [3, 25]. We attribute this to the +fact that the distance between these electrodes is slightly +larger than for the other junctions so that no Josephson +supercurrent is obtained. However, the junction between +L and R acts as a shunt resistor taking care that the +switching to the superconducting state is non-hysteretic. +The differential resistance RRB measured between R and +B electrodes, depicted in Figs 2(c) and (d), shows a sim- +ilar behaviour as RLB, i.e. featuring also an extended +superconducting range due a compensation provided by +part of ILR. The tilt of the superconducting range indi- +cated by the dashed line in Figure 2(c) is lower compared +to Fig. 2(a) since now ILR is the compensating current. +Simulations +The experimental results are modeled by assuming +a network of two resistively and capacitively shunted +Josephson (RCSJ) junctions coupled by a resistor RC, +as illustrated in Fig. 3(a). +Solving the related system +of differential equations numerically, in analogy to what +was presented in previous works [3, 4], we simulate the + +3 +400nm +ILB +IRB +a +b +B +L +R +VLB +VRB +FIG. 1. Rendering of a selective-area grown three-terminal Josephson junction and false color scanning electron micrograph +with circuit: (a) The three-terminal junction is composed of the silicon substrate (gray bottom layer), the first hard mask +composed out of a silicon oxide (white)/ silicon nitride (blue) layer (as indicated by the labels). On top of this another hard +mask layer composed of silicon oxide (white) and silicon nitride (blue) is deposited and patterned as a shadow mask. The +topological insulator (red) is grown selectively into the first hard mask trench and the shadow mask is used for the definition +of the junction in the metal deposition (silver) step. (b) False-color scanning electron micrograph of the in-situ prepared three- +terminal junction device. Niobium contacts (cyan) are deposited on top of the TI (red). The measurement configuration is also +indicated. +FIG. 2. +Differential resistance maps: (a) shows RLB as a +function of the bias currents ILB and IRB at 25 mK with cor- +responding line cuts given in (b). In (c) the differential re- +sistance map of RRB is depicted with a selection of line cuts +given in (d). The dashed lines in a and (c) indicate the super- +conducting regions of compensating bias currents. The differ- +ential resistances was measured by using lock-in technique, +i.e. RLB = ∆VLB/∆ILB and RRB = ∆VRB/∆IRB. +behaviour of the experimental system (information about +the procedure see Supplementary Material). The results +of the simulations are shown in Figs. 3(b) to (e), where +the differential resistance RLB is given as a function of +the bias currents ILB and IRB. +The model describes the experiment well by reproduc- +ing the Josephson supercurrent along the inclined lines +originating from compensating currents from both elec- +trodes with a superconducting region at the center. The +inclination is determined by the coupling resistance RC. +In Figs. 3(b) and (c), the coupling resistance was taken +as RC = 4 · RLB, with RLB = 40 Ω which results in the +same tilt as observed experimentally. Taking these val- +ues into account the normal state resistance is given by +RN = 6/5 · RLB = 48 Ω. In our simulations for the crit- +ical current and for the Steward-McCumber parameter +we assumed Ic = 538 nA and βc = (2e/ℏ)IcR2 +NC = 0.1, +respectively, with c the junction capacitance. We found +that the superconducting state in the junction between +R and B leads to some weak feature as a similar line in- +clined towards horizontal orientation. Note, that for this +line RLB is non-zero, as the supercurrent in the other +junction only partly reduces the current in the junction +between L and B and hence only partially reduces the +voltage drop. A noticeable difference between experiment +and simulation is that in the measurements the extension +of the superconducting state observed along the inclined +line (cf. Fig. 2(a) is decreased compared to the simula- +tion depicted in Fig. 3(b). As discussed by Draelos et +al. [21], this effect can be explained by dissipation in the +neighboring junction being in the normal state resulting +in an effective heating, in particular for junctions with +small dimensions. In our simulation the direct coupling +between the different junctions was neglected. As shown +by Arnault et al. [4], including coupling results in a more +complex contour of the critical current area. If the cou- +pling resistance becomes very small, i.e. RC → 0, the +observed lines in the differential resistance shift towards +the diagonal (cf. Figs. 3(d) and (e). Thus, both junctions +are maximally correlated to both current biases ILB and +IRB. +Temperature dependence +In Figs. 4(a) to (f) the differential resistance maps are +shown for RLB and RRB measured at temperatures of + +b +a +150 +80 +0.5 +(μA) +(U) +60 +100 +0.0 +RLB +RLB +B +40 +50 +-0.5 +20 +0 +0 +-0.5 0.0 0.5 +-0.5 0.0 +0.5 +ILB (μA) +ILB (μA) +d +c +80 +0.5 +(U) +(μA) +60 +(U) +50 +0.0 +RRB +RRB +RB +40 +-0.5 +20 +0 +0 +-0.5 0.0 +0.5 +-0.5 +0.0 +0.5 +ILB (μA) +IRB (μA)200 nm* +EHT = 5.00 kV +Signal A = InLens +Date :1 Feb 2020 +HNF +Mag = 102.02 K X +WD= 5.0mm +Tilt = +0.0 ° +LE01550VPSiN4 +JLB +JRB +RN +IRB +VRB +J +C +RN +ILB +VLB +J +C +RC +a +b +c +d +e +FIG. 3. +Numerical simulation of different coupling scenar- +ios: (a) The three-terminal circuit is modeled by two RCSJ +shunted Josephson junctions JLB and JRB (green and blue +dashed line boxes), which are each modeled by a resistor RN, +a capacitor c, and an ideal Josephson junction J. Currents +ILB and IRB are supplied via current sources while the volt- +age drops VLB and VRB across the junctions are measured. +Both junctions are coupled via a coupling resistance RC. (b) +Differential resistance RLB as a function of current biases +for a realistic scenario for RC close to the one extracted in +the experiment: RN = 40 Ω, RC = 160 Ω, Ic = 538 nA, +βc = (2e/ℏ)IcR2 +NC = 0.1. The zero resistance range is ob- +served as a tilted line due to a compensation by a part of +IRB. Additionally, the influence of the second junction is ob- +served as a similar line close to horizontal orientation. The +corresponding line cuts indicated in (b) are presented in (c). +The scenario for a very small coupling resistance (RC → 0) +is shown as a color map of RLB and selected line cuts in (d) +and (e). +100 mK, 200 mK, and 800 mK. One finds that with in- +creasing temperature the area of the central supercon- +ducting region shrinks. This is in accordance with the +temperature dependence of the critical current of a sin- +gle Nb/Bi4Te3/Nb reference junction, as shown in the +Supplementary Material. It is noteworthy that the super- +conducting feature along the inclined lines basically does +not change with increasing temperature. This can be ex- +plained by the fact, that the dissipation in the neighbor- +ing junction already leads to an increased temperature +being larger than the substrate temperature. +0.50.0 0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +a +T=100 mK +0.50.0 0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +b +T=100 mK +0.50.0 0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +c +T=200 mK +0.50.0 0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +d +T=200 mK +0.50.0 0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +e +T=800 mK +0.50.0 0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +f +T=800 mK +0 +50 +100 +RLB ( ) +0 +50 +100 +RRB ( ) +50 +150 +250 +RLB ( ) +0 +50 +100 +150 +RRB ( ) +10 +20 +30 +RLB ( ) +10 +20 +RRB ( ) +FIG. 4. Differential resistance maps at various temperatures: +Left column (a), (c), (e) shows the differential resistance RLB, +right column (b), (d), (f) RRB, accordingly. +The temper- +atures displayed in the rows from up to down are 100 mK, +200 mK, and 800 mK, respectively. +rf characteristics +Next the radio frequency response of the system is in- +vestigated in order to confirm that the experiment is de- +scribed well by Josephson junction physics and to analyze +the rf response of the Josephson current. This is done by +first choosing a frequency and an amplitude for the rf ir- +radiation so that both junctions show a large rf response +in the differential resistance. Subsequently the same DC +bias sweeps are performed as in the prior experiments. +Figures 5(a) and (b) show Shapiro step measurements of +the differential resistances RLB and RRB, respectively, as +a function of bias currents ILB and IRB. The differential +resistances are calculated by numerical differentiation. +Differential resistances obtained by lock-in amplifier mea- +surements can be found in the Supplementary Material. +The rf frequency frf and the according power was set to +5.8 GHz and 0 dBm, respectively. The differential resis- +tances show clear intercrossing stripe-like patterns which +can be attributed to the presence of Shapiro steps con- +firming the presence of a Josephson supercurrent. The +intercrossing parallel stripes indicating a coupling of both +junctions. By calculating the related voltage drop we find +that for both junctions the Shapiro steps are located at +integer multiples, n = 1, 2, 3 . . . , of the characteristic + +5 +0.5 0.0 +0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +a +0.5 0.0 +0.5 +ILB ( A) +0.5 +0.0 +0.5 +IRB ( A) +b +20 +40 +60 +RRB ( ) +0 +20 +40 +60 +RLB ( ) +FIG. 5. Shapiro step measurements at 5.8 GHz: (a) Numer- +ically determined differential resistance RLB as a function of +ILB and IRB at 5.8 GHz and rf power of 0 dBm. (b) Corre- +sponding map of the differential resistance RRB. +voltage V0 = hfrf/2e. +In Figs. 6(a) and (b) the differential resistance maps +of RLB and RLB, now taken at 8.5 GHz at 0 dBm, are +depicted, respectively. +Here, the color maps are plot- +ted as a function of the normalized voltages VLB/V0 and +VRB/V0. On first sight one finds that the Shapiro step +pattern is more pronounced in RLB. We attribute this to +a stronger coupling of the rf signal compared to the neigh- +bouring junction due to spatial variations of the rf field. +As for the measurements at 5.8 GHz a coupling of both +junctions, although weaker, is observed. Our experimen- +tal results of Shapiro step measurements are supported +by comparison to simulations based on the previously in- +troduced RCSJ model. In Supplementary Figures 4(a) +and (b) maps of the simulated values of RLB and RLB +as a function of the normalized bias voltages are shown. +There, one finds that the coupling by RC results in a +weak cross coupling of the Shapiro signal resulting in in- +tercrossing stripe-like patterns of different contrast. +A closer inspection of the resistance map presented in +Fig. 6(a) reveals that apart from the integer Shapiro steps +also half-integer Shapiro steps, e.g. at n = 1/2, are ob- +served. The half integer steps are also clearly resolved +in the averaged value of RLB along VLB/V0 shown in +Fig. 6(a). In single Josephson junctions such fractional +steps are interpreted by assuming a skewed current-phase +relationship [39–41] (a simulation for this case using our +model is provided in the Supplementary Material). More +specifically for multi-terminal junctions the rf response +of superconductivity induced into normal metal has been +studied previously by Duvauchelle et al. [20]. Here, half- +integer steps have been found and interpreted as a feature +due to the presence of coherent quartet states. +How- +ever, in Fig. 2 we did not find indications of quartet +states, which would be visible by a feature in the dif- +ferential resistance at opposite voltage drops on the left +and right terminal [18]. Other experimental observations +of such fractional steps in multi-terminal junctions are in- +terpreted on the basis of highly connected nonlinear net- +works of Josephson junctions, where (due to the higher +phase space) different transitions of the phase particle in +the washboard potential are possible [4]. However, since +fractional Shapiro steps were observed in single junctions +made with similar materials [42], we favor the explana- +tion based on a skewed current-phase relationship, which +can be attributed to contributions of quasi-ballistic trans- +port. +In our measurements under rf radiation we did +not find indications of missing odd Shapiro steps, as pre- +dicted when Majorana bound states are present in topo- +logical junctions [2, 43]. Probably, for our samples the +narrow width of the Bi4Te3 ribbons prevents the forma- +tion of these states, since due to the finite Berry phase +a magnetic field along the junctions is required to gain +a gap closure for the coherent surface states around the +nanoribbon cross section [36]. +Magnetic field response +The junction characteristics were also analyzed in a +perpendicularly oriented magnetic field B⊥. In Fig. 7(a) +the magnetic field dependence RLB is plotted as a func- +tion of B⊥ and ILB, while IRB is kept at zero. +One +clearly observes a Fraunhofer-like interference pattern of +the switching current, i.e. +the boundary between the +red superconducting areas and the areas with finite resis- +tance. The blue line in Fig. 7(a) indicates the according +fitting based on the Fraunhofer interference relation. The +close resemblance of the experimental data to an ideal +Fraunhofer pattern points towards a relatively homoge- +neous distribution of the supercurrent density. From the +fit we extract a period of about ∆B =14 mT, which corre- +sponds to a junction area of 152×103 nm2. Relating these +values to the dimensions of the left junction JLB one finds +that the period is about a factor of ten smaller than ex- +pected. Based on the actual junction size of 200×72 nm2 +a period of 144 mT is expected for a h/2e flux periodic- +ity. +We attribute the discrepancy to the experimental +period to a pronounced flux focusing effect, where the +magnetic field is expelled from the edge regions of the +superconducting electrodes and bundled in the junction +area. As a matter of fact, a comparably large flux fo- +cusing effect was previously observed in similar planar +Josephson junctions based on topological insulators and +Nb superconducting electrodes [36]. +In Fig. 7(b) the magnetic field dependence RRB is +shown as a function of B⊥ and IRB at ILB = 0. Once +again, a Fraunhofer-like interference is observed, al- +though with a smaller period, i.e. +an larger effective +area where the magnetic flux is picked up. The reason +for the difference compare to the measurements shown +in Fig. 7(a) might be some inhomogeneity in the super- +current density in the junction. Finally, the RLB maps +are scanned diagonally, i.e. +ILB = IRB, as shown in +Fig. 7(c). +Here, once again a regular Fraunhofer pat- +tern is observed, which is almost identical to the pat- +tern shown in Fig. 7(d), indication, that the current IRB +through the neighboring junction basically has not effect. + +6 +FIG. 6. Shapiro step measurements at 8.5 GHz: (a) Numerically determined differential resistance RLB as a function of the +normalized voltage drops VLB/V0 and VRB/V0 at 8.5 GHz and rf power of 0 dBm, with V0 = hfrf/2e. The blue curves represent +the averaged signal along VLB/V0 and VRB/V0, respectively. The dashed lines indicate the half-integer steps. (b) Corresponding +map of the differential resistance RRB. +a +b +c +FIG. 7. Differential resistances under perpendicular magnetic field sweep: (a) shows a map of RLB as a function of B⊥ and ILB +for IRB = 0. (b) represents the corresponding map of RRB as a function of B⊥ and IRB for ILB = 0. In (c) RLB is plotted with +the sweep current chosen to be ILB = IRB, which corresponds a sweep along the diagonal in the current plane. In all cases a +standard Fraunhofer pattern is fitted indicated as blue lines. +IV. +CONCLUSION +We have succeeded in extending the previously devel- +oped in situ fabrication technology for Josephson junc- +tions to a working more complex design of a three- +terminal junction. Analysis of the transport experiments +shows that our system indeed behaves like a coupled net- +work of Josephson junctions in DC transport, rf response, +as well as magnetic field response. This is the first re- +port on the topological multi-terminal devices where an +interaction between the individual Josephson junctions is +observed. Moreover, the observation of fractional steps +in the rf response opens a window that provides a first +insight into the novel physics of this type of device. On +a more technical level, our results demonstrate the re- +alization of more complex devices required for network +structures in topological quantum circuits. +Further investigations and detailed understanding of +such a system are crucial for the realization of complex +topological quantum systems. In future, similar exper- +iments with more intricate circuit designs and super- +conducting phase controlled measurements will be per- +formed. The complexities in the junction characteristics +arose from the selected weak-link material Bi4Te3. +In +future experiments we plan to incorporate conventional +three-dimensional topological insulators, e.g. +Bi2Te3, +Sb2Te3, Bi2Se3, and the topological Dirac semimetal ex- +hibited by the correctly tuned BixTey stoichiometric al- +loy. +ACKNOWLEDGEMENT +We thank Herbert Kertz for technical assistance as +well as Kristof Moors and Roman Riwar for fruitful +discussion. This work was partly funded by the Deutsche +Forschungsgemeinschaft (DFG, German Research Foun- +dation) under Germany’s Excellence Strategy—Cluster +of Excellence Matter and Light for Quantum Computing + +b +e +← 80 +fss +60 +60 +2 +2 +3 +40B +40 +1 +1 +R +R +R +0 +0 +20 +-20 +-1 +-1 +-2 +-2 +0 +-2 +0 +2 +-2 +0 +2 +LB(Vo) +VLB(Vo)500 +100 +80 +60 +0 +40 +20 +-500: +-25 +0 +25 +BL( +(mT)d +80 +500 +ILB =IRB (nA) +60 +(U) +0 +40 +KLB +R +20 +-500 +-25 +0 +25 +BL( +(mT)A +n +B500 +100 +80 +(U) +60 +0 +RRB +40 +20 +-500 +-25 +0 +25 +B→ (mT)(nA) +RBB +R() +RB +R(nA) +IRB += +BA +B7 +(ML4Q) EXC 2004/1—390534769, the German Federal +Ministry +of +Education +and +Research +(BMBF) +via +the Quantum Futur project “MajoranaChips” (Grant +No. +13N15264) within the funding program Photonic +Research Germany, as well as the Bavarian Ministry of +Economic Affairs, Regional Development and Energy +within Bavaria’s High-Tech Agenda Project “Bausteine +f¨ur +das +Quantencomputing +auf +Basis +topologischer +Materialien +mit +experimentellen +und +theoretischen +Ans¨atzen“ (grant allocation no. 07 02/686 58/1/21 1/22 +2/23). +[1] A. +Kitaev, +Fault-tolerant +quantum +computation +by +anyons, Annals of Physics 303, 2 (2003). +[2] T. 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Rosenbach, Quantum transport and induced super- +conductivity in selectively deposited topological insulator +devices, Dissertation, RWTH Aachen University, Aachen +(2021). +[43] F. Dom´ınguez, O. Kashuba, E. Bocquillon, J. Wieden- +mann, R. S. Deacon, T. M. Klapwijk, G. Platero, L. W. +Molenkamp, B. Trauzettel, and E. M. Hankiewicz, Joseph- +son junction dynamics in the presence of 2π- and 4π- +periodic supercurrents, Phys. Rev. B 95, 195430 (2017). + +1 +Supplementary Material: Supercurrent in Bi4Te3 Topological Material-Based Three-Terminal +Junctions +SI. +SINGLE JUNCTION MEASUREMENTS +As a reference a single Nb/Bi4Te3/Nb junction was measured. The junction has a length of 140 nm and a width +of 500 nm. In Supplementary Figure S1(a) the current voltage characteristics is shown at temperatures in the range +from 30 mK to 0.77 K. +At lowest temperature a critical current of 750 nA is obtained. +In contrast to the three +terminal junction, here, a hysteretic behaviour is observed, which can be explained by the missing shunt for the single +Josephson junction. We attribute the hysteresis to heating resulting in a lower return current Ir compared to Ic [S1]. +The critical current monotonously decreases with temperature with some kink around 0.4 K. The latter might be +attributed to a switching from a more diffusive to a more ballistic transport in the weak link [S2]. +Supplementary Figure S1. Current-voltage characteristics of a single Nb/Bi4Te3/Nb junction: (a) Current-voltage charac- +teristics at temperatures ranging from 30 mK to 0.77 K. (b) Critical current Ic as well as return current Ir as a function of +temperature. +SII. +RCSJ MODEL FOR A THREE-TERMINAL JUNCTION +The characteristics of our three-terminal junctions is simulated by employing a two-dimensional resistively and +capacitively shunted Josephson junction (RCSJ) Ansatz in analogy to what was presented in previous works [S3, S4]. +In Fig. 3(a) in the main text the corresponding network is depicted including two resistively and capacitively shunted +Josephson junctions with the normal state resistance RN and the capacitance C. We assume two identical junctions +each having a critical current of Ic. The junctions are connected by a coupling resistor RC representing the non- +superconducting junction between electrodes L and R. Following the RCSJ Ansatz the characteristics of the three- +terminal junction can be described by a set of coupled differential equations of the form: +ILB +Ic += sin(ϕLB) + dϕLB +d�τ ++ βc +d2ϕLB +d�τ 2 ++ RN +RC +�dϕRB +d�τ +− dϕLB +d�τ +� +, +(S1) +IRB +Ic += sin(ϕRB) + dϕRB +d�τ ++ βc +d2ϕRB +d�τ 2 +− RN +RC +�dϕRB +d�τ +− dϕLB +d�τ +� +, +(S2) +with ϕLB and ϕRB the phase differences between junctions JLB and JRB, respectively, �τ = t/τJ the normalized time, +τJ = Φ0/(2πIcRN), with Φ0 = h/2e the magnetic flux quantum, and βc = (2e/ℏ)IcR2 +NC the Stewart-McCumber +parameter [S5]. The equations are similar to the standard RCSJ model for a single junction, except of the last term, +which introduces the current through the resistor, coupling the two junctions. This current is a result of the voltage +difference between the two junctions and the coupling resistance. For RC → ∞ the coupling term goes to zero, leading +to two individual junctions (decoupled system) and for RC → 0 the system is dominated by the coupling term. + +a +b +40 +750 +X +X +500 +Ic +20 +X +250 +(Λr) +0 +0 +-20 +-250 +X +X +X +X +X +X +-40 +-500 +750 +C +-60 +—500 +-250 +0 +250 +500 +0.0 +0.2 +0.4 +0.6 +0.8 +-1000 -750 +T (K) +I (nA)2 +SIII. +SHAPIRO STEPS IN THREE-TERMINAL JUNCTION EXPERIMENTS +The differential resistances RLB and RRB exposed to an rf radiation with a frequency of 5.8 GHz at 0 dBm recorded +as a function of the applied DC currents are presented in Supplementary Figures S2(a) and (b). In contrast to the +corresponding figure, which was gained by numerical differentiation, here, the resistance is directly taken using a +lock-in amplifier. In Supplementary Figures S3 the corresponding measurements at a frequency of 5.8 GHz at 0 dBm +are shown. +Supplementary Figure S2. Shapiro Step response at 5.8 GHz: (a) shows the measured differential resistance across the first +junction RLB as a function of the direct current ILB and IRB across the junction. +(b) shows RRB for the same current +constellation. +Supplementary Figure S3. Shapiro Step response at 8.5 GHz: a shows the measured differential resistance across the first +junction RLB as a function of the direct current ILB and IRB across the junction. b shows RRB for the same current constellation. +SIV. +SHAPIRO STEPS IN THREE-TERMINAL JUNCTION SIMULATION +Using the model described in Supplementary Information SII the Shapiro response was simulated by adding an +oscillation contribution ij,rf sin(2πfrft), j = LB, RB, to the dc bias currents. The simulated differential resistances +RLB and RRB as a function of the normalized voltage drops at a frequency of 8.5 GHz are presented in Supplementary +Figures S4(a) and (b). One finds that the Shapiro response is strong in the corresponding junctions, while the coupling +from the neighboring junction is weak. +In order to simulate the appearance of the fractional Shapiro steps a non-sinusoidal current-phase relationship was +assumed for the Josephson junction by including a sin(2ϕ) contribution. In Supplementary Figures S5(a) and (b) the + +b +a +1000 +1000 +68.0 +72 +60.8 +64 +500 +53.6 +500 +56 +46.4 +48 +(nA) +(nA) +(nA) +(nA) +39.2 +40 +0 +0 +RB +9 +32.0 +32 +R +R +24.8 +24 +-500 +-500 +17.6 +16 +10.4 +8 +3.2 +-1000 +-1000- +0 +-1000 -500 +0 +500 +1000 +-1000 -500 +0 +500 +1000 +'LB(nA) +'Lb(nA)b +a +0.5 +0.5 +60 +60 +IRB(nA) +(nA) +C +0.0 +40 +0.0 +40 +LB +B +R +R +20 +-0.5 +-0.5 +20 +0 +-0.5 0.0 0.5 +-0.5 0.0 0.5 +ILB (nA) +ILB (nA)3 +Supplementary Figure S4. Shapiro step simulations at 8.5 GHz: (a) Numerically determined differential resistance RLB as a +function of the normalized voltage drops VLB/V0 and VRB/V0 at 8.5 GHz. The blue curves represent the averaged signal along +VLB/V0 and VRB/V0, respectively. (b) Corresponding map of the differential resistance RRB with the blue curves representing +the averaged differential resistance along VLB/V0 and VRB/V0, respectively. +respective simulation outcomes RLB and RRB for junctions JLB and JRB are shown as a function of bias currents. +One finds that by increasing the sin 2ϕ contribution fractional steps appear. +[S1] H. Courtois, M. Meschke, J. T. Peltonen, and J. P. Pekola, Origin of hysteresis in a proximity Josephson junction, Physical +Review Letters 101, 067002 (2008). +[S2] P. Sch¨uffelgen, D. Rosenbach, C. Li, T. W. Schmitt, M. Schleenvoigt, A. R. Jalil, S. Schmitt, J. K¨olzer, M. Wang, +B. Bennemann, U. Parlak, L. Kibkalo, S. Trellenkamp, T. Grap, D. Meertens, M. Luysberg, G. Mussler, E. Berenschot, +N. Tas, A. A. Golubov, A. Brinkman, T. Sch¨apers, and D. Gr¨utzmacher, Selective area growth and stencil lithography for +in situ fabricated quantum devices, Nature Nanotechnology 14, 825 (2019). +[S3] G. V. Graziano, J. S. Lee, M. Pendharkar, C. J. Palmstrøm, and V. S. Pribiag, Transport studies in a gate-tunable +three-terminal Josephson junction, Phys. Rev. B 101, 054510 (2020). +[S4] E. G. Arnault, T. F. Q. Larson, A. Seredinski, L. Zhao, S. Idris, A. McConnell, K. Watanabe, T. Taniguchi, I. Borzenets, +F. Amet, and G. Finkelstein, Multiterminal inverse ac Josephson effect, Nano Letters 21, 9668 (2021). +[S5] M. Tinkham, Introduction to Superconductivity (Dover Publications, New York, 2004). + +e +b +100 +100 +80 +80 +2 +2 +(5) +60 +60 +1 +L +RB +B +Lo +R +0 +0 +m + 40 +B +40 +V +-1 +-1 +20 + 20 +-2 +-2 +0 +0 +-2 +0 +2 +-2 +0 +2 +VB / Vo +VLB / Vo4 +Supplementary Figure S5. Simulation of Shapiro maps at 8.5 GHz with 2φ term: (a) differential resistance of the first junction +Shapiro steps as a function of ILB, IRB with an equal contribution of a sin 2φ-term in the system, (b) the same for the second +junction. (c) and (d) show the same after doubling the sin 2φ contribution in the system and (e) and (f) show the same after +doubling the contribution of (c) and (d). + +a +e +1000 +1000 +1000 +(n' +.u. +(nA) +(nA) +0 +0 +0 +9 +R +R +-1000 +-1000 +-1000 +-1000 +1000 +-1000 +1000 +-1000 +0 +0 +0 +1000 +ILB(nA) +ILB(nA) +I LB(nA) +b +d +1000 +1000 +1000 +RRB(a.u.) +IRB(nA) +(nA) +IRB(nA) +0 +0 +0 +R +R +-1000 +-1000 +-1000 +-1000 +0 +1000 +-1000 +0 +1000 +-1000 +0 +1000 +ILB(nA) +ILB(nA) +ILB(nA) \ No newline at end of file diff --git a/btE4T4oBgHgl3EQfPgwg/content/tmp_files/2301.04973v1.pdf.txt b/btE4T4oBgHgl3EQfPgwg/content/tmp_files/2301.04973v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6d56d3a54a39cecebcfede8969058dbd3ec4a321 --- /dev/null +++ b/btE4T4oBgHgl3EQfPgwg/content/tmp_files/2301.04973v1.pdf.txt @@ -0,0 +1,1288 @@ +Desynchronizing two oscillators while stimulating and observing only one +Erik T.K. Maua) and Michael Rosenblumb) +Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, +Germany +(Dated: January 13, 2023) +Synchronization of two or more self-sustained oscillators is a well-known and studied phenomenon, appearing +both in natural and designed systems. In some cases, the synchronized state is undesired, and the aim is +to destroy synchrony by external intervention. In this paper, we focus on desynchronizing two self-sustained +oscillators by short pulses delivered to the system in a phase-specific manner. We analyze a non-trivial case +when we cannot access both oscillators but stimulate only one. +The following restriction is that we can +monitor only one unit, be it a stimulated or non-stimulated one. First, we use a system of two coupled +Rayleigh oscillators to demonstrate how a loss of synchrony can be induced by stimulating a unit once per +period at a specific phase and detected by observing consecutive inter-pulse durations. Next, we exploit the +phase approximation to develop a rigorous theory formulating the problem in terms of a map. We derive +exact expressions for the phase – isostable coordinates of this coupled system and show a relation between +the phase and isostable response curves to the phase response curve of the uncoupled oscillator. Finally, we +demonstrate how to obtain phase response information from the system using time series and discuss the +differences between observing the stimulated and unstimulated oscillator. +Keywords: control of synchrony, phase response, phase reduction +Synchronization is a natural phenomenon ob- +served when oscillators interact. +In some cir- +cumstances, a synchronized state is undesired or +even harmful. In recent decades, much research +has been conducted to develop open and closed- +loop control techniques to control synchrony in +a system by external intervention. +This paper +focuses on a special example motivated by a neu- +roscience application. We treat two coupled os- +cillators with a restriction that stimulation does +only influence one of them directly. +Another +constraint is that we have observational access +to only one unit, the stimulated or the other. +Our objective is to destroy synchrony by pulsatile +stimulation, and we achieve this goal by deliver- +ing pulses each time the observed oscillator at- +tains a pre-selected trigger phase. +We demon- +strate how to recognize a desired desynchronized +state in practice by observing the elapsed time be- +tween consecutive phase-triggered pulses. Based +on the assumptions of weakly coupled phase oscil- +lators and short pulses, we develop a theoretical +framework to describe the system’s dynamics in +response to this stimulation protocol in terms of +a dynamical map. +This formulation utilizes the +phase-isostable description of oscillatory dynam- +ics. +We use that to derive a relation between +the response curves of the individual oscillator +and the coupled system. Our theoretical results +are supported by direct numerical simulations of +an example system with coupling functions con- +taining higher harmonic terms. +We discuss the +a)erikmau@uni-potsdam.de +b)mros@uni-potsdam.de +approach’s optimization for monitoring the stim- +ulated and the unstimulated oscillator. +Subject +to optimization is the choice of a proper trigger +phase and the strength and polarity of the pulses. +We demonstrate how to extract the required in- +formation from observations of the system and +highlight the approach’s limitations. +I. +INTRODUCTION +Synchronization of oscillatory sources can be benefi- +cial or harmful. Examples of the desired synchrony are +power grids’ functioning1–4 and atrial pacemaker cells’ +coordinated activity5,6. On the contrary, Parkinson’s dis- +ease and epilepsy are often related to an adverse effect of +synchrony in large neuronal populations7–11. Numerous +model studies suggested various techniques for the con- +trol of synchrony to cope with this adverse effect 12–24. +These studies exploited models of (infinitely) many or +several25 mean-field coupled limit-cycle oscillators and +assumed that the control input affects the whole pop- +ulation or at least its significant part(s). The feedback +techniques relied on observing the collective dynamics. A +general approach called synchronization engineering26,27 +also implies access to all network units. +Here, we consider a particular control problem and pro- +pose a method to desynchronize two limit-cycle oscilla- +tors. Our study is motivated by a neuroscience problem +formulated by Azodi-Avval and Gharabaghi28, who mod- +eled the effect of phase-specific neuromodulation by deep +brain stimulation on the synchronized activity of two +brain areas. Treating these areas as macroscopical os- +cillators, they assumed that measurements from both os- +cillators were available and exploited the technique from +arXiv:2301.04973v1 [nlin.CD] 12 Jan 2023 + +2 +Ref.29 to determine the phase response curve (PRC) for +one of the units. +Knowledge of the PRC allows stim- +ulation at the most sensitive phase and thus provides +a way to efficient desynchronization; however, the PRC +obtained from observation of two interacting units gener- +ally differs from the phase response to external stimula- +tion. As another relevant and motivating application, we +mention studies of circadian rhythms using the so-called +forced desynchrony protocol30. For example, de la Igle- +sia et al.31 exposed rats to an artificial light-dark rhythm +with a period of 22 hours and found that the rats’ activ- +ity pattern split into the entrained rhythm and another +one with a period significantly larger than 24 hours. This +splitting may indicate an enforced desynchronization of +individual circadian oscillators. +We +elaborate +on +the +idea +by +Azodi-Avval +and +Gharabaghi28 and suggest a minimal setup where we +achieve desynchronization by observing and perturbing +only one unit. We consider two versions of the approach, +where we monitor either the stimulated oscillator or the +other. Having in mind a possible neuroscience applica- +tion, we exploit a pulsatile perturbation delivered ap- +proximately once per oscillatory cycle. We remark that +models of two coupled phase oscillators with open-loop +pulsatile stimulation have been studied in Refs.32–34. We +also mention that Montaseri et al.35,36 used a feedback +controller design inspired by the role of astrocytes in +neural information processing to desynchronize two os- +cillators. However, Refs.35,36 assumed that both systems +could be observed and stimulated. Finally, we recall that +Pyragas et al.37 and Tukhlina et al.38 considered syn- +chrony suppression in a model of two interacting oscil- +lator populations, one used for sensing and another for +stimulation. This model can be treated as two coupled +macroscopic oscillators. However, desynchronization on +the level of subpopulations means quenching of macro- +scopic oscillators, while our study aims to keep systems +oscillating but destroy their synchrony. +This article is structured as follows: +First, we il- +lustrate the problem formulation and the detection of +stimulation-induced desynchronization using two coupled +Rayleigh oscillators in Section II. In Section III, we de- +velop a theoretical framework for two weakly coupled os- +cillators, describing phase-specific stimulation of the sys- +tem in terms of a dynamical map. Our theoretical analy- +sis exploits the phase – isostable representation of oscilla- +tory dynamics39. Section IV shows a relation between the +phase and isostable response curves of the synchronized +oscillatory dynamic and the phase response curve of an +uncoupled oscillator and thus complements the theoreti- +cal analysis. Here we also discuss possible approaches to +obtain the phase response curve from time series data of +only one oscillator. Finally, Section V discusses a strat- +egy to optimize the simulation by minimizing the total +intervention in the system, as well as open problems and +limitations of our approach. +II. +ILLUSTRATION OF THE APPROACH +The general theory says that phase dynamics of two +weakly coupled limit-cycle oscillators can be illustrated +by the motion of an overdamped particle in an inclined +potential, see, e.g.,40 and Fig. 1. +The particle at rest +in a potential well corresponds to the synchronous state +with the phase difference ϕ1 − ϕ2 = const. Thus, the +desynchronization problem reduces to kicking the particle +down the potential, inducing phase slips, i.e., relatively +rapid jumps where the phase difference changes by ±2π. +(Certainly, one can kick the particle to move it up, but +this action requires stronger stimulation and, therefore, +is less efficient.) For that purpose, we consider relatively +rare pulses applied approximately once per oscillation pe- +riod. Suppose each pulse shifts the particle toward the +local maximum. +Between two consecutive stimuli, the +particle tends to return to equilibrium. This consider- +ation shows that there shall be a critical value of the +pulse strength such that the phase shifts accumulate and +the particle eventually moves from its stable equilibrium +position over the maximum to the following equilibrium +position. This way, the phase difference changes by 2π +(phase slip). The continuing stimulation evokes the next +phase slip, and so on. +0 +1 +2 +3 +4 +Figure 1. The dynamics of the phase difference between two +weakly coupled oscillators can be illustrated by the motion +of an overdamped particle in an inclined potential, plotted +here for the case ω1 < ω2. A synchronous state corresponds +to a particle trapped in a minimum of the potential. Stim- +uli applied at a proper phase can shift the particle from the +equilibrium position and eventually move it to the next po- +tential well, decreasing the phase difference ϕ1−ϕ2 by 2π, i.e., +inducing a phase slip. We aim to design a stimulation that +permanently causes phase slips, thus destroying synchrony. +We demonstrate the approach exploiting the system +of two coupled Rayleigh oscillators perturbed by a pulse +stimulation: +¨x1 − µ(1 − ˙x2 +1) ˙x1 + ω2 +1x1 = ε(x2 − x1) + p(t) , +(1) +¨x2 − µ(1 − ˙x2 +2) ˙x2 + ω2 +2x2 = ε(x1 − x2) . +(2) + +3 +Parameters are µ = 2, ω1 = 0.98, ω2 = 1.02, ε = 0.2. +The perturbation p(t) is a pulse train, p(t) = � +k P(tn), +where P(tn) is a finite-length pulse applied at the instant +tn. +We note that we label the stimulated unit as the +first for definiteness. +Next, without loss of generality, +we choose ω1 < ω2; to treat the opposite choice ω1 > ω2, +one has to choose another stimulation phase, as discussed +below. +We now discuss the determination of the stimulation +times tn. Suppose we observe x1(t). We define threshold- +crossing events tn as the instants when x1(tn) = x0 and +˙x1(tn) is either always positive or always negative; here, +x0 is the threshold value. (The proper choice of x0 and +condition for ˙x1 is discussed below in Section IV.) We +apply pulses at tn with the following additional restric- +tion. Suppose for definiteness that we choose the con- +dition ˙x1 > 0. If the pulse applied at tn reduces x1(t), +then after a very short time interval ∆t ≪ T0, where +T0 is the period of synchronous oscillation, x1(t) again +achieves the threshold value x0. We neglect this thresh- +old crossing and wait till the next one so that the inter- +vals τn = tn+1 − tn are of the order of T0. We denote +the return times τn as partial periods of the first oscil- +lator. The formulated condition can be easily explained +in terms of the oscillator’s phase. Indeed, the threshold +condition x1(tn) = x0 corresponds to achieving a certain +phase ϕ0. Stimulation can decrease ϕ0; thus, for the sub- +sequent stimulation, we wait until the oscillator’s phase +becomes ϕ0 + 2π. A similar consideration applies when +we monitor x2. +6.3 +6.32 +6.34 +6.5 +7 +7.5 +(a) +0 +200 +400 +600 +800 +1000 +time +-4 +-3 +-2 +-1 +(b) +Figure 2. (a) Partial periods of the stimulated Rayleigh oscil- +lator vs. stimulation times, for weak, I = 2, and strong, I = 4 +stimulation (blue diamonds, left vertical axis, and red circles, +right vertical axis, respectively). +In both cases, the stimu- +lation changes the period. +However, when the stimulation +amplitude is below a certain threshold, the two coupled os- +cillators remain synchronized, as seen from the (proto)phase +difference depicted in (b). +If the stimulation is sufficiently +strong, it induces phase slips; the occurrence of phase slips +can be traced from the oscillations of τn. +We illustrate the effect of stimulation by plotting the +partial periods τn vs. tn in Fig. 2a, for x0 = 1, ˙x1 > +0. Panel (b) shows the protophase difference41. We use +rectangular pulses of duration ∆ = 0.01 and amplitude +I. Inspecting the plot, we conclude that oscillation of +τn indicates phase slips and, hence, a desynchronizing +action.42 +C + +7.5 +(a) +6.5 +6 +-3 +20 x10 +15 +(b) +10 +5 +0 +0 +1 +2 +I +. , . +.. . +. .. +. . ' +... +� : • t +0 +=;: ; ;' 1;: i':• +! +l•I +I +I +I +: +:, +I I +I +' +I +i ! I +3 +,, +4 +Figure 3. Illustration of the case when the first Rayleigh os- +cillator is stimulated by a pulse whenever the phase of the +second one attains a specific fixed value. In (a), we plot val- +ues of the second unit periods τn vs. +the stimulation am- +plitude I (red dots); for better visibility, we also show the +minimal and maximal values of τn for each I (blue circles). +For I ≲ 2.55 we have τmin = τmax, what means that the +system remains synchronized. For I ≳ 2.55, the partial peri- +ods τn oscillate, indicating phase slips and loss of synchrony. +The loss of synchrony is confirmed in panel (b), where we +demonstrate the difference Ω of oscillator frequencies that is +non-zero for I ≳ 2.55. +Figure 3 depicts the case when we observe the second +oscillator. Thus, we define the partial periods, now for +the second oscillator, as τn = tn+1 − tn via the events +tk when x2(tn) crosses a certain threshold, e.g., in the +positive direction. +Omitting the first 50 intervals, we +plot τn, τmin = min(τn), and τmax = max(τn), n > 50, +for different values of the pulse amplitude I. We used +x0 = −1; other parameters are the same as in Fig. 2. +We see that sufficiently strong stimulation results in os- +cillatory behavior of τn, which means the appearance of +phase slips, and, hence, desynchronization. Thus, we can +desynchronize the system by stimulating only one of two +synchronous oscillators while observing any of these two. +We support this conclusion with theoretical analysis in +the next Section. + +4 +III. +DESYNCHRONIZING BY PULSE STIMULATION: +THEORY +It is well-known that, +for sufficiently weak cou- +pling, phase dynamics of two interacting units obey the +Kuramoto-Daido equations: +˙ϕ1 = ω1 + C1(ϕ1 − ϕ2) + Z(ϕ1)p(t) , +(3) +˙ϕ2 = ω2 + C2(ϕ2 − ϕ1) , +(4) +where C1,2 are coupling functions. Here, we assume for +definiteness that ω1 < ω2 and that stimulation p(t) af- +fects the first oscillator. The last term in Eq. (3) describes +the stimulation, where p(t) is the external force, and the +phase response curve (PRC) of the uncoupled oscillator +Z(ϕ1) quantifies the sensitivity of the unit to perturba- +tion. We will consider separately two cases where we ob- +serve either the first or the second oscillator. Therefore, +introducing the phase difference η = ϕ1 − ϕ2 we re-write +Eqs. (3,4) as equations for η, ϕi, where either i = 1 or +i = 2: +˙η = f(η) + Z(ϕi + δi2η)p(t) , +(5) +˙ϕi = gi(η) + δi1Z(ϕi + δi2η)p(t) . +(6) +Here, δi2 is the Kronecker symbol, g1(η) = ω1 + C1(η), +g2(η) = ω2 + C2(−η), and f(η) = g1(η) − g2(η). Note +that PRC Z remains the function of ϕ1 = ϕi + δi2η. +Suppose there are no perturbations, p(t) = 0. Then, +Eq. (5) reduces to ˙η = f(η). The dynamics of this equa- +tion are well-studied. Depending on the parameters, it +has either asynchronous solution ˙η < 0 or synchronous, +phase-locked solution ˙η = 0. In the latter case, one or +several pairs of stable and unstable fixed points exist. We +present the theory for the case when there exists only one +stable fixed point η∗ = const, f ′(η∗) < 0, and discuss a +possible extension to the general case in Section V. Asyn- +chronous solutions correspond to quasiperiodic trajecto- +ries on the two-torus spanned by ϕ1, ϕ2. In contrast, the +existence of stable and unstable fixed points in Eq. (5) +means the appearance of stable and unstable limit cycles +on the torus. +Consider the stable limit cycle on the two-torus. The +frequency of this synchronous solution is ω = gi(η∗). +Next, we define the phase on the limit cycle and in its +vicinity. +We emphasize that the phase of the uncou- +pled oscillator ϕi is not the true asymptotic phase of the +synchronous solution of the coupled system because its +time derivative is not a constant but depends on η, see +Eq. (6). Thus, in the context of the coupled system, we +treat ϕi as the protophase (angle variable). Using the +ansatz Φ(ϕi, η) = ϕi + δi2η∗ + Fi(η) with an additional +condition Fi(η∗) = 0, we require ˙Φ = ω and obtain +˙Φ(ϕi, η) = ˙ϕi + F ′ +i(η) ˙η = gi(η) + F ′ +i(η)f(η) = ω . +Solving this equation for F ′ +i and integrating, we obtain43: +Φ(ϕi, η) = ϕi + δi2η∗ + +� η +η∗ +ω − gi(s) +f(s) +ds . +(7) +Using f(η) = g1(η) − g2(η), it is easy to check that +Φ(ϕ1, η) = Φ(ϕ2, η), i.e., the definition of phase does +not depend on the chosen protophase. On the limit cycle +(η = η∗), we have Φ = ϕ1 = ϕ2 + η∗, meaning phase and +protophase coincide up to a constant shift. We remind +that by construction, Φ(ϕi, η∗) = ϕ1, i.e. the protophase +ϕ1 coincides with Φ on the limit cycle. +Before proceeding with a separate analysis of the cases +i = 1 (the first oscillator is observed) and i = 2 (the sec- +ond unit is observed), we conclude the theoretical con- +sideration by the following remark. +For the attractive +cycle on the torus, η − η∗ = ψ describes the deviation +from the stable solution; hence, the variable η plays the +role of the amplitude. +In a small vicinity of the limit +cycle44, we then write ˙ψ = f ′(η∗)ψ = κψ and interpret +ψ as the isostable variable39. We return to the phase – +isostable representation of the synchronized dynamics in +Section IV. +A. +Stimulating and observing the same oscillator +Here, we assume we observe the first unit and compute +the intervals between the stimuli. We recall that we stim- +ulate each time the phase of the first oscillator attains +some fixed value ϕ0. Let the variable η immediately be- +fore the n-th stimulus is ηn. We assume instantaneous +phase shift due to the δ-kick, i.e., P(tn) = qδ(t − tn), so +that ϕ1 = ϕ0 → ϕ0 +A and η → η+A, where the instan- +taneous phase shift A = qZ(ϕ0) and q is the amplitude of +the δ-pulse. As before, we denote the time between the +n-th and n + 1-th kick by τn. Between the stimuli, we +deal with autonomous dynamics. Hence, τn is obtained +by +τn = +� ηn+1 +ηn+A +ds +f(s) , +(8) +and the phase Φ within this time interval grows by ωτn. +We thus write +Φ(ϕ0 + A, ηn + A) + ωτn = Φ(ϕ0 + 2π, ηn+1) . +(9) +Exploiting the definition of phase from Eq. (7), we obtain +the equation +A + +� ηn+A +η∗ +ω − g1(s) +f(s) +ds + ωτn = 2π + +� ηn+1 +η∗ +ω − g1(s) +f(s) +ds . +(10) +By inserting the expression of τn from Eq. (8) into this +formula, we finally obtain +2π − A − +� ηn+1 +ηn+A +g1(s) +f(s) ds = 0 . +(11) +This equation defines a one-dimensional map ηn+1 = +F(ηn) with the parameter A. We iterate this map, start- +ing from η0 = η∗ and solving Eq. (11) numerically45, for + +5 +a fixed kick strength A. +Using the obtained values of +ηn, we integrate numerically Eq. (8) and obtain τn. We +remind that the sequence of intervals τn can easily be +measured in an experiment. +In Section II, we have demonstrated that depending on +the stimulation strength, the sequence τn either saturates +or oscillates, see Fig. 2. The former case means that the +map ηn+1 = F(ηn) has a fixed point ˆη(A) with an obvi- +ous condition ˆη(0) = η∗. We denote the corresponding +interval ˆτ(A), where ˆτ(0) = 2π/ω. +For small A both ηn + A and ηn+1 are close to η∗ and +we can write the first-order approximation of Eq. (10). +For this purpose, we use ω = g1(η∗) and compute +limη→η∗(ω − g1(s))/f(s) using the L’Hospital’s rule. We +obtain: +A − g′ +1(η∗) +f ′(η∗)(ηn + A − η∗) + ωτn = 2π − g′ +1(η∗) +f ′(η∗)(ηn+1 − η∗) . +(12) +In the following, we define γ = 1 − g′ +1(η∗)/f ′(η∗). The +approximation (12) yields the intervals τn in the vicinity +of η∗, i.e., for small kick strength A as +τn = 1 +ω [2π − γA + (γ − 1)(ηn+1 − ηn)] . +(13) +Imposing the fixed point condition ηn = ηn+1 and insert- +ing A = qZ(ϕ0) we obtain an expression for ˆτ as +ˆτ = 1 +ω (2π − γqZ(ϕ0)) . +(14) +We remark that the direction of convergence to that fixed +point depends on the sign of γ − 1. There may be a τn +in the transient that is larger or smaller than both ˆτ and +2π/ω. +B. +Stimulating the first oscillator while observing the second +one +Now, we use the events ϕ2 = ϕ0 as a trigger for stimu- +lation. Again, we aim to describe the dynamic via a one- +dimensional map ηn+1 = F(ηn). The effect of the kick is +now ϕ1 → ϕ1 + qZ(ϕ1) = ϕ1 + qZ(ϕ0 + η) and, hence, +η → η + qZ(ϕ0 + η). The evoked shift of η depends on η +itself and is not constant as in the previous case. Thus, +we cannot combine the kick action q, the trigger phase +ϕ0, and the response Z(ϕ0) into a constant phase shift, +but have to treat it as a function qZ(ϕ0 + η) evaluated +at ηn. For convenience, we denote ˜Z(η) := qZ(ϕ0 + η). +Accordingly, the interval τn between two kicks is +τn = +� ηn+1 +ηn+ ˜ +Z(ηn) +ds +f(s) . +(15) +Proceeding as in the previous case, we write, similarly +to Eq. (9): +Φ(ϕ0, ηn + ˜Z(ηn)) + ωτn = Φ(ϕ0 + 2π, ηn+1) . +(16) +Finally, we obtain the equation +2π − +� ηn+1 +ηn+ ˜ +Z(ηn) +g2(s) +f(s) ds = 0 +(17) +that defines the map ηn+1 = F(ηn) depending on func- +tion ˜Z. +Similarly to the previous case, we find an approximate +expression for τn in the limit of weak kicks leaving the +phase difference close to η∗. We approximate Eq. (16) by +−g′ +2(η∗) +f ′(η∗)(ηn + ˜Z(ηn) − η∗) + ωτn = 2π − g′ +2(η∗) +f ′(η∗)(ηn+1 − η∗) . +(18) +Note that −g′ +2(η∗)/f ′(η∗) equals the above defined con- +stant γ used in the previous case of monitoring the first +oscillator. This can be checked by inserting the original +coupling functions C1 and C2 into f, g1, g2. For τn we +obtain +τn = 1 +ω (2π − γ ˜Z(ηn) + γ(ηn+1 − ηn)) . +(19) +In the limit of small q we conclude ˜Z(ηn) = qZ(φ0+ηn) ≈ +qZ(φ0+η∗). Thus for the fixed point ˆτ, we obtain a result +similar to that of the first case: +ˆτ = 1 +ω (2π − γZ(φ0 + η∗)q) . +(20) +Compared to Eq. (14), the only difference is the argument +of Z. In the case of the first oscillator being monitored, +it is ϕ0, and in the current case, it is ϕ0 + η∗. We re- +mark that by the definition of phase via Eq. (7), in both +cases we have Φ0 = ϕ0 + δi2η∗. Thus, in both cases the +expression for ˆτ in the limit of small q reads +ˆτ = 1 +ω (2π − γqZ(Φ0)) , +(21) +In the following, we will test the derived dynamical +map F for a model of coupled phase oscillators and com- +pare it to a direct simulation with both finite-size and +Dirac kicks. +C. +An example: coupled phase oscillators +We consider two phase oscillators with coupling func- +tions containing higher harmonics terms +˙ϕ1 = ω1 + ε sin(ϕ2 − ϕ1) + σ sin(2(ϕ2 − ϕ1)) + Z(ϕ1)p(t) , +˙ϕ2 = ω2 + ε sin(ϕ1 − ϕ2) + β sin(3(ϕ1 − ϕ2)) , +(22) +with the parameters ω1 = 0.98, ω2 = 1.02, ε = 0.05, +σ = 0.02 and β = −0.01. For the response curve Z, we +choose a simple sine function Z(ϕ1) = sin(ϕ1). +Thus, the relevant functions for the map F read +g1(η) = ω1 − ε sin(η) − σ sin(2η), g2(η) = ω2 + ε sin(η) + + +6 +β cos(3η) and f(η) = g1(η) − g2(η). For the chosen pa- +rameters, the system attains a stable phase difference +η∗ ≈ −0.12π. Thus frequency, Floquet exponent, and +PRC prefactor follow as ω ≈ 1.01, κ ≈ −0.11, and +γ ≈ 0.30. +We perform the stimulation experiment by monitor- +ing either the first or the second oscillator. The results +are depicted in Fig. 4 and Fig. 5, respectively. In both +cases, the proposed theory for the iterated mapping F +corresponds to the direct simulation with Dirac kicks to +a large extent. Both agree with the direct simulation by +kicks of finite duration ∆ for small q; however, the results +differ for large |q|. This discrepancy is due to the differ- +ence in the effect of stimulating with the amplitude I +for time ∆ starting at ϕ0, compared to an instantaneous +shift of I∆Z(ϕ0). +We remark, that since we define the phase Φ on the +limit cycle as Φ = ϕ1, we have Φ0 = ϕ0 if the first +oscillator triggers the stimulation at ϕ1 = ϕ0 and Φ0 = +ϕ0 − η∗ if the second oscillator triggers it at ϕ2 = ϕ0. +In the first numerical experiment, we monitor the first +oscillator. +We choose Φ0 = ϕ0 = 3π/2 as the trigger +phase since it corresponds to an extremum of Z. +We +observe the appearance of phase slips for q ≳ 0.5. Since +we do not observe phase slips for equally strong negative +pulses, we conclude the favorable polarity of the phase +shift to be negative (positive kicks at negative PRC value +Z(ϕ0) < 0). This conclusion corresponds to our choice +ω1 < ω2. +Monitoring the second oscillator, we experiment with +two different trigger phases Φ0 = 3π/2 (ϕ0 = Φ0 + η∗ ≈ +1.62π) and Φ0 = 1.88π (ϕ0 ≈ 2π). +Even though +Φ0 = 3π/2 yields an extremum of Z, we do not observe +phase slips in the shown range of kick actions q, neither +for positive nor for negative kicks, see Fig. 5(a). However, +for the value Φ0 = 1.88π, we observe the appearance of +phase slips in an interval of q. For finite-sized kicks of +∆ = 10−5, phase slips occur for 0.49 ≲ q ≲ 0.71. For the +Dirac kicks, both for the mapping and the direct simu- +lation, the interval of phase slips is narrower: it starts +at q ≳ 0.53 and ends at q ≲ 0.56. For sufficiently large +q, a new fixed point is formed. This happens due to the +dependence of the kick-induced phase shift on the phase +difference η. In contrast to the case of monitoring the first +oscillator, here, the kick-induced phase shift can change +its sign depending on the phase difference ηn. The kick +is strong enough for the first few iterations to bring the +system out of its potential well. As the system then tends +to relax to the next equilibrium value η∗−2π and reaches +the next trigger point ϕ2 = ϕ0 + 2π, the kick acts in the +opposite direction and brings the system up the potential +wall again. In this way, the system gets trapped, and a +fixed point establishes. For practical purposes of avoid- +ing that scenario, we mention the possibility of pausing +the stimulation after one phase slip or varying the kick +strength randomly. +Such behavior is not possible if we monitor the first os- +cillator, at least if there exists only one stable phase dif- +ference η∗ of the unperturbed coupled system: Since the +kick-induced phase shift does not depend on the phase +difference η (at least for Dirac kicks), and thus is con- +stant for a given trigger phase ϕ0, it will constantly shift +the phase difference in the same direction (the evoked +phase shift A = const). +Thus, if the kicks are strong +enough to induce a phase slip once, they will continue +causing them. +6.0 +6.5 +7.0 +7.5 +τ +(a) +0 +159 +n +6.5 +7.0 +τn +(b) +−0.6 +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +q = I∆ +0.00 +0.05 +0.10 +|Ω| +(c) +τn: Dirac kicks +τn: map dynamics +ˆτ = 2π +ω − γ +ωZ(Φ0)q +finite size kicks (∆ =1e-05) +finite size kicks (∆ =0.001) +finite size kicks (∆ =0.01) +Figure 4. Dynamics of the kicked phase oscillator system (22). +Here, we monitor the first oscillator and deliver kicks at +Φ0 = ϕ0 = 3π/2. Panel (a) depicts the bifurcation diagram +for the asymptotic behavior of inter-kick intervals τn. The val- +ues of τn for n ≥ 50 are shown as a function of the kick action +q for a direct simulation of Dirac kicks (orange crosses), and +the iteration of the map F (purple circles). The approximate +expression (21) for the fixed point ˆτ is drawn as a black dashed +line. For small kick actions, the τn converge to a fixed point +in first-order approximation given by ˆτ. +Phase slips occur +for sufficiently large values of q. An example in (b), depicts +the inter-kick durations τn for q ≈ 0.51 (this value is marked +with a dotted line in (a) and (c)). Panel (c) shows the bi- +furcation diagram for the time-averaged frequency difference +|Ω| of both oscillators (time averaging over 80 kicks). Data +points correspond to direct simulations of Dirac kicks (orange +crosses) and finite-sized kicks with pulse widths ∆ = 10−5 +(olive lower triangles), ∆ = 10−3 (green right triangles), and +∆ = 10−2 (dark blue upper triangles). The emergence of a +non-zero value of |Ω| coincides with the disappearance of the +stable fixed point and onset of oscillatory dynamics for τn in +(a). +IV. +FINDING THE PROPER PHASE FOR STIMULATION +In the previous section, we have shown that sufficiently +intense pulses can induce phase slips if delivered con- +secutively each time the monitored oscillator attains a +pre-selected target phase ϕ0. However, the critical kick +strength of these pulses to achieve phase slips depends +on ϕ0, and for some disadvantageous ϕ0, it might not +work at all. +This section illustrates the determination +of a proper target phase for that stimulation protocol, + +7 +6.00 +6.25 +6.50 +τ +(a) +−0.5 +0.0 +0.5 +q = I∆ +6.0 +6.5 +τ +(b) +τn: finite size kicks +τn: Dirac kicks +τn: map dynamics +ˆτ = 2π +ω − γ +ωZ(Φ0)q +Figure 5. Bifurcation diagrams of the kicked phase oscillator +system (22) when monitoring the second oscillator, for the +cases of Φ0 = 3π/2 (ϕ0 ≈ 1.62π, panel (a)) and Φ0 = 1.88π +(ϕ0 ≈ 0, panel (b)). The values of τn for n ≥ 50 are depicted +as a function of the kick action q for direct simulation of +finite-size kicks (green triangles; pulse duration ∆ = 10−5 and +amplitude I = q/∆), Dirac kicks (orange crosses), and the +iteration of the map F (purple circles). The expression (21) +for the fixed point ˆτ is drawn as a black dashed line. Phase +slips do not occur in (a). +In (b) phase slips occur only in +an interval of kick actions q which is different for Dirac and +finite-sized kicks. +which leads to phase slips for as weak pulses as possible. +A. +Phase and isostable response curves +Following Section III, we consider the dynamics of the +synchronized system as a limit-cycle oscillation. Corre- +spondingly, this oscillation can be characterized by the +phase response curve (PRC) Z. In general, this curve +differs from the PRC of the uncoupled oscillator, i.e., +Z ̸= Z. Z contains information on how external stimu- +lation shifts the phase of the synchronous oscillation Φ. +Next, the deviation from the limit cycle of the synchro- +nized system is quantified by the isostable response curve +(IRC) I46. As discussed in Section III, the deviation is +η − η∗, i.e., it corresponds to the deviation of the phase +difference η = ϕ1 −ϕ2 from its stable value. The descrip- +tion in terms of PRC and IRC is valid if the system is on +or very close to the limit cycle when stimulated. For a +detailed explanation, see39,47. +We derive the PRC Z from the gradient of Φ and the +PRC of the uncoupled oscillators, both evaluated at the +limit cycle: +Z(Φ) = (∂ϕiΦ · δi1Z(ϕ1) + ∂ηΦ · Z(ϕ1)) |η=η∗ . +(23) +With the partial derivatives ∂ϕiΦ|η=η∗ += +δi1 and +∂ηΦ|η=η∗ = −g′ +i(η∗)/f ′(η∗), see Eq. (7), we conclude +Z(Φ) = γZ(Φ) . +(24) +Thus, the PRC Z generally differs from the response +curve of the first oscillator Z by a factor of γ. +This +factor γ is characteristic of the coupled system and can +potentially take any real value, including 0. Similarly, we +derive the IRC by +I(Φ) = (∂ϕiψ · δi1Z(ϕi) + ∂ηψ · Z(ϕ1)) |η=η∗ . +(25) +Here, the partial derivative with respect to ϕi vanishes +(∂ϕiψ = 0) since the isostable variable ψ depends on the +phase difference η only. The partial derivative with re- +spect to the phase difference ∂ηψ|η=η∗ is some constant +that depends on the chosen scaling of the isostable vari- +able. Thus, the IRC is proportional to the response curve +Z and thus also to Z: +I(Φ) ∝ Z(Φ) ∝ Z(Φ) . +(26) +To desynchronize the two oscillators, we want to push +their phase difference η as far away from its value η∗ in +the locked state as possible. Hence, we want to maximize +the response in the isostable variable ψ, which is achieved +by stimulating the system at a phase that maximizes the +IRC I. By relation (26), we have to look for the extrema +of Z or Z to obtain the extrema of the IRC. In the fol- +lowing part of this Section, we will discuss the practical +aspects of PRC inference. +B. +PRC inference for coupled Rayleigh oscillators +To demonstrate the PRC inference for the system of +two coupled Rayleigh oscillators (2), examined in Sec- +tion II, we choose the observable x1 and assign phase +values from 0 to 2π to one period of the unperturbed os- +cillation, mapping threshold values of x1 to phases. Thus, +instead of operating with phases, we can use the signal +values; see the solid gray line in Fig. 6. +As a benchmark, we exploit the standard approach and +apply consecutively single pulses at different phases ϕ +(i.e., at different signal thresholds) and wait until the +system returns to the same state for the k-th time; we +denote this time interval as Tk. +Since we are dealing +with a weakly stable system48, it may be necessary to +wait several periods to ensure that the system has relaxed +back to the limit cycle sufficiently close. The PRC then +computes as +Z(ϕ) = 2π +q +kT0 − Tk +T0 +, +(27) +where T0 = 2π/ω is the natural period of the system, and +q is the action of the pulse. +A more practical way to infer the PRC is to exploit the +newly developed IPID-1 technique49,50. This technique +uses the observed scalar time series and known pulsatile + +8 +external stimulation to infer PRC via a direct fit of the +Winfree equation. See51 for the code of implementation. +The standard technique requires at least k · m periods +of the oscillation to obtain m data points of the PRC. For +example, to compute the PRC via the standard technique +in Fig. 6, we used k = 20. IPID-1 needs a substantially +shorter observation time to conveniently depict the entire +PRC due to the least squares fit. In addition, IPID-1 +does not rely on a specially designed stimulation protocol +that hits a certain target phase. For example, adding a +Poissonian process to the stimulation period suffices. The +requirement for IPID-1 is that the time series of both an +observable of the system and the external stimulation +are known. The results of the inferred PRC using the +standard and IPID-1 methods are depicted and compared +in Fig. 6. +0 +π +2π +ϕ +−2 +0 +2 +Z, Z, x1 +Z +Z +Z (IPID-1) +observable x1 +Figure 6. Comparison of PRC inference techniques for the +system of coupled Rayleigh oscillators (2), the parameters re- +main as specified in Sec. II. The solid orange curve with upper +triangles and the dashed purple curve, respectively, depict the +resulting PRCs Z from the standard technique and the IPID- +1 method. For comparison, the teal curve with lower triangles +illustrates the PRC of the first Rayleigh oscillator Z for the +uncoupled case ε = 0, obtained by the standard method. In +contrast to the relation (24) between the PRCs of the cou- +pled and uncoupled system in the phase oscillator model, the +curves differ not only in scale but also are slightly shifted. +The IPID-1-inferred PRC for the coupled system correctly +reproduces the PRC’s shape but not the scaling. However, +the latter is not important for our approach. The solid gray +curve is the observable x1 of the coupled system; it provides +a map to translate threshold crossings into phases. +In the more difficult case of observing the second os- +cillator (which is not directly stimulated), the IPID- +1 method failed, for our example, yielding a vanishing +PRC. However, the standard method is still applicable +in that case. +C. +Are the PRCs extrema optimal targets for +phase-triggered stimulation? +Let us assume that we obtained the exact PRC Z and +thus have perfect knowledge about phases (i.e., thresh- +olds) at which the system is displaced most efficiently +from the limit cycle. Does that mean we have found the +best phase to trigger external pulses? +In the case of monitoring the first oscillator, it indeed +does. +As we have seen in Sec. III, the evoked phase +shift is constant if the stimuli are applied at the same +ϕ1 every time. +Moreover, selecting the extrema of Z +ensures the maximal phase shift. What remains to be +determined is whether the kicks shall be positive or neg- +ative, i.e., whether advancing or delaying the system is +more efficient in causing phase slips. +For our choice, +ω1 < ω2, slowing the first oscillator by negative phase +shifts was the favorable choice. For the opposite case, it +would be vice versa. We remark that Fig. 2 shows the +coupled Rayleigh system for a phase-specific stimulation +each time x1 crosses the threshold x0 = 1 from below. +This threshold corresponds to a phase of ϕ1 ≈ 0.18π, see +Fig. 6, and is close to the minimum of Z. Thus, it is +an excellent choice to induce phase slips for comparably +small positive kick actions q. +The opposite case of monitoring the second oscillator +is more involved. The reason is that the induced phase +shifts following a pulse are not constant as in the previous +case. +By selecting a trigger phase ϕ2 = ϕ0, the kick- +induced phase shift also depends on the phase difference +η, see Sec. III B. Thus, even when ϕ0 is most effective on +the limit cycle at η∗, it might lose this efficiency for the +new phase difference ˆη that establishes as a result of the +consecutive kicks. We do not yet see a practical way to +overcome this issue just by knowing the PRC. It might +still be a good idea to start exploring efficient phases +close to the extrema of Z since these at least guarantee +the most significant possible displacement from the limit +cycle for the first few kicks. To avoid a trapping scenario +as described in Sec. III C and shown in Fig. 5 we mention +the possibility to add a stochastic process to the pulse +action q. +V. +DISCUSSION +In this article, we have demonstrated how a system +of two synchronized oscillators can be desynchronized by +short pulses applied to only one of both in a phase-specific +manner. We focused on the restriction of having access +to the observation of only one of the two units. For both +cases of observing the stimulated and the unstimulated +oscillator, we showed the efficiency of this approach for +a well-chosen trigger phase. +We developed a theoreti- +cal framework for the approximation of weakly coupled +phase oscillators. This framework allowed us to derive an +exact expression for the phase of the coupled system. We +used it to establish a relation between the coupled sys- + +9 +tem’s phase response and the individual oscillator’s phase +response curve. This relation can be used to find efficient +trigger phases for a phase-specific stimulation protocol. +In particular, we discuss the optimization of the stim- +ulation. The first issue is the polarity of the pulse’s ac- +tion, which determines whether an induced phase shift +advances or delays the phase of the stimulated oscilla- +tor. We know that phase delays are favorable if the stim- +ulated oscillator is slower than the unstimulated one in +the absence of coupling (which is the case in the examples +shown in this article). Vice versa, if it were faster, phase +advances are favorable. However, the induced phase shift +is the product of both action and phase response at the +trigger phase. Thus, the same pulse can induce advancing +and delaying shifts if delivered at different phases. Hence, +to account for that consideration, knowledge about the +PRC up to a positive factor is also required. +We re- +mark that to determine which direction is favorable for +the induced phase shift, it must be known whether the +stimulated oscillator is faster or slower than the other. +Since that is unknown a priori, we suggest testing both +polarities for a given phase with a high phase response +in absolute value. +Another issue is how to minimize the number of pulses +required to induce a phase slip. We remind that evoked +phase slip means that the system escapes the basin of +the locally stable phase difference and then evolves to- +ward the next potential minimum, see Fig. 1. Obviously, +having reached the local maximum of the potential, the +system tends to the next equilibrium state by itself. It +does not need additional pulses driving it in that direc- +tion. Thus, pausing stimulation after passing the maxi- +mum excludes unnecessary intervention and also avoids a +trapping scenario described in Section IV for the case of +monitoring the unstimulated oscillator. The underlying +problem is to detect the instant of passing over the bar- +rier. While the emergence of an oscillating pattern for +τn unambiguously reveals phase slips, see, e.g., Fig. 2a +and Fig. 4b, we do not know how to detect the barrier +crossing from this pattern. This task remains an open +problem for future research. +In the following, we comment on the limitations of the +theoretical description of our approach. +Our consider- +ations rely on weak-coupling approximation with Dirac +pulse stimulation. Thus, strongly coupled systems can +differ from the phase description used here. +Also, the +effects of very strong or long stimuli might not be accu- +rately described by the derived dynamical map. +Within our theoretical framework, several questions re- +main unanswered. +First, we do not see a straightfor- +ward data-driven way to predict the critical action to in- +duce phase slips. If the dynamical equations are known, +the critical action can be found by numerically solving +Eqs. (11), (17) for a fixpoint as a function of action q. +The boundaries of existence then mark the critical ac- +tions. We rely on continuously increasing the action for +unknown dynamical equations until phase slips appear. +Another issue is the optimal trigger phase for the case +of monitoring the unstimulated oscillator. As outlined +in Section IV, it is not necessarily an extremum of the +phase response curve that leads to phase slips at all, let +alone in the most efficient way. +We do not yet see a +practical solution apart from trying out different phases +in the vicinity of an extremum of the phase response. +Another assumption we made throughout this article +was the uniqueness of the system’s stable phase differ- +ence equilibria. In principle, multiple stable equilibrium +states are possible, corresponding to multiple local min- +ima of the potential in Fig. 1. We will discuss such a +case now. Unlike the case of a unique stable state, where +the system reenters the basin of attraction if the unstable +equilibrium is crossed, the system finds itself in the basin +of attraction of another stable phase difference. Thus, +system quantities like the frequency, the Floquet expo- +nent, and the PRC scaling factor γ can change. +The +individual phase response Z remains constant, though. +This new basin might be impossible to leave with the +same kicks that kicked it there in the first place. If we +monitor the stimulated oscillator, increasing the kick ac- +tion will eventually suffice to leave the basin. Repeating +this procedure for potentially more stable states will re- +sult in a kick action large enough to leave all basins and +thus induce phase slips: a repeating visit of all basins. +There is no such guarantee for monitoring the unstimu- +lated oscillator, and we cannot exclude that it might be +necessary to change the trigger phase depending on the +current basin. +ACKNOWLEDGMENTS +E.T.K.M. +acknowledges +financial +support +from +Deutsche Forschungsgemeinschaft (DFG, German Re- +search Foundation), Project-ID 424778381 – TRR 295. +We thank Prof. A. 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However, achieving the required spectral resolution requires +a relatively long time series. +43For the actual computation of Φ it is important to avoid inte- +grating over a singularity f(s) = 0, e.g., at the unstable phase +difference. +44We note that one can adequately introduce the isostable variable +as ψ := +f(η) +κ +exp +�� η +η∗ +κ−f′(s) +f(s) +ds +� +so that equation ˙ψ = κψ is +valid in the whole basin of attraction of the limit cycle. Intro- +duced in this way, ψ generally differs from η − η∗ if the quantity +is not small. +45From +SciPy52, +we +exploit +the +integration +algo- +rithm +scipy.integrate.quad +and +root-finding +algorithm +scipy.optimize.root with solver method hybr (modified Powell +hybrid method). We define the r.h.s. of Eq. (11) as a function +of ηn+1 with parameters A and ηn. We obtain ηn+1 = FA(ηn) +by calling the root-finding on this function with initial guess +ηn + A. For faster computation, we circumvent to execute the +root-finding algorithm every time we call FA by computing +FA(η) for a sufficiently large set of η-values once and fitting this +to a finite Fourier series. +46Similarly to PRC, the IRC I is defined as the infinitesimal re- +sponse in the isostable variable ψ on the limit cycle. It is a func- +tion of phase and enters the equation for the isostable variable +as ˙ψ = κψ + I(Φ)p(t). +47D. Wilson and B. Ermentrout, SIAM Journal on Applied Dy- +namical Systems 17, 2516 (2018). +48For the chosen parameter value µ = 2, individual oscillators +are strongly stable, but the limit cycle of the coupled system +is weakly stable. +49R. Cestnik and M. Rosenblum, Scientific Reports 8, 13606 (2018). +50R. Cestnik, E. T. K. Mau, and M. Rosenblum, New Journal of +Physics 24, 123012 (2022). +51R. Cestnik and E. T. K. Mau, “IPID-1,” (2022). +52P. +Virtanen, +R. +Gommers, +T. +E. +Oliphant, +M. +Haber- +land, T. Reddy, D. Cournapeau, E. Burovski, P. 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Vázquez-Baeza, Nature +Methods 17, 261 (2020). + diff --git a/btE4T4oBgHgl3EQfPgwg/content/tmp_files/load_file.txt b/btE4T4oBgHgl3EQfPgwg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b7109e46f425acffdca26e8cbaea7ee73420d54 --- /dev/null +++ b/btE4T4oBgHgl3EQfPgwg/content/tmp_files/load_file.txt @@ -0,0 +1,967 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf,len=966 +page_content='Desynchronizing two oscillators while stimulating and observing only one Erik T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Maua) and Michael Rosenblumb) Department of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 24/25, D-14476 Potsdam-Golm, Germany (Dated: January 13, 2023) Synchronization of two or more self-sustained oscillators is a well-known and studied phenomenon, appearing both in natural and designed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In some cases, the synchronized state is undesired, and the aim is to destroy synchrony by external intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In this paper, we focus on desynchronizing two self-sustained oscillators by short pulses delivered to the system in a phase-specific manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We analyze a non-trivial case when we cannot access both oscillators but stimulate only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The following restriction is that we can monitor only one unit, be it a stimulated or non-stimulated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' First, we use a system of two coupled Rayleigh oscillators to demonstrate how a loss of synchrony can be induced by stimulating a unit once per period at a specific phase and detected by observing consecutive inter-pulse durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Next, we exploit the phase approximation to develop a rigorous theory formulating the problem in terms of a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We derive exact expressions for the phase – isostable coordinates of this coupled system and show a relation between the phase and isostable response curves to the phase response curve of the uncoupled oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Finally, we demonstrate how to obtain phase response information from the system using time series and discuss the differences between observing the stimulated and unstimulated oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Keywords: control of synchrony, phase response, phase reduction Synchronization is a natural phenomenon ob- served when oscillators interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In some cir- cumstances, a synchronized state is undesired or even harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In recent decades, much research has been conducted to develop open and closed- loop control techniques to control synchrony in a system by external intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This paper focuses on a special example motivated by a neu- roscience application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We treat two coupled os- cillators with a restriction that stimulation does only influence one of them directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Another constraint is that we have observational access to only one unit, the stimulated or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Our objective is to destroy synchrony by pulsatile stimulation, and we achieve this goal by deliver- ing pulses each time the observed oscillator at- tains a pre-selected trigger phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We demon- strate how to recognize a desired desynchronized state in practice by observing the elapsed time be- tween consecutive phase-triggered pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Based on the assumptions of weakly coupled phase oscil- lators and short pulses, we develop a theoretical framework to describe the system’s dynamics in response to this stimulation protocol in terms of a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This formulation utilizes the phase-isostable description of oscillatory dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We use that to derive a relation between the response curves of the individual oscillator and the coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Our theoretical results are supported by direct numerical simulations of an example system with coupling functions con- taining higher harmonic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We discuss the a)erikmau@uni-potsdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='de b)mros@uni-potsdam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='de approach’s optimization for monitoring the stim- ulated and the unstimulated oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Subject to optimization is the choice of a proper trigger phase and the strength and polarity of the pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We demonstrate how to extract the required in- formation from observations of the system and highlight the approach’s limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' INTRODUCTION Synchronization of oscillatory sources can be benefi- cial or harmful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Examples of the desired synchrony are power grids’ functioning1–4 and atrial pacemaker cells’ coordinated activity5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' On the contrary, Parkinson’s dis- ease and epilepsy are often related to an adverse effect of synchrony in large neuronal populations7–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Numerous model studies suggested various techniques for the con- trol of synchrony to cope with this adverse effect 12–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' These studies exploited models of (infinitely) many or several25 mean-field coupled limit-cycle oscillators and assumed that the control input affects the whole pop- ulation or at least its significant part(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The feedback techniques relied on observing the collective dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A general approach called synchronization engineering26,27 also implies access to all network units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Here, we consider a particular control problem and pro- pose a method to desynchronize two limit-cycle oscilla- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Our study is motivated by a neuroscience problem formulated by Azodi-Avval and Gharabaghi28, who mod- eled the effect of phase-specific neuromodulation by deep brain stimulation on the synchronized activity of two brain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Treating these areas as macroscopical os- cillators, they assumed that measurements from both os- cillators were available and exploited the technique from arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='04973v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='CD] 12 Jan 2023 2 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='29 to determine the phase response curve (PRC) for one of the units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Knowledge of the PRC allows stim- ulation at the most sensitive phase and thus provides a way to efficient desynchronization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' however, the PRC obtained from observation of two interacting units gener- ally differs from the phase response to external stimula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As another relevant and motivating application, we mention studies of circadian rhythms using the so-called forced desynchrony protocol30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For example, de la Igle- sia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='31 exposed rats to an artificial light-dark rhythm with a period of 22 hours and found that the rats’ activ- ity pattern split into the entrained rhythm and another one with a period significantly larger than 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This splitting may indicate an enforced desynchronization of individual circadian oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We elaborate on the idea by Azodi-Avval and Gharabaghi28 and suggest a minimal setup where we achieve desynchronization by observing and perturbing only one unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We consider two versions of the approach, where we monitor either the stimulated oscillator or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Having in mind a possible neuroscience applica- tion, we exploit a pulsatile perturbation delivered ap- proximately once per oscillatory cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We remark that models of two coupled phase oscillators with open-loop pulsatile stimulation have been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='32–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We also mention that Montaseri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='35,36 used a feedback controller design inspired by the role of astrocytes in neural information processing to desynchronize two os- cillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='35,36 assumed that both systems could be observed and stimulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Finally, we recall that Pyragas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='37 and Tukhlina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='38 considered syn- chrony suppression in a model of two interacting oscil- lator populations, one used for sensing and another for stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This model can be treated as two coupled macroscopic oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, desynchronization on the level of subpopulations means quenching of macro- scopic oscillators, while our study aims to keep systems oscillating but destroy their synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This article is structured as follows: First, we il- lustrate the problem formulation and the detection of stimulation-induced desynchronization using two coupled Rayleigh oscillators in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In Section III, we de- velop a theoretical framework for two weakly coupled os- cillators, describing phase-specific stimulation of the sys- tem in terms of a dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Our theoretical analy- sis exploits the phase – isostable representation of oscilla- tory dynamics39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Section IV shows a relation between the phase and isostable response curves of the synchronized oscillatory dynamic and the phase response curve of an uncoupled oscillator and thus complements the theoreti- cal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Here we also discuss possible approaches to obtain the phase response curve from time series data of only one oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Finally, Section V discusses a strat- egy to optimize the simulation by minimizing the total intervention in the system, as well as open problems and limitations of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' ILLUSTRATION OF THE APPROACH The general theory says that phase dynamics of two weakly coupled limit-cycle oscillators can be illustrated by the motion of an overdamped particle in an inclined potential, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=',40 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The particle at rest in a potential well corresponds to the synchronous state with the phase difference ϕ1 − ϕ2 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, the desynchronization problem reduces to kicking the particle down the potential, inducing phase slips, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', relatively rapid jumps where the phase difference changes by ±2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (Certainly, one can kick the particle to move it up, but this action requires stronger stimulation and, therefore, is less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=') For that purpose, we consider relatively rare pulses applied approximately once per oscillation pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Suppose each pulse shifts the particle toward the local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Between two consecutive stimuli, the particle tends to return to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This consider- ation shows that there shall be a critical value of the pulse strength such that the phase shifts accumulate and the particle eventually moves from its stable equilibrium position over the maximum to the following equilibrium position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This way, the phase difference changes by 2π (phase slip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The continuing stimulation evokes the next phase slip, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 0 1 2 3 4 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The dynamics of the phase difference between two weakly coupled oscillators can be illustrated by the motion of an overdamped particle in an inclined potential, plotted here for the case ω1 < ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A synchronous state corresponds to a particle trapped in a minimum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Stim- uli applied at a proper phase can shift the particle from the equilibrium position and eventually move it to the next po- tential well, decreasing the phase difference ϕ1−ϕ2 by 2π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', inducing a phase slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We aim to design a stimulation that permanently causes phase slips, thus destroying synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We demonstrate the approach exploiting the system of two coupled Rayleigh oscillators perturbed by a pulse stimulation: ¨x1 − µ(1 − ˙x2 1) ˙x1 + ω2 1x1 = ε(x2 − x1) + p(t) , (1) ¨x2 − µ(1 − ˙x2 2) ˙x2 + ω2 2x2 = ε(x1 − x2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (2) 3 Parameters are µ = 2, ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='98, ω2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='02, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The perturbation p(t) is a pulse train, p(t) = � k P(tn), where P(tn) is a finite-length pulse applied at the instant tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We note that we label the stimulated unit as the first for definiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Next, without loss of generality, we choose ω1 < ω2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' to treat the opposite choice ω1 > ω2, one has to choose another stimulation phase, as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We now discuss the determination of the stimulation times tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Suppose we observe x1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We define threshold- crossing events tn as the instants when x1(tn) = x0 and ˙x1(tn) is either always positive or always negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' here, x0 is the threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (The proper choice of x0 and condition for ˙x1 is discussed below in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=') We apply pulses at tn with the following additional restric- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Suppose for definiteness that we choose the con- dition ˙x1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' If the pulse applied at tn reduces x1(t), then after a very short time interval ∆t ≪ T0, where T0 is the period of synchronous oscillation, x1(t) again achieves the threshold value x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We neglect this thresh- old crossing and wait till the next one so that the inter- vals τn = tn+1 − tn are of the order of T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We denote the return times τn as partial periods of the first oscil- lator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The formulated condition can be easily explained in terms of the oscillator’s phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Indeed, the threshold condition x1(tn) = x0 corresponds to achieving a certain phase ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Stimulation can decrease ϕ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' thus, for the sub- sequent stimulation, we wait until the oscillator’s phase becomes ϕ0 + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A similar consideration applies when we monitor x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='32 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 (a) 0 200 400 600 800 1000 time 4 3 2 1 (b) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (a) Partial periods of the stimulated Rayleigh oscil- lator vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' stimulation times, for weak, I = 2, and strong, I = 4 stimulation (blue diamonds, left vertical axis, and red circles, right vertical axis, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In both cases, the stimu- lation changes the period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, when the stimulation amplitude is below a certain threshold, the two coupled os- cillators remain synchronized, as seen from the (proto)phase difference depicted in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' If the stimulation is sufficiently strong, it induces phase slips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' the occurrence of phase slips can be traced from the oscillations of τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We illustrate the effect of stimulation by plotting the partial periods τn vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' tn in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 2a, for x0 = 1, ˙x1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Panel (b) shows the protophase difference41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We use rectangular pulses of duration ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='01 and amplitude I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Inspecting the plot, we conclude that oscillation of τn indicates phase slips and, hence, a desynchronizing action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='42 C 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 (a) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 6 3 20 x10 15 (b) 10 5 0 0 1 2 I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='. .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=" ' ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' � : • t 0 =;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=': ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content="' 1;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=": i':• !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=" l•I I I I : :, I I I ' I i !" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' I 3 ,, 4 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Illustration of the case when the first Rayleigh os- cillator is stimulated by a pulse whenever the phase of the second one attains a specific fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In (a), we plot val- ues of the second unit periods τn vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' the stimulation am- plitude I (red dots);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' for better visibility, we also show the minimal and maximal values of τn for each I (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For I ≲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='55 we have τmin = τmax, what means that the system remains synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For I ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='55, the partial peri- ods τn oscillate, indicating phase slips and loss of synchrony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The loss of synchrony is confirmed in panel (b), where we demonstrate the difference Ω of oscillator frequencies that is non-zero for I ≳ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Figure 3 depicts the case when we observe the second oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, we define the partial periods, now for the second oscillator, as τn = tn+1 − tn via the events tk when x2(tn) crosses a certain threshold, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', in the positive direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Omitting the first 50 intervals, we plot τn, τmin = min(τn), and τmax = max(τn), n > 50, for different values of the pulse amplitude I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We used x0 = −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' other parameters are the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We see that sufficiently strong stimulation results in os- cillatory behavior of τn, which means the appearance of phase slips, and, hence, desynchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, we can desynchronize the system by stimulating only one of two synchronous oscillators while observing any of these two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We support this conclusion with theoretical analysis in the next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' DESYNCHRONIZING BY PULSE STIMULATION: THEORY It is well-known that, for sufficiently weak cou- pling, phase dynamics of two interacting units obey the Kuramoto-Daido equations: ˙ϕ1 = ω1 + C1(ϕ1 − ϕ2) + Z(ϕ1)p(t) , (3) ˙ϕ2 = ω2 + C2(ϕ2 − ϕ1) , (4) where C1,2 are coupling functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Here, we assume for definiteness that ω1 < ω2 and that stimulation p(t) af- fects the first oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (3) describes the stimulation, where p(t) is the external force, and the phase response curve (PRC) of the uncoupled oscillator Z(ϕ1) quantifies the sensitivity of the unit to perturba- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We will consider separately two cases where we ob- serve either the first or the second oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Therefore, introducing the phase difference η = ϕ1 − ϕ2 we re-write Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (3,4) as equations for η, ϕi, where either i = 1 or i = 2: ˙η = f(η) + Z(ϕi + δi2η)p(t) , (5) ˙ϕi = gi(η) + δi1Z(ϕi + δi2η)p(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (6) Here, δi2 is the Kronecker symbol, g1(η) = ω1 + C1(η), g2(η) = ω2 + C2(−η), and f(η) = g1(η) − g2(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Note that PRC Z remains the function of ϕ1 = ϕi + δi2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Suppose there are no perturbations, p(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (5) reduces to ˙η = f(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The dynamics of this equa- tion are well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Depending on the parameters, it has either asynchronous solution ˙η < 0 or synchronous, phase-locked solution ˙η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the latter case, one or several pairs of stable and unstable fixed points exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We present the theory for the case when there exists only one stable fixed point η∗ = const, f ′(η∗) < 0, and discuss a possible extension to the general case in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Asyn- chronous solutions correspond to quasiperiodic trajecto- ries on the two-torus spanned by ϕ1, ϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In contrast, the existence of stable and unstable fixed points in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (5) means the appearance of stable and unstable limit cycles on the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Consider the stable limit cycle on the two-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The frequency of this synchronous solution is ω = gi(η∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Next, we define the phase on the limit cycle and in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We emphasize that the phase of the uncou- pled oscillator ϕi is not the true asymptotic phase of the synchronous solution of the coupled system because its time derivative is not a constant but depends on η, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, in the context of the coupled system, we treat ϕi as the protophase (angle variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Using the ansatz Φ(ϕi, η) = ϕi + δi2η∗ + Fi(η) with an additional condition Fi(η∗) = 0, we require ˙Φ = ω and obtain ˙Φ(ϕi, η) = ˙ϕi + F ′ i(η) ˙η = gi(η) + F ′ i(η)f(η) = ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Solving this equation for F ′ i and integrating, we obtain43: Φ(ϕi, η) = ϕi + δi2η∗ + � η η∗ ω − gi(s) f(s) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (7) Using f(η) = g1(η) − g2(η), it is easy to check that Φ(ϕ1, η) = Φ(ϕ2, η), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', the definition of phase does not depend on the chosen protophase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' On the limit cycle (η = η∗), we have Φ = ϕ1 = ϕ2 + η∗, meaning phase and protophase coincide up to a constant shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We remind that by construction, Φ(ϕi, η∗) = ϕ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' the protophase ϕ1 coincides with Φ on the limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Before proceeding with a separate analysis of the cases i = 1 (the first oscillator is observed) and i = 2 (the sec- ond unit is observed), we conclude the theoretical con- sideration by the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For the attractive cycle on the torus, η − η∗ = ψ describes the deviation from the stable solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' hence, the variable η plays the role of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In a small vicinity of the limit cycle44, we then write ˙ψ = f ′(η∗)ψ = κψ and interpret ψ as the isostable variable39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We return to the phase – isostable representation of the synchronized dynamics in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Stimulating and observing the same oscillator Here, we assume we observe the first unit and compute the intervals between the stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We recall that we stim- ulate each time the phase of the first oscillator attains some fixed value ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Let the variable η immediately be- fore the n-th stimulus is ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We assume instantaneous phase shift due to the δ-kick, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', P(tn) = qδ(t − tn), so that ϕ1 = ϕ0 → ϕ0 +A and η → η+A, where the instan- taneous phase shift A = qZ(ϕ0) and q is the amplitude of the δ-pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As before, we denote the time between the n-th and n + 1-th kick by τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Between the stimuli, we deal with autonomous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Hence, τn is obtained by τn = � ηn+1 ηn+A ds f(s) , (8) and the phase Φ within this time interval grows by ωτn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We thus write Φ(ϕ0 + A, ηn + A) + ωτn = Φ(ϕ0 + 2π, ηn+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (9) Exploiting the definition of phase from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (7), we obtain the equation A + � ηn+A η∗ ω − g1(s) f(s) ds + ωτn = 2π + � ηn+1 η∗ ω − g1(s) f(s) ds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (10) By inserting the expression of τn from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (8) into this formula, we finally obtain 2π − A − � ηn+1 ηn+A g1(s) f(s) ds = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (11) This equation defines a one-dimensional map ηn+1 = F(ηn) with the parameter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We iterate this map, start- ing from η0 = η∗ and solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (11) numerically45, for 5 a fixed kick strength A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Using the obtained values of ηn, we integrate numerically Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (8) and obtain τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We remind that the sequence of intervals τn can easily be measured in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In Section II, we have demonstrated that depending on the stimulation strength, the sequence τn either saturates or oscillates, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The former case means that the map ηn+1 = F(ηn) has a fixed point ˆη(A) with an obvi- ous condition ˆη(0) = η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We denote the corresponding interval ˆτ(A), where ˆτ(0) = 2π/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For small A both ηn + A and ηn+1 are close to η∗ and we can write the first-order approximation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For this purpose, we use ω = g1(η∗) and compute limη→η∗(ω − g1(s))/f(s) using the L’Hospital’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We obtain: A − g′ 1(η∗) f ′(η∗)(ηn + A − η∗) + ωτn = 2π − g′ 1(η∗) f ′(η∗)(ηn+1 − η∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (12) In the following, we define γ = 1 − g′ 1(η∗)/f ′(η∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The approximation (12) yields the intervals τn in the vicinity of η∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', for small kick strength A as τn = 1 ω [2π − γA + (γ − 1)(ηn+1 − ηn)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (13) Imposing the fixed point condition ηn = ηn+1 and insert- ing A = qZ(ϕ0) we obtain an expression for ˆτ as ˆτ = 1 ω (2π − γqZ(ϕ0)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (14) We remark that the direction of convergence to that fixed point depends on the sign of γ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' There may be a τn in the transient that is larger or smaller than both ˆτ and 2π/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Stimulating the first oscillator while observing the second one Now, we use the events ϕ2 = ϕ0 as a trigger for stimu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Again, we aim to describe the dynamic via a one- dimensional map ηn+1 = F(ηn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The effect of the kick is now ϕ1 → ϕ1 + qZ(ϕ1) = ϕ1 + qZ(ϕ0 + η) and, hence, η → η + qZ(ϕ0 + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The evoked shift of η depends on η itself and is not constant as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, we cannot combine the kick action q, the trigger phase ϕ0, and the response Z(ϕ0) into a constant phase shift, but have to treat it as a function qZ(ϕ0 + η) evaluated at ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For convenience, we denote ˜Z(η) := qZ(ϕ0 + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Accordingly, the interval τn between two kicks is τn = � ηn+1 ηn+ ˜ Z(ηn) ds f(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (15) Proceeding as in the previous case, we write, similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (9): Φ(ϕ0, ηn + ˜Z(ηn)) + ωτn = Φ(ϕ0 + 2π, ηn+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (16) Finally, we obtain the equation 2π − � ηn+1 ηn+ ˜ Z(ηn) g2(s) f(s) ds = 0 (17) that defines the map ηn+1 = F(ηn) depending on func- tion ˜Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Similarly to the previous case, we find an approximate expression for τn in the limit of weak kicks leaving the phase difference close to η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (16) by −g′ 2(η∗) f ′(η∗)(ηn + ˜Z(ηn) − η∗) + ωτn = 2π − g′ 2(η∗) f ′(η∗)(ηn+1 − η∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (18) Note that −g′ 2(η∗)/f ′(η∗) equals the above defined con- stant γ used in the previous case of monitoring the first oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This can be checked by inserting the original coupling functions C1 and C2 into f, g1, g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For τn we obtain τn = 1 ω (2π − γ ˜Z(ηn) + γ(ηn+1 − ηn)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (19) In the limit of small q we conclude ˜Z(ηn) = qZ(φ0+ηn) ≈ qZ(φ0+η∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus for the fixed point ˆτ, we obtain a result similar to that of the first case: ˆτ = 1 ω (2π − γZ(φ0 + η∗)q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (20) Compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (14), the only difference is the argument of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the case of the first oscillator being monitored, it is ϕ0, and in the current case, it is ϕ0 + η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We re- mark that by the definition of phase via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (7), in both cases we have Φ0 = ϕ0 + δi2η∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, in both cases the expression for ˆτ in the limit of small q reads ˆτ = 1 ω (2π − γqZ(Φ0)) , (21) In the following, we will test the derived dynamical map F for a model of coupled phase oscillators and com- pare it to a direct simulation with both finite-size and Dirac kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' An example: coupled phase oscillators We consider two phase oscillators with coupling func- tions containing higher harmonics terms ˙ϕ1 = ω1 + ε sin(ϕ2 − ϕ1) + σ sin(2(ϕ2 − ϕ1)) + Z(ϕ1)p(t) , ˙ϕ2 = ω2 + ε sin(ϕ1 − ϕ2) + β sin(3(ϕ1 − ϕ2)) , (22) with the parameters ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='98, ω2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='02, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='05, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='02 and β = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For the response curve Z, we choose a simple sine function Z(ϕ1) = sin(ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, the relevant functions for the map F read g1(η) = ω1 − ε sin(η) − σ sin(2η), g2(η) = ω2 + ε sin(η) + 6 β cos(3η) and f(η) = g1(η) − g2(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For the chosen pa- rameters, the system attains a stable phase difference η∗ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='12π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus frequency, Floquet exponent, and PRC prefactor follow as ω ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='01, κ ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='11, and γ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We perform the stimulation experiment by monitor- ing either the first or the second oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The results are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In both cases, the proposed theory for the iterated mapping F corresponds to the direct simulation with Dirac kicks to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Both agree with the direct simulation by kicks of finite duration ∆ for small q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' however, the results differ for large |q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This discrepancy is due to the differ- ence in the effect of stimulating with the amplitude I for time ∆ starting at ϕ0, compared to an instantaneous shift of I∆Z(ϕ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We remark, that since we define the phase Φ on the limit cycle as Φ = ϕ1, we have Φ0 = ϕ0 if the first oscillator triggers the stimulation at ϕ1 = ϕ0 and Φ0 = ϕ0 − η∗ if the second oscillator triggers it at ϕ2 = ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the first numerical experiment, we monitor the first oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We choose Φ0 = ϕ0 = 3π/2 as the trigger phase since it corresponds to an extremum of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We observe the appearance of phase slips for q ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Since we do not observe phase slips for equally strong negative pulses, we conclude the favorable polarity of the phase shift to be negative (positive kicks at negative PRC value Z(ϕ0) < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This conclusion corresponds to our choice ω1 < ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Monitoring the second oscillator, we experiment with two different trigger phases Φ0 = 3π/2 (ϕ0 = Φ0 + η∗ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='62π) and Φ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='88π (ϕ0 ≈ 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Even though Φ0 = 3π/2 yields an extremum of Z, we do not observe phase slips in the shown range of kick actions q, neither for positive nor for negative kicks, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, for the value Φ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='88π, we observe the appearance of phase slips in an interval of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For finite-sized kicks of ∆ = 10−5, phase slips occur for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='49 ≲ q ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For the Dirac kicks, both for the mapping and the direct simu- lation, the interval of phase slips is narrower: it starts at q ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='53 and ends at q ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For sufficiently large q, a new fixed point is formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This happens due to the dependence of the kick-induced phase shift on the phase difference η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In contrast to the case of monitoring the first oscillator, here, the kick-induced phase shift can change its sign depending on the phase difference ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The kick is strong enough for the first few iterations to bring the system out of its potential well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As the system then tends to relax to the next equilibrium value η∗−2π and reaches the next trigger point ϕ2 = ϕ0 + 2π, the kick acts in the opposite direction and brings the system up the potential wall again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In this way, the system gets trapped, and a fixed point establishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For practical purposes of avoid- ing that scenario, we mention the possibility of pausing the stimulation after one phase slip or varying the kick strength randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Such behavior is not possible if we monitor the first os- cillator, at least if there exists only one stable phase dif- ference η∗ of the unperturbed coupled system: Since the kick-induced phase shift does not depend on the phase difference η (at least for Dirac kicks), and thus is con- stant for a given trigger phase ϕ0, it will constantly shift the phase difference in the same direction (the evoked phase shift A = const).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, if the kicks are strong enough to induce a phase slip once, they will continue causing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 τ (a) 0 159 n 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 τn (b) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='8 q = I∆ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='10 |Ω| (c) τn: Dirac kicks τn: map dynamics ˆτ = 2π ω − γ ωZ(Φ0)q finite size kicks (∆ =1e-05) finite size kicks (∆ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='001) finite size kicks (∆ =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='01) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Dynamics of the kicked phase oscillator system (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Here, we monitor the first oscillator and deliver kicks at Φ0 = ϕ0 = 3π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Panel (a) depicts the bifurcation diagram for the asymptotic behavior of inter-kick intervals τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The val- ues of τn for n ≥ 50 are shown as a function of the kick action q for a direct simulation of Dirac kicks (orange crosses), and the iteration of the map F (purple circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The approximate expression (21) for the fixed point ˆτ is drawn as a black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For small kick actions, the τn converge to a fixed point in first-order approximation given by ˆτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Phase slips occur for sufficiently large values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' An example in (b), depicts the inter-kick durations τn for q ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='51 (this value is marked with a dotted line in (a) and (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Panel (c) shows the bi- furcation diagram for the time-averaged frequency difference |Ω| of both oscillators (time averaging over 80 kicks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Data points correspond to direct simulations of Dirac kicks (orange crosses) and finite-sized kicks with pulse widths ∆ = 10−5 (olive lower triangles), ∆ = 10−3 (green right triangles), and ∆ = 10−2 (dark blue upper triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The emergence of a non-zero value of |Ω| coincides with the disappearance of the stable fixed point and onset of oscillatory dynamics for τn in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' FINDING THE PROPER PHASE FOR STIMULATION In the previous section, we have shown that sufficiently intense pulses can induce phase slips if delivered con- secutively each time the monitored oscillator attains a pre-selected target phase ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, the critical kick strength of these pulses to achieve phase slips depends on ϕ0, and for some disadvantageous ϕ0, it might not work at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This section illustrates the determination of a proper target phase for that stimulation protocol, 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='50 τ (a) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 q = I∆ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='5 τ (b) τn: finite size kicks τn: Dirac kicks τn: map dynamics ˆτ = 2π ω − γ ωZ(Φ0)q Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Bifurcation diagrams of the kicked phase oscillator system (22) when monitoring the second oscillator, for the cases of Φ0 = 3π/2 (ϕ0 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='62π, panel (a)) and Φ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='88π (ϕ0 ≈ 0, panel (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The values of τn for n ≥ 50 are depicted as a function of the kick action q for direct simulation of finite-size kicks (green triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' pulse duration ∆ = 10−5 and amplitude I = q/∆), Dirac kicks (orange crosses), and the iteration of the map F (purple circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The expression (21) for the fixed point ˆτ is drawn as a black dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Phase slips do not occur in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In (b) phase slips occur only in an interval of kick actions q which is different for Dirac and finite-sized kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' which leads to phase slips for as weak pulses as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Phase and isostable response curves Following Section III, we consider the dynamics of the synchronized system as a limit-cycle oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Corre- spondingly, this oscillation can be characterized by the phase response curve (PRC) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In general, this curve differs from the PRC of the uncoupled oscillator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', Z ̸= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Z contains information on how external stimu- lation shifts the phase of the synchronous oscillation Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Next, the deviation from the limit cycle of the synchro- nized system is quantified by the isostable response curve (IRC) I46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As discussed in Section III, the deviation is η − η∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', it corresponds to the deviation of the phase difference η = ϕ1 −ϕ2 from its stable value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The descrip- tion in terms of PRC and IRC is valid if the system is on or very close to the limit cycle when stimulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For a detailed explanation, see39,47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We derive the PRC Z from the gradient of Φ and the PRC of the uncoupled oscillators, both evaluated at the limit cycle: Z(Φ) = (∂ϕiΦ · δi1Z(ϕ1) + ∂ηΦ · Z(ϕ1)) |η=η∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (23) With the partial derivatives ∂ϕiΦ|η=η∗ = δi1 and ∂ηΦ|η=η∗ = −g′ i(η∗)/f ′(η∗), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (7), we conclude Z(Φ) = γZ(Φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (24) Thus, the PRC Z generally differs from the response curve of the first oscillator Z by a factor of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This factor γ is characteristic of the coupled system and can potentially take any real value, including 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Similarly, we derive the IRC by I(Φ) = (∂ϕiψ · δi1Z(ϕi) + ∂ηψ · Z(ϕ1)) |η=η∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (25) Here, the partial derivative with respect to ϕi vanishes (∂ϕiψ = 0) since the isostable variable ψ depends on the phase difference η only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The partial derivative with re- spect to the phase difference ∂ηψ|η=η∗ is some constant that depends on the chosen scaling of the isostable vari- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, the IRC is proportional to the response curve Z and thus also to Z: I(Φ) ∝ Z(Φ) ∝ Z(Φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (26) To desynchronize the two oscillators, we want to push their phase difference η as far away from its value η∗ in the locked state as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Hence, we want to maximize the response in the isostable variable ψ, which is achieved by stimulating the system at a phase that maximizes the IRC I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' By relation (26), we have to look for the extrema of Z or Z to obtain the extrema of the IRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the fol- lowing part of this Section, we will discuss the practical aspects of PRC inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' PRC inference for coupled Rayleigh oscillators To demonstrate the PRC inference for the system of two coupled Rayleigh oscillators (2), examined in Sec- tion II, we choose the observable x1 and assign phase values from 0 to 2π to one period of the unperturbed os- cillation, mapping threshold values of x1 to phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, instead of operating with phases, we can use the signal values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' see the solid gray line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As a benchmark, we exploit the standard approach and apply consecutively single pulses at different phases ϕ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', at different signal thresholds) and wait until the system returns to the same state for the k-th time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' we denote this time interval as Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Since we are dealing with a weakly stable system48, it may be necessary to wait several periods to ensure that the system has relaxed back to the limit cycle sufficiently close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The PRC then computes as Z(ϕ) = 2π q kT0 − Tk T0 , (27) where T0 = 2π/ω is the natural period of the system, and q is the action of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A more practical way to infer the PRC is to exploit the newly developed IPID-1 technique49,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This technique uses the observed scalar time series and known pulsatile 8 external stimulation to infer PRC via a direct fit of the Winfree equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' See51 for the code of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The standard technique requires at least k · m periods of the oscillation to obtain m data points of the PRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For example, to compute the PRC via the standard technique in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 6, we used k = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' IPID-1 needs a substantially shorter observation time to conveniently depict the entire PRC due to the least squares fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In addition, IPID-1 does not rely on a specially designed stimulation protocol that hits a certain target phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For example, adding a Poissonian process to the stimulation period suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The requirement for IPID-1 is that the time series of both an observable of the system and the external stimulation are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The results of the inferred PRC using the standard and IPID-1 methods are depicted and compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 0 π 2π ϕ −2 0 2 Z, Z, x1 Z Z Z (IPID-1) observable x1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Comparison of PRC inference techniques for the system of coupled Rayleigh oscillators (2), the parameters re- main as specified in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The solid orange curve with upper triangles and the dashed purple curve, respectively, depict the resulting PRCs Z from the standard technique and the IPID- 1 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For comparison, the teal curve with lower triangles illustrates the PRC of the first Rayleigh oscillator Z for the uncoupled case ε = 0, obtained by the standard method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In contrast to the relation (24) between the PRCs of the cou- pled and uncoupled system in the phase oscillator model, the curves differ not only in scale but also are slightly shifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The IPID-1-inferred PRC for the coupled system correctly reproduces the PRC’s shape but not the scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, the latter is not important for our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The solid gray curve is the observable x1 of the coupled system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' it provides a map to translate threshold crossings into phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the more difficult case of observing the second os- cillator (which is not directly stimulated), the IPID- 1 method failed, for our example, yielding a vanishing PRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, the standard method is still applicable in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Are the PRCs extrema optimal targets for phase-triggered stimulation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Let us assume that we obtained the exact PRC Z and thus have perfect knowledge about phases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', thresh- olds) at which the system is displaced most efficiently from the limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Does that mean we have found the best phase to trigger external pulses?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the case of monitoring the first oscillator, it indeed does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As we have seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' III, the evoked phase shift is constant if the stimuli are applied at the same ϕ1 every time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Moreover, selecting the extrema of Z ensures the maximal phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' What remains to be determined is whether the kicks shall be positive or neg- ative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', whether advancing or delaying the system is more efficient in causing phase slips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For our choice, ω1 < ω2, slowing the first oscillator by negative phase shifts was the favorable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For the opposite case, it would be vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We remark that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 2 shows the coupled Rayleigh system for a phase-specific stimulation each time x1 crosses the threshold x0 = 1 from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This threshold corresponds to a phase of ϕ1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='18π, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 6, and is close to the minimum of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, it is an excellent choice to induce phase slips for comparably small positive kick actions q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The opposite case of monitoring the second oscillator is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The reason is that the induced phase shifts following a pulse are not constant as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' By selecting a trigger phase ϕ2 = ϕ0, the kick- induced phase shift also depends on the phase difference η, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, even when ϕ0 is most effective on the limit cycle at η∗, it might lose this efficiency for the new phase difference ˆη that establishes as a result of the consecutive kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We do not yet see a practical way to overcome this issue just by knowing the PRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' It might still be a good idea to start exploring efficient phases close to the extrema of Z since these at least guarantee the most significant possible displacement from the limit cycle for the first few kicks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' To avoid a trapping scenario as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' III C and shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 5 we mention the possibility to add a stochastic process to the pulse action q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' DISCUSSION In this article, we have demonstrated how a system of two synchronized oscillators can be desynchronized by short pulses applied to only one of both in a phase-specific manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We focused on the restriction of having access to the observation of only one of the two units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For both cases of observing the stimulated and the unstimulated oscillator, we showed the efficiency of this approach for a well-chosen trigger phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We developed a theoreti- cal framework for the approximation of weakly coupled phase oscillators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This framework allowed us to derive an exact expression for the phase of the coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We used it to establish a relation between the coupled sys- 9 tem’s phase response and the individual oscillator’s phase response curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This relation can be used to find efficient trigger phases for a phase-specific stimulation protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In particular, we discuss the optimization of the stim- ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The first issue is the polarity of the pulse’s ac- tion, which determines whether an induced phase shift advances or delays the phase of the stimulated oscilla- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We know that phase delays are favorable if the stim- ulated oscillator is slower than the unstimulated one in the absence of coupling (which is the case in the examples shown in this article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Vice versa, if it were faster, phase advances are favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, the induced phase shift is the product of both action and phase response at the trigger phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, the same pulse can induce advancing and delaying shifts if delivered at different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Hence, to account for that consideration, knowledge about the PRC up to a positive factor is also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We re- mark that to determine which direction is favorable for the induced phase shift, it must be known whether the stimulated oscillator is faster or slower than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Since that is unknown a priori, we suggest testing both polarities for a given phase with a high phase response in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Another issue is how to minimize the number of pulses required to induce a phase slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We remind that evoked phase slip means that the system escapes the basin of the locally stable phase difference and then evolves to- ward the next potential minimum, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Obviously, having reached the local maximum of the potential, the system tends to the next equilibrium state by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' It does not need additional pulses driving it in that direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, pausing stimulation after passing the maxi- mum excludes unnecessary intervention and also avoids a trapping scenario described in Section IV for the case of monitoring the unstimulated oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The underlying problem is to detect the instant of passing over the bar- rier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' While the emergence of an oscillating pattern for τn unambiguously reveals phase slips, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 2a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 4b, we do not know how to detect the barrier crossing from this pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This task remains an open problem for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In the following, we comment on the limitations of the theoretical description of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Our consider- ations rely on weak-coupling approximation with Dirac pulse stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, strongly coupled systems can differ from the phase description used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Also, the effects of very strong or long stimuli might not be accu- rately described by the derived dynamical map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Within our theoretical framework, several questions re- main unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' First, we do not see a straightfor- ward data-driven way to predict the critical action to in- duce phase slips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' If the dynamical equations are known, the critical action can be found by numerically solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (11), (17) for a fixpoint as a function of action q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The boundaries of existence then mark the critical ac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We rely on continuously increasing the action for unknown dynamical equations until phase slips appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Another issue is the optimal trigger phase for the case of monitoring the unstimulated oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' As outlined in Section IV, it is not necessarily an extremum of the phase response curve that leads to phase slips at all, let alone in the most efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We do not yet see a practical solution apart from trying out different phases in the vicinity of an extremum of the phase response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Another assumption we made throughout this article was the uniqueness of the system’s stable phase differ- ence equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' In principle, multiple stable equilibrium states are possible, corresponding to multiple local min- ima of the potential in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We will discuss such a case now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Unlike the case of a unique stable state, where the system reenters the basin of attraction if the unstable equilibrium is crossed, the system finds itself in the basin of attraction of another stable phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Thus, system quantities like the frequency, the Floquet expo- nent, and the PRC scaling factor γ can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' The individual phase response Z remains constant, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This new basin might be impossible to leave with the same kicks that kicked it there in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' If we monitor the stimulated oscillator, increasing the kick ac- tion will eventually suffice to leave the basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Repeating this procedure for potentially more stable states will re- sult in a kick action large enough to leave all basins and thus induce phase slips: a repeating visit of all basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' There is no such guarantee for monitoring the unstimu- lated oscillator, and we cannot exclude that it might be necessary to change the trigger phase depending on the current basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' ACKNOWLEDGMENTS E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' acknowledges financial support from Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation), Project-ID 424778381 – TRR 295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Gharabaghi for inspiring discussions.' metadata={'source': 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+page_content=' Tukhlina and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Rosenblum, Journal of Biological Physics 34, 301 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 39D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Wilson and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Moehlis, Physical Review E 94, 052213 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 40A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Pikovsky, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Rosenblum, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (Cambridge University Press, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 41In this plot, we operate with θ, which is the polar angle in the x, ˙x plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' This variable (protophase) differs from the true phase on the time scale of a single period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' this difference is not essential here since we are interested in the presence or absence of phase slips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 42Destruction of synchrony, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', a transition from periodic to quasiperiodic motion, can be traced in the power spectrum of x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' However, achieving the required spectral resolution requires a relatively long time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 43For the actual computation of Φ it is important to avoid inte- grating over a singularity f(s) = 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=', at the unstable phase difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 44We note that one can adequately introduce the isostable variable as ψ := f(η) κ exp �� η η∗ κ−f′(s) f(s) ds � so that equation ˙ψ = κψ is valid in the whole basin of attraction of the limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Intro- duced in this way, ψ generally differs from η − η∗ if the quantity is not small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 45From SciPy52, we exploit the integration algo- rithm scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='integrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='quad and root-finding algorithm scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='root with solver method hybr (modified Powell hybrid method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We define the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' (11) as a function of ηn+1 with parameters A and ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' We obtain ηn+1 = FA(ηn) by calling the root-finding on this function with initial guess ηn + A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' For faster computation, we circumvent to execute the root-finding algorithm every time we call FA by computing FA(η) for a sufficiently large set of η-values once and fitting this to a finite Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 46Similarly to PRC, the IRC I is defined as the infinitesimal re- sponse in the isostable variable ψ on the limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' It is a func- tion of phase and enters the equation for the isostable variable as ˙ψ = κψ + I(Φ)p(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 47D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Wilson and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Ermentrout, SIAM Journal on Applied Dy- namical Systems 17, 2516 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 48For the chosen parameter value µ = 2, individual oscillators are strongly stable, but the limit cycle of the coupled system is weakly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 49R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Cestnik and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Rosenblum, Scientific Reports 8, 13606 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 50R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Cestnik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Mau, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Rosenblum, New Journal of Physics 24, 123012 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 51R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Cestnik and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Mau, “IPID-1,” (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' 52P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Virtanen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Gommers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Oliphant, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Haber- land, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Reddy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Cournapeau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Burovski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Peterson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Weckesser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Bright, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' van der Walt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Brett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Wilson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Millman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Mayorov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Nelson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Jones, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Kern, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Larson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Carey, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Polat, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Feng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Moore, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Van- derPlas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Laxalde, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Perktold, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Cimrman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Henriksen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Quintero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Harris, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Archibald, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Ribeiro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Pedregosa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' van Mulbregt, SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='0 Contributors, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Vi- jaykumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Bardelli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Rothberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Hilboll, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Kloeckner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Scopatz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Lee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Rokem, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Woods, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Fulton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Mas- son, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Häggström, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Fitzgerald, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Nicholson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Hagen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Pasechnik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Olivetti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Martin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Wieser, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Silva, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Lenders, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Wilhelm, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Young, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Price, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Ingold, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Allen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Audren, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Probst, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Dietrich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Sil- terra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Halchenko, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} +page_content=' Vázquez-Baeza, Nature Methods 17, 261 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE4T4oBgHgl3EQfPgwg/content/2301.04973v1.pdf'} diff --git a/cNE2T4oBgHgl3EQfwgg2/vector_store/index.faiss b/cNE2T4oBgHgl3EQfwgg2/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a22a3d8ef3f2f7b26e4fc8bded3639031765dc71 --- /dev/null +++ b/cNE2T4oBgHgl3EQfwgg2/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Modeling social resilience: +Questions, answers, open problems +Frank Schweitzer1,2,∗, Georges Andres1, Giona Casiraghi1, Christoph Gote1,3, +Ramona Roller1, Ingo Scholtes3,4, Giacomo Vaccario1, Christian Zingg1 +1Chair of Systems Design, ETH Zurich, Switzerland +2Complexity Science Hub, Vienna, Austria +3Department of Informatics, University of Zurich, Switzerland +4Chair of Computer Science XV, Julius-Maximilians-Universität Würzburg, Germany +Abstract +Resilience denotes the capacity of a system to withstand shocks and its ability to re- +cover from them. We develop a framework to quantify the resilience of highly volatile, non- +equilibrium social organizations, such as collectives or collaborating teams. It consists of four +steps: (i) delimitation, i.e., narrowing down the target systems, (ii) conceptualization, i.e., +identifying how to approach social organizations, (iii) formal representation using a combi- +nation of agent-based and network models, (iv) operationalization, i.e. specifying measures +and demonstrating how they enter the calculation of resilience. Our framework quantifies +two dimensions of resilience, the robustness of social organizations and their adaptivity, and +combines them in a novel resilience measure. It allows monitoring resilience instantaneously +using longitudinal data instead of an ex-post evaluation. +1 +Introduction +Why do some social organizations succeed to persist and thrive in the presence of crises and +shocks, while others fail under the same conditions? They have different levels of resilience that +can be most generally described as a system’s capacity to withstand shocks and its ability to +recover from them. Such a description already implies different features: +(i) resilience is a systemic property, as opposed to a property of system elements, +(ii) resilience is not restricted to a specific system, it rather seems to be a general property of +different systems, +(iii) resilience is described as a response to a shock, i.e., it can be only recognized in the presence +of shocks, or perturbations, +(iv) resilience is not a static property because shocks and recovery imply time dependent pro- +cesses. +∗Corresponding author, fschweitzer@ethz.ch +1/54 +arXiv:2301.00183v1 [cs.SI] 31 Dec 2022 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Resilience has to consider not only the magnitude of shocks, but also different types of shocks the +system has to absorb. The same system can be robust to, e.g., the impact of an earthquake, but +not to the spreading of a disease. More importantly, to be resilient a system also needs to have +the ability to recover. A system may be robust even in the presence of various perturbations, but +once it is impacted by a critical shock it will not recover, so it cannot be seen as resilient. +Both the response to a shock and ability to recover are system specific and therefore difficult +to generalize. This also hampers a more precise or formal definition of resilience. We should +not expect that there is a universal concept of resilience, applicable to various types of systems +[25, 116, 177]. In fact, resilience concepts diverge across and sometimes even within scientific +disciplines [6, 9, 42]. +We therefore do not provide a review of existing resilience concepts and refer to the literature +already available [9, 53, 86]. Instead, with our paper we want to broadly inspire researchers +from different scientific disciplines who already study social organizations by providing new +modeling perspectives. The analysis of collaborative teams and collectives is a core topic of +social psychology [14] and organizational theory [71]. Hence, case studies inform about, e.g., +collective decisions, coordination and conflict resolution in teams. +But the models used are most often descriptive models, not generative models. Descriptive models +include statistical models, e.g., regression models, or database models, e.g., conceptual entity- +relationship models that indeed resemble our knowledge graphs (see Figure 9). In organizational +psychology there are mental models of teams and team members to describe perceived relation- +ships or the collective representation of knowledge. Descriptive models try to include as much +detail as possible. But generative models try to include as much detail as necessary to generate +a macro-social behavior from the micro dynamics of the constitutive individuals. This method- +ological approach is advocated in analytical sociology [75]. +Agent-based and network models belong to the class of generative models. Stochastic actor- +oriented models (SAOM) [162] and exponential random graph models (ERGM) [95, 101, 133] +aim at combining agent-based and network approaches [48]. They also belong to the class of data- +driven models, using methods similar to logistic regression. With their focus on link prediction +to detect reciprocity, transitivity or homophily they are less suited to study systemic properties +of social systems, such as resilience. Computational issues, in particular scalability, assumptions +about utility maximization of actors and problems in model specification prevent a broader range +of applications [102]. +But there are better solutions. In this overview paper we want to sketch a framework how to utilize +them for the study of social organizations. This framework provides interfaces for mining larger +and more fine grained data about interactions between individuals (Section 6.1), an analytically +tractable network ensemble to avoid computational issues (Section 4.2), statistical methods to +2/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +infer signed relations and significant interactions from observed data (Section 4.2) and formal +ways to estimate the social impact of agents involved in these relations (Section 5.1). +But most of all, this framework allows to calculate robustness and adaptivity instantaneously, to +estimate the resilience of the organization. The attention is shifted from the micro configurations, +the dyads and triads, towards the macro-properties of social networks. These are no longer +reduced to topological features, but involve a dynamic component to describe the response to +shocks, namely the potential for change in an organization. With this the formal modeling of +social organizations can be moved to a new level. It will also impact research on resilience which +is dominated by two paradigmatic views, engineering and ecological resilience (Sections 2.2, 2.3). +So far, theoretical research on resilience and empirical research on resilience indicators have +been largely segregated [138]. Many studies restrict their focus on specific systems, in particular +engineered systems, ecological, or urban systems [43, 72, 79, 116, 157]. There are also studies +about the resilience of socio-economic systems and organizations [86, 98]. As we detail below, they +are of little help for the problems discussed in this paper, for two reasons: (i) When referring to +social systems, most often our modern human society is addressed [100]. This bears a complexity +way too large to be captured in a formal modeling approach and restricts the discussion to a +discourse level. (ii) Our research interest are social organizations at smaller scale. Organizational +resilience studies have pointed out “factors” for improving the resilience of such systems, e.g., +integration or redundancy [110, 158] or social capital [91, 127, 159]. But they do not instruct us +what to do if such factors shall be modeled and quantified. +A major aim of this paper is to provide a framework to overcome this research gap. To de- +velop a broader foundation, we will specifically address the problems of conceptualizing social +organizations. One main issue is their volatility, i.e., the continuous change of their structure +and dynamics that makes it difficult to define stability, to measure the impact of shocks, or to +distinguish recovery from change. A second and probably more ambitious aim is to revise the +premature anxiety-laden understanding of “shocks” and “breakdowns” that dominate the discus- +sion of societal resilience. To cope with social organizations, we need to shift the focus from the +fear to breakdown towards the faith to recover. +2 +What do we know about resilience? +In order to investigate the resilience of social organizations, we should first take a look at two of +the most prominent resilience concepts in engineering and in ecology. They may provide already +good starting points to formalize robustness and adaptivity. If so, we have to test weather such +formalization could be utilized to model social organizations. But even if that is not the case +we learn from the shortcomings about requirements for social resilience concepts. This helps to +further clarify the underlying assumptions. +3/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Questions +• Is there a difference between robustness, stability, and resilience? +• What is the relation between adaptivity and recovery? +• How is robustness related to existing systemic risk measures? +• Should a resilient system always return to equilibrium? +2.1 +Robustness and adaptivity +Constituting dimensions. +While the concrete meaning of resilience may vary across scientific +disciplines, there are also conceptual commonalities. As pointed out in Figure 1, resilience bears +relations to concepts of robustness and adaptivity. The former relates to the property of a system +to withstand shocks, the latter to its ability to overcome their impact. Robustness represents the +structural and adaptivity the dynamic dimension of resilience. However, we need to understand +how they constitute resilience as a function defined on these two dimensions. +Resilience +Robustness: Structural +Absorptive [54, 174] +Buffer [87] +Coping [90] +Redundant [19] +Robust [19] +Adaptivity: Dynamic +Adaptive [54, 90] +Learning [87] +Recoverability [54] +Self-organization [87] +Transformative [90] +Figure 1: Examples of resilience factors used in the literature, which we assigned to the struc- +tural and the dynamic dimension of resilience without claiming consistency across different +systems and shocks. +Topology. +We argue that both dimensions need to be combined to explain resilience, but +most often they have been studied as stand-alone concepts. For instance, the robustness of inter- +connected systems is defined with respect to the failure of nodes or links. Robustness measures +then estimate the size of failure cascades after a shock [4, 12, 37, 126]. This approach uses the +complex networks perspective that we also utilize in our framework. But it assumes that simple +4/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +topological features, like connectedness, are sufficient to describe the robustness of a system. This +leaves out the dependence on individual properties which becomes a problems when applying +the approach to social systems. +Equilibrium. +In addition to topological robustness, other robustness measures build on dy- +namical stability. It analyzes how small perturbations of an assumed equilibrium state affect a +system. If the system is able to regain its equilibrium state, it is robust at least against this type +of perturbations. This approach has several limitations when applying it to social organizations. +It requires to know the dynamics of the system, to estimate its response to shocks. Further, the +underlying assumption of different types of equilibrium states can hardly be justified for very +volatile social systems. +Response. +Adaptivity is often simply understood as dynamics, which neglects the quality of +change. Adaptive systems do not simply react to a shock. Their response aims at preserving the +system’s functionality, to ensure its persistence. Recovery after a shock therefore implies a certain +directedness which depends on the type of shock. Most important, the system needs to have +several options to adapt even to unexpected challenges. Therefore, adaptivity should estimate +how many options exist in a given situation. This points towards the problems of quantification +and measurement discussed later. +Creative destruction. +Additionally, simply adopting existing concepts of robustness and +adaptivity may lead to misconceptions about social systems. Quite often, the understanding +of these concepts builds on a negative perception of shocks. This ignores the role of “creative +destruction” that, according to the economist Joseph Schumpeter, is instrumental for renewing +and further developing the economy. Stable systems do not evolve. Therefore, challenging their +stability is one of the driving forces of evolution. This also regards social organizations. The +leave of established members is not only a threat, it is also an opportunity for newcomers. +Creative organizations often respond to shocks with innovations. Therefore, the discussion about +robustness and adaptivity should not just focus on maintaining the status quo. Resilience means +to cope with change in a sustainable manner. +2.2 +Infrastructure systems +Critical functionality. +With respect to critical technical infrastructures such as power grids +or communication networks, the design of resilient systems has been studied extensively [4, 21, 83, +161, 164, 166]. Here resilience, sometimes also called “resiliency”, refers to a system’s capacity to +“maintain an acceptable level of service in the presence of [. . . ] challenges” [161]. Such challenges +5/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +include software and hardware faults, human mistakes, malicious attacks or large-scale natural +disasters. +This concept of resilience is goal-oriented, preserving a system’s functionality after the shock +as illustrated in Figure 2. A power system has a defined functionality which remains as long as +no critical shocks are caused by internal malfunction, e.g., lack of maintenance, or by external +disruptions, e.g., an earthquake. If the system has to “absorb” a shock, this functionality is +partially or entirely destroyed because the system lost its robustness, to some degree. +Time +Critical Functionality +Plan +Absorb +Recover +Adapt +Figure 2: The common perception of resilience for engineered systems. +Adaptivity. +We note that functionality is assumed here as a function of an underlying network +structure, in this case the power grid. Consequently, the focus is on the robustness of this under- +lying structure. Adaptivity refers to possible changes in the processes running on this structure, +and not to the structure itself. This can be illustrated by the following example: During the 9/11 +attacks in 2001, the Internet infrastructure in downtown Manhattan was largely destroyed. This +posed a severe shock to the global Internet because of (i) the termination of transatlantic cables +in the basement of the World Trade Center (WTC) and (ii) the failure of the major Internet +exchange point NYIIX next to the WTC, which was responsible for 70 % of transatlantic traffic. +Despite these combined failures, the attack caused only minor disruptions and the global routing +infrastructure continued to operate normally within a few hours [46, 164]. +This resilience of the Internet was not obtained by the robustness of the hardware, which was +heavily affected in the above example, but by the adaptive capacity of the dynamics on the +network. However, redundancies built into the underlying network, i.e., a level of robustness, +were essential to allowing for the quick rerouting of the communication flow, i.e., adaptivity. +Hence, a minimum level of robustness can be a precondition for the adaptive capacity. +6/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Recovery. +What distinguishes resilience from robustness is the ability to recover. Infrastruc- +ture systems do not recover by themselves [63]. They have to be rebuilt by human activity, +reestablishing the functionality and often even improving it. Thus, attributing the ability to +recover to the infrastructure would reduce the concept of resilience to absurdity. Every bridge +that was rebuilt after a collapse would then be “resilient”. In engineering, hence resilience is often +used as a synonym for robustness. For example, Dinh et al. [43] define a “resilient reactor” as one +achieved by “[using] a tank designed to withstand high pressures and temperatures”. However, +once a tank has exploded, it has no ability to recover a functional state. +Systemic risk. +Quantifying the impact of shocks requires to model the system’s response. +Only such models allow to estimate whether large parts of the system will be destroyed, which is +commonly denoted as systemic risk. Based on the theory of extreme events [41], one can calculate +probabilities for rare, but severe external shocks, e.g., floods or earthquakes. Using engineering +knowledge about materials [120] and constructions, then allows to propose critical values for +system properties, e.g., the minimum width of walls, or the maximum height of buildings, etc. +Systemic risk, from this perspective, is reduced to the risk that an external event with a critical +magnitude may occur. Consequently, robustness is defined only with respect to such rare events, +for instance, “the ability of power systems to withstand low-probability high-impact incidents in +an efficient manner” [92]. +Loss estimation. +In infrastructure systems, such as transportation networks, it is common +to measure robustness by employing loss estimation models. These models evaluate the ability of +the system elements to withstand a given level of stress or demand without suffering degradation +or loss of function [19]. Quantification allows to control the robustness of such systems to some +degree, e.g., counterbalancing shocks by a central control station. Engineered systems are designed +systems, top to bottom. Their functions are clearly defined, and therefore the impact of a shock +can be estimated. +Failure cascades. +The focus on external extreme events neglects another major cause for +systemic collapse, namely the amplification of a few small failures inside the system into a failure +cascade. If such cascades reach a critical size, they can also destroy the system from inside. +Robustness in this case has to be defined with respect to (i) the probability that a few system +elements fail, and (ii) the mechanisms that can amplify such failures, e.g., by redistributing load. +Both may depend, in addition to internal conditions, also on external feedback processes. Large +scale blackouts in the U.S. have been explained by such failure cascades in the power grid [4, 178]. +To study failure cascades we need an appropriate representation of the engineered system. For +power grids, communication networks or transportation infrastructure, the network approach +7/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +is used most often [181]. System elements, e.g., transformers or responders, are represented by +nodes, and their physical connections by links in the network. Many models of failure cascades +test the robustness of networks by removing nodes or links either in a random or a targeted +manner [21, 170] and measure whether the network breaks down into disconnected components +after such attacks. A modular network is robust if shocks trigger small failure cascades [8, 18]. +Targeted attacks, however, usually result in large failures as the network becomes disconnected +[10, 156]. +Amplifying mechanisms. +Removal tests reduce the problem of systemic risk to mere topo- +logical properties. To understand amplifying mechanisms inside a systems, we need models for +the dynamics inside the nodes, but also models for the exchange of quantities between nodes [22– +24, 107]. That means we need a framework that couples the dynamics of the network, i.e., the +failure of nodes or links, with the dynamics on the network, i.e., the rerouting of communication +or the redistribution of load. +Without such a framework, we cannot understand the robustness of the system, even less its +adaptivity, and hence, its resilience. Extreme value theory [35] can at best estimate the probability +that large-scale failure cascades happen, whereas network models can provide an explanation why +they happen. Resilience, as a systemic property, cannot be reduced to the robustness against +extreme shocks, it has to be derived from a broader perspective that helps to understand why +and when small events can be turned into big disasters. +2.3 +Ecological systems +The classic view. +Historically, the notion of resilience first appeared in ecology. Ecosystems +constantly change under dynamic processes of renewal and reorganization. Therefore, resilience +concepts do not primarily focus on robustness as a static property of the system, but rather on +the adaptivity as the dynamic component of resilience. For Holling [81], resilience is based on the +“ability of a system to return to an equilibrium state after a temporary disturbance”. Because +shocks and perturbations are unavoidable, emphasis is on the survival, or the persistence, of the +ecological system, regardless of the impact of a shock [176]. +Dynamical systems. +Such a notion of resilience essentially builds on the theory of dynamical +systems and its concept of “stability”. In different scientific areas, e.g., nonlinear dynamics, control +theory, physico-chemical reaction kinetics, or biological pattern formation, a system is said to +be stable if it returns to the equilibrium state after a shock. Approaches to assess resilience in +biological and engineered systems are based on this idea [93]. +8/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Most systems are in fact metastable, i.e., they are stable in the presence of small perturbations, but +become unstable if these perturbations exceed a certain critical level. Then, instead of returning +to the previous equilibrium state, the system moves away from it, possibly to another equilibrium +state, as Figure 3 indicates. This new equilibrium state corresponds to a “different” system, i.e., +to a system that has evolved and adapted to the new situation. This aspect has been studied +as robust adaption, combining the notions of robustness and adaptivity. There are analogies to +concepts of phase transitions in physics and chemistry and regime shifts in social and biological +systems. +shock +low recovery rate +State +Potential +Figure 3: Mechanical analogy for a metastable equilibrium. This figure is adapted from Fig- +ure 1 in [104]. +The meaning of recovery. +In the resilience concept of ecological systems recovery may have +different connotations dependent on whether a system returns to a previously attained stable +state or to a different one. Even a previous state would be gradually different because evolving +systems would never reach identical states. +In more general terms, resilience can be implicitly defined by “the amount of disturbance that an +ecosystem could withstand without changing self-organized processes and structures (defined as +alternative stable states)” [72]. To capture differences in the quality of a resilient ecosystem, the +additional factors of ecological “vulnerability” and “sensitivity” have been proposed [40, 115, 135]. +They consider that species are more sensitive to perturbations in certain time periods or under +special conditions. This complements the notion of “stability” of ecosystems or “robustness” of +engineered systems. +Time scales. +What matters for ecological resilience is the time scale of recovery. If it would +take forever to return to a previous state, the system is not resilient. Therefore, “the inverse of +9/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +the length of time required for an ecosystem to return to near-normal” [70] is often used as a +resilience measure. But resilience itself can also be timescale specific [25, 39]. On a daily timescale, +the system may return to a near-normal state. But on a time scale of decades, the system may +lose this ability. +Hence, resilience cannot be reduced to some concept of “stability”, the dimension of adaptivity is +essential. Resilient ecosystems maintain a critical balance between stability and instability, the +former providing a level of robustness against small perturbations, the latter allowing them to +adapt to changing environmental conditions [176]. +Connectedness and potential. +To further grasp the generic mechanisms underlying ecolog- +ical resilience, two system properties have been proposed: potential and connectedness. “Potential +sets limits to what is possible - it determines the number of alternative options for the future” +[82]. Hence, potential resembles our notion of adaptivity. Connectedness, on the other hand, is +related to robustness. “Connectedness is assumed to increase over time, leading to high internal +control and limited potential to cope with disturbances” [82]. This is an important remark as it +points out to the fact that robustness above a certain critical value may have a negative impact +on resilience. “When connectedness is low, resilience is high because the system can vary over a +wide range of states and respond to disturbances in many different ways. When connectedness, +however, is high, ecosystem resilience is low because the system is more tightly organized and +has fewer options for responding to disturbances” [69]. We will return to this argument when +discussing our own concept of social resilience. +Adaptive cycles. +It was argued that resilience changes in a cyclic manner because an increase +in connectedness may lead to a breakdown of the system: “Naturally such over-connected systems +crash into a release period, where they have the potential to reorganize, thereby coping with dis- +turbances. This development is believed to be cyclic” [69]. We note that the ability to reorganize +is given only if during “release phases” of low connectivity the potential, i.e., the adaptivity, is +high. This is in line with our arguments for the recovery of social organizations discussed below. +So far, adaptive cycles have not been found empirically: “Because of its very general nature, the +concept of the adaptive cycle should be considered a metaphor [25] or thinking tool rather than +a testable scientific theory” [69]. We may add here that with our approach we move the adaptive +cycle from a metaphor to a testable concept, which is also accessible to formal modeling. +Couplings between adaptive cycles on different temporal and spatial scales may lead to a nested +hierarchy, called panarchy [82]. +10/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Losing resilience. +Instead of specifying properties for resilient states, we could address the +complementary question, namely about the properties that indicate a loss of resilience or about +states that need to be avoided. This reveals critical conditions for stabilizing feedback cycles, +critical magnitudes for perturbations or critical levels of diversity. To identify factors that lead to +the loss of resilience, non-parametric regression models or machine learning tools, e.g., symbolic +regression, can be used to analyze data. +Early warning signals for losing resilience can be obtained from time series. A slower recovery +rate from perturbations, known as critical slowing down [104], right before a tipping point is +a possible indicator. Also an increase in the auto-correlation of systemic variables, e.g., order +parameters, indicates the vicinity of a regime shift. +Coupling to social systems. +Insights into ecological resilience are limited if the dynamics +of ecological systems is predominantly driven by the coupling to the human sphere [11]. On the +global scale social and ecological systems are, in fact, coupled inseparably, mainly because of hu- +man interventions. In recent years, this has triggered integrative research studying the resilience +of social-ecological systems [51, 175, 176]. It relates ecological resilience with the resilience of +societies responding to environmental challenges that originate from the ecological systems into +which they are embedded. +Conclusions +Different scientific disciplines have their own understanding of resilience. We should not +expect to find a universal resilience measure. The key question is [25]: Resilience of what to +what? Answering this question requires to have a model of the respective system. Resilience +concepts for social organizations may benefit from ecological concepts, because issues of +time scales, different equilibria and adaptive cycles are already addressed. +3 +Why are social systems different? +Existing concepts of resilience from engineering or ecology do not seem to provide the best basis +for social resilience. Although formal approaches exist, we cannot simply reuse them. Before +developing suitable alternatives, we may first clarify what makes social systems different from +other types of systems. This leads us to more fundamental questions about defining systems and +models. What seems to be a detour at this point will later allow us to better ground our notion +of social resilience from a methodological perspective. +11/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Questions +• What types of social systems do we want to investigate? +• What characterizes social organizations and collectives? +• What is the focus of organizational resilience? +• What modeling consequences entails the complex systems approach? +3.1 +Social organizations +No model of society. +The need to understand social resilience is often motivated by the many +crises that our societies face, today [167]. Their vulnerabilities have been widely recognized, +ranging from pandemics to political polarization, from climate change to budget crises, from +infrastructure breakdown to poverty migration. Consequently, the resilience of societies cannot +be decoupled from the resilience of ecosystems, political systems, infrastructure systems, financial +and economic systems, etc. While these connections cannot be denied, they raise a methodological +question that, unfortunately, is not addressed with the same emphasis: How should we model all +of these interdependencies? +Four steps towards a model. +We aim at a quantitative understanding of social systems. +Therefore, we specify the resilience problem in a tractable manner in four steps that are sum- +marized in Figure 4. The first step is delimitation: Which types of systems should be specifically +investigated, and which ones not? This question is discussed in the remainder of Section 3.1. In +Section 3.2, we further distinguish our problem from existing concepts of organizational resilience. +The second step is conceptualization: Which approaches should we use to describe social systems? +Which of the many possible features will we focus on? This is discussed in Section 3.3. Only this +clarification will enable the third step, representation: To build a model means to represent the +system and its elements in a formal manner. In Sections 4.1, 4.2 we introduce different network +concepts as candidates to represent properties of social systems. In Section 5, eventually, we +address the fourth step, operationalization: How do we specify measures such that they can +be calculated? Again, we introduce different solutions to choose from. Finally, in Section 6, we +comment on the data required for the fourth step. +Systems of systems. +For methodological reasons we distinguish social resilience from the +resilience of societies. The latter is not the target of our formal modeling. Society is a system +of systems, as described above. To understand the resilience of societies requires modeling the +interaction of these systems [130]. +12/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Step 1: Delimination +• Type of system: Social organization +• Example: Collaborative teams +• Focus on relations between individuals +Step 2: Conceptualization +• Complex adaptive system +• Resilience as an emergent property +• Self-organization, no time scale separation +Step 3: Representation +• Combine agent-based and network models +• Multi-edge networks, statistical ensembles +• Signed relations, propensities of interaction +Step 4: Operationalization +• Social impact and importance of agents +• Structural/topological robustness +• Resilience: Robustness & adaptivity +Data Collection +• Extract interaction data from repositories +• Network regression: Propensities +• Analyzing temporal data: Causal relations +Figure 4: Our framework to quantify social resilience +Models for systems of systems exist only in rudimentary form [130]. Particular emphasis was +given to the coupling between socio-economic systems and ecological systems. More recently, also +the coupling between climate systems and socio-economic systems and/or ecological systems is +captured by formal models. Most of the systemic relations that constitute a “society” are not +formalized at all [76]. +Collectives. +We have to restrict our investigation of social resilience to clearly defined social +entities rather than “social systems” in general. Our focus are social organizations, or collectives. +With the term social organization we refer to formal or informal groups of interrelated individuals +who pursue a collective goal and who are embedded into an environment [80, 124]. +For illustrative purposes, our running examples are project teams, in particular teams of software +developers [80, 139, 182]. These teams face numerous shocks during their development. Compet- +ing products, technical evolution, organizational problems, lack of motivation or resources put +up challenges and let them fail quite often. Their common goal, namely to develop a software +for a certain scope, is important to distinguish this type of social system from a collection of a +hundred persons who, for example, use the subway without being interrelated or contributing +13/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +to a common good. Collective goals in most cases go beyond the mere survival [47, 124], which +would characterize ecological systems. +Beyond individual resilience. +The term “social” specifically refers to human individuals +in the following, although many animal societies also have a remarkable degree of social or- +ganization. A broad range of psychological studies focuses on the resilience of an individual +[7, 45, 94, 171]. Given that we want to understand resilience as a systemic property, psychologi- +cal resilience would require us to model the individual as a system [108]. That could be possible, +but is not aligned with our aims. +Instead, we are interested in collectives of many individuals, for which we use the term social +organization synonymously. We assume collectives of the order of 102 individuals, small enough +that the impact of a few individuals can still matter, but large enough to distinguish individual +and collective in a meaningful manner. This implies that resilience, as a systemic property, is +neither identical to, nor the mere combination of the psychological resilience of its members. +3.2 +Organizational resilience +Different from engineering, in psychology and organizational science the concept of resilience +focuses on the dynamic component, i.e., adaptivity as the system’s ability to cope with shocks. +But in social organizations resilience does not require to return to a previous state. Hence, +resilience is generally seen as “the ability of groups or communities to cope with external stresses +and disturbances as a result of social, political and environmental change” [2]. We note that in +such a definition shocks primarily result from other systems an organization is embedded in, +rather than from internal processes, which is the main focus of our concept of social resilience. +Community resilience. +Examples for this outside orientation are studies of citizen commu- +nities in urban or rural areas [2, 109]. Their response to natural hazards or disasters [19, 38] or to +climate change [111] is of particular interest. Community resilience generally refers to the ability +of communities to cope and adjust to stresses and to engage community resources to overcome +adversity [121, 159]. An organization should not only persist after a disturbance, but also manage +to strengthen its capability for future adjustments [165]. The ability to transform challenges into +advantages is known as transformational resilience. +Whether a new state is resilient may depend on specific positive outcomes that need to be +achieved [123]. Hence, resilience comprises more than just persistence: “The capacity of actors to +access capitals in order to – not only cope with and adjust to adverse conditions (that is, reactive +capacity) – but also search for and create options (that is, proactive capacity), and thus develop +increased competence (that is, positive outcomes) in dealing with a threat” [123]. +14/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Resilience factors. +It is an open question why and how some organizations manage to thrive +and enhance core capabilities when faced with a crisis, while others fail [173]. Recent research +highlights three factors that influence the resilience of a social system: (i) its vulnerability (or +susceptibility) to disruptions, (ii) its level of anticipation, foresight, or situational awareness for +such vulnerabilities, and (iii) its adaptive capacity, flexibility or fluidity which allows to mitigate +vulnerabilities or respond to disruptions [62, 114]. +To better understand the role of these factors, most empirical resilience studies have followed a +“hindsight approach”, focusing on organizations which have recovered from a shock and trans- +formed crises into advantages [55, 103, 132, 180]. For such social organizations, Sutcliffe and +Vogus [165] define organizational resilience as the ability to maintain a “positive adjustment +under challenging conditions”. +Adaptive capacity. +A system’s capacity to adapt in a constantly changing environment is +also referred to as adaptive capacity [56, 160]. From a social science perspective, the adaptive +capacity is expressed in a number of different ways, for instance in terms of the ability to learn +and store knowledge, the ability to anticipate disruptive events, the level of creativity in problem +solving, or the dynamics of organizational structures [52, 160]. Some of these aspects have been +assessed by means of survey research designs, such as the learning capability [36], situational +awareness, creativity [114], or the fluidity of structures [62]. +Missing macro-variables. +Most notions of resilience proposed above reveal their limitations +when it comes to quantification [100, 105]. Two problems need to be solved, (i) to define a measure +for resilience that considers also the dynamics of the system, and (ii) to measure the defined +variables against available data. Many studies of social resilience, e.g., in disaster management +[38, 88, 109, 121, 129, 136, 168], monitor resources for basic needs or survey social well-being. +But we lack macro-variables to describe social organizations, e.g., to measure their adaptive +capacity and their elasticity. Such variables exist in economics, for instance productivity and +efficiency measures, but also for ecological systems, e.g., biomass production or recovery rates. +In engineering functional resilience can be computed through the integral below the function of +performance [20, 129]. +In absence of these variables, tools to derive early-warning signals, e.g., the critical slowing down +or the increase of auto-correlations mentioned above, cannot be applied. Therefore, one of our +aims is to provide such macro-variables for robustness and adaptivity and to show how they can +be monitored over time. These variables will help separating the ability to resist a shock from +the capacity to recover. +15/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +3.3 +Conceptualization +In Section 3.1 we have already processed the first step towards a model of social resilience, namely +delimitation. The second step, conceptualization, now requires us to specify how to approach +highly dynamic social organizations. This particularly regards the modeling framework that +shall foster our understanding of the micro-processes to explain social resilience. +Different conceptual frameworks. +Ecological systems are often modeled using concepts +from system dynamics [147], where species are described by densities. Interactions between dif- +ferent species in a food web are then formally expressed by coupled differential equations. This +approach does not focus on individuals, but mostly this is also not needed. +Models of engineered systems often use concepts from control theory. This allows to steer system +elements, e.g., transformers, from a central perspective, but requires to have precise models of +such elements and their relations to others. This is often the case because engineered systems +are designed systems. +Both of these modeling approaches cannot be applied to social organizations the way we see +them. They are much more volatile, more adaptive in response to shocks and, most importantly, +have no defined reference state. Therefore, in the following we specify what concepts we will use +to describe their structure and dynamics. +Complex systems. +We start from the insight that social organizations are complex adaptive +systems. They comprise a larger number of interacting system elements, commonly denoted +as agents. Taking the complex system perspective implies that systemic properties, such as +resilience, need to be understood as emerging from the interaction of agents. +Hence, we have to develop a bottom-up perspective for social resilience, starting from the micro, +or agent, level rather than from the macro, or systemic, level. This is in line with the method- +ological principles of analytical sociology [75], which aims at explaining macro-social phenomena +from the micro-processes that generate them [49]. +Agent-based and network models. +To formalize both the dynamics of agents and their +relations, we combine agent-based modeling with temporal multi-layer network models. The +agents, as the nodes of the network, are characterized by different properties, such as status, +roles, knowledge, opinions, which depend on other agents and can change over time. Furthermore, +agents are heterogeneous. They can be of different types and even within one type their properties +are not identical. For instance, agents’ function and efficiency in solving tasks vary across agents. +We therefore have to model agents explicitly, to overcome approaches solely based on topological +features to describe the functioning of a social system. +16/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Agents’ interactions and their social relations are captured in different network layers, which +evolve over time. This requires us to also model interactions explicitly. In particular, we have +to distinguish random from meaningful interactions and to find ways to infer roles and social +relations from interaction data. This paves the ground for a statistical approach based on net- +work ensembles, which also provides the interface for data-driven modeling, which we discuss in +section 4.2. +Finite systems. +Our approach explicitly addresses the finite number of agents. Focusing on +larger systems would have the advantage that we could calculate simple statistical measures, e.g., +averages, to overcome details. For the type of social organizations discussed here details matter +and are therefore explicitly addressed. We need to consider individuals and discrete events instead +of continuous variables characterizing a whole system, such as densities. +Work teams, online chat groups, or school classes differ from large social systems not only in their +interaction structures or perceived goals, they also differ in size. Emergent phenomena of social +systems, such as coherence or cooperation, depend on size. Large systems necessarily behave +differently from smaller ones because regime shifts or phase transitions can occur. Therefore our +models for social resilience are not expected to describe very large social systems, e.g., political +parties or urban populations. +In small systems, such as collectives, stochastic influences can have a larger relative impact on +the dynamics. Further, path dependent processes in the evolution of these systems cannot be +ignored. Local effects, such as neighborhood relations become important. Therefore, known limit +cases of formal modeling, such as the mean-field approach in which all agents interact in a similar +manner, cannot be readily applied to collectives. Instead, we need to build agent-based models +that reflect agents’ heterogeneity. +Self-organized systems. +An important difference to, e.g., technical systems, is the level of +adaptivity in social organizations which cannot be simply reduced to “dynamics”. Instead, emerg- +ing structures in social systems feed back on the interaction of agents and cause further change, +often denoted as second-order emergence [147]. This is related to co-evolution and learning, which +occurs on the individual and on the organizational level. +The outcome of these dynamics can hardly be predicted. Social organizations cannot be com- +pletely controlled and agents cannot be forced to behave in a predictable manner when facing +changes. Instead of central control distributed influences and self-organization play a major role. +In general, agents respond to changes both in intended and in unintended ways [144]. This makes +the response of social organizations to internal or external shocks so difficult to model. +17/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Time +Critical Functionality +Plan +Absorb +Recover +Adapt +Figure 5: Problems defining a resilient state for volatile organizations. +No separation of time scales. +Social organizations are very volatile systems which makes it +almost impossible to define a reference state, as Figure 5 illustrates. More importantly, we have to +account for the fact that the absorption of shocks and the subsequent recovery cannot be clearly +separated as in Figure 2. Instead, changes of robustness and adaptivity follow instantaneously. +This is a noted difference to ecological systems where the time scale of adaptivity is usually +much larger and an out-of-equilibrium state can be clearly separated from the equilibrium and +the relaxation time scale is well defined. +Conclusions +Our notion of social resilience focuses on social organizations and teams. To develop a formal +model we adopt the viewpoint of complex adaptive systems. Modeling resilient societies +would require a system dynamics approach, instead. Existing concepts of organizational +resilience mostly take a management perspective. We aim instead to model resilience bottom +up, as an emerging property of organizations. Our framework will combine agent-based and +network models. +4 +How shall we model social organizations? +We continue to go from the general, i.e., delimitation and conceptualization, to the particular, +now addressing the problem of system representation. Once we agreed upon the complex systems +approach with its agent-based and network models, the biggest hurdle is to turn these concepts +into formal structures. Instead of presenting just one solution, we have to prepare for a broader +perspective. The following descriptions should therefore be seen as alternatives for choosing +formal approaches. In Section 4.2, when we introduce network ensembles, we want to highlight +possible options of utilizing ensembles. +18/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Questions +• Why is system representation a central problem for modeling? +• What are the differences between the various network representations? +• Why do we need a network ensemble? How shall it be used? +• What types of dynamics do we consider for social organizations? +4.1 +Network representation +Various options. +There is not the one way to construct a network, not only because of +different topologies. There are different types of networks, as we outline below. Which network +type is most suited to represent the organization depends on the context and the available +information, i.e., data. +One could argue that we should indeed start our discussion with the latter, to explain what +data we got. This is denoted as the supply driven approach in Section 6.1. We instead follow +the demand driven approach, which requires us to first identify what data we will need. Hence, +before collecting data from an empirical system, a suitable formal system representation has to +be chosen. Only then the question about the minimal set of data needed should be addressed. +Link properties. +Networks are one way of representing complex systems. The nodes of the +network are the agents, and links aij between nodes i and j represent their relations or interac- +tions. Figure 6 shows one example. The network approach focuses on the topological structure, +which can be conveniently summarized in an adjacency matrix A with the entries aij. Links +between agents are usually directed, e.g., agent i assigns a task to agent j and aij ≠ aji, repeated, +e.g., there are multiple links between the same pair of agents, aij ≥ 1, and time bound, aij(t), +i.e., they have to respect causal ordering or bursts of activities. +Network inference builds on the assumption that the topological structure encodes information +about agents, i.e., individuals. Utilizing this information could reduce the model complexity +because it allows for operationalizing the structure and dynamics of social organizations. But +studying the network topology would reveal hidden information about individuals and collectives +only to some degree. Therefore, the network approach has to be extended by explicit models of +agents. +Links versus signed relations. +The network reconstruction is most often based on inter- +action data to determine links, aij. Interactions may be frequent and short-lived. What matters +for the resilience of organizations are rather the social relations ωij between agents. These are +19/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +72 +69 +58 +95 +52 +71 +6 +73 +42 +79 +49 +4 +53 +107 +91 +92 +76 +46 +32 +59 +93 +40 +83 +81 +39 +100 +66 +70 +68 +89 +18 +104 +30 +65 +44 +101 +103 +88 +27 +56 +20 +63 +57 +7 +28 +77 +21 +85 +80 +67 +108 +23 +24 +50 +48 +55 +96 +61 +78 +25 +9 +51 +94 +19 +87 +98 +16 +13 +29 +36 +31 +2 +74 +45 +11 +86 +8 +90 +64 +105 +26 +47 +97 +10 +35 +14 +17 +37 +75 +22 +3 +34 +41 +43 +102 +38 +84 +15 +99 +12 +5 +62 +54 +60 +106 +33 +82 +1 +Figure 6: Collaboration network of software developers. Larger link width indicates more in- +teractions. Node color codes individual importance, measured by coreness values as a proxy of +network integration. Blue colors correspond to higher coreness. +generally signed relations, i.e., they have positive or negative signs. It takes time to establish +social relations and they usually change on a longer time scale. Compared to interaction data, +data about signed relations is rare. Therefore we need methods to infer signed relations from +interaction data, as described below. +Signed relations crucially impact the robustness of a social network. The theory of struc- +tural balance [64, 73] considers triads involving three agents (see Figure 7). A network is +assumed to be robust, i.e., stable, if it contains balanced triads. To determine the balanced +state, the classical approach only takes the signs of the signed relations into account, Sijk = +sign(wij) sign(wik) sign(wkj). If Sijk = 1, triads are balanced, if Sijk = −1, they are unbalanced +and have the tendency to change into balanced triads as Figure 7 shows. The line index [73] mea- +sures the minimal amount of signs that need to be changed to turn all unbalanced into balanced +triads and can therefore serve as a measure of structural robustness, which is explained later. +Bipartite networks. +One of the main challenges in modeling social organizations comes +from the vast heterogeneity not only in the agents’ properties, but also in their interactions. +The notion of a link, i.e., a direct interaction, is already an abstraction. Taking the developer +example, collaboration means that two developers work on the same code. This would be most +appropriately represented as a bipartite network between different entities, the developers and the +pieces of code (see Figure 8 ). The collaboration network then is a projection, where developers +have a direct link if they have changed the same piece of code. A second network results from +the projection on the code; two pieces of code are connected if they were changed by the same +developer. +20/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Figure 7: Unbalanced triad and the three ways to obtain a balanced triad. +Multi-edge networks. +If two developers collaborate on more than one piece of code, nodes +in each network projection can be connected by more than one link, i.e., we have a multi-edge +network (see Figure 15). From this network we are able to construct an important network +ensemble, as we discuss below. +D1 +D2 +D3 +D4 +D5 +D6 +D7 +D8 +D9 +D10 +P1 +P2 +P3 +P4 +P5 +P6 +P7 +P8 +(a) +D1 +D2 +D3 +D4 +D5 +D6 +D7 +D8 +D9 +D10 +(b) +P1 +P2 +P3 +P4 +P5 +P6 +P7 +P8 +(c) +Figure 8: (a) Bipartite network of developers (P) and software code (D). (b) Projected collabo- +ration network of developers. (c) Projected network of code changed. +Knowledge graphs. +Following these considerations, the starting point for representing or- +ganizations by means of networks is not the social network between agents, which is already a +reduction. Instead we have to start from a relational graph, also known as knowledge graph, that +visualizes the various ways of connecting individuals, as shown in Figure 9. +21/54 + +2 ++ ++ +Paradise +State +.1 +k ++ +2 ++ ++ ++ +1 +k +k +1 +Polarized +State ++ +.1 +kF. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +User +Repository +File +Organization +Commit +Addition +Deletion +Issue +fork +star +watch +own +own +belong to +follow +link to +make +contain +contain +modify +modify +consist of +consist of +open +comment on +close +link to +belong to +belong to +Figure 9: Relational graph of software development activities on GitHub +Multi-layer networks. +From a knowledge graph we construct different projections, each of +which creates its own network. These networks are combined in a multilayer network, as shown +in Figure 10. In each layer the nodes and their interactions are different. If the nodes are the +same in each layer, but the links represent different types of interactions (e.g., friendship, work +relations) this is known as a multiplex network. Hence, we have now intra-layer links within each +layer and inter-layer links between layers [57]. +The multilayer network is accessible to mathematical investigations, by representing the topo- +logical structure as tensors. This allows to apply methods of spectral analysis [3, 184]. +Hypergraphs. +A noted limitation of networks is the decomposition of any type of interactions +into bilateral interactions between two agents. For instance, in a group of five agents this proce- +dure results in ten links. To overcome this limitation in modeling group interactions, we resort +to hypergraphs [13, 125]. This is a special type of higher-order networks, in which higher-order +nodes contain groups of simultaneously interacting agents. Links between higher-order nodes +then capture group interactions. +Similar to multi-layer networks, higher-order networks can have levels of increasing order. The +first order would be then the standard network, the second order level contains groups of two +agents, the third order groups of three agents, and so forth. This way hypergraphs allow to model +22/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Figure 10: Multilayer network with intra-layer and inter-layer links. +interactions between groups of different sizes by means of inter-layer links. For the formation and +the dissolution of groups, however, more refined dynamic models are needed [50, 149]. +4.2 +Network ensembles +Probabilistic approach. +A network representation of the collective constructed from avail- +able data will be only one possible realization and not necessarily a very typical one. Ideally, +we would need a probability distribution that assigns to all possible networks a probability to +occur. Such a network ensemble is largely determined by the constraints of agents to form links. +Figure 11 shows sample networks from such an ensemble. +Figure 11: Six networks sampled from a network ensemble. They look similar, but differ in +their details. +If no link constraints are taken into account but only the total number of nodes, n, and links, +m, we would arrive at a very large ensemble of random networks that all have the same n and +m and the same probability to occur. The network constructed from data will be part of this +ensemble, but it is statistically indistinguishable from the other networks, most of which will look +23/54 + +VF. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +very different. Hence, we need to incorporate more information to restrict the network ensemble, +and to increase the probability for our reconstructed network in comparison to others. +Generalized Hypergeometric Ensemble of Graphs (gHypEG). +To model multi-edge +networks characterized by heterogenous constraints, we have proposed gHypEG [30], a broad +class of analytically tractable statistical ensembles of finite, directed, and multi-edge networks. +It introduces dyadic link propensities Ωij, which capture the preference of nodes to form links. +Precisely, the ratio Ωij/Ωik is the odds to draw a link (i,j) rather than a link (i,k). The propen- +sities reflect social mechanisms such as homophily or reciprocity [17]. Furthermore, gHypEG can +incorporate formal assignments to classes or communities [27]. To do so, it employs propensities +ΩB +kl for links between nodes i,j that are in different “blocks”, i.e., communities, k,l. +gHypEG has the benefit of being defined by closed form probability distributions. Thanks to this, +we are able to calculate the weights for all incorporated features by means of efficient numerical +Maximum Likelihood Estimation (MLE), without the need of expensive Markov Chain Monte +Carlo (MCMC) simulations. +Network regression. +The challenge to obtain the propensities Ωij can be mastered by means +of a multiplex network regression [26]. Each network layer l encodes different types of known +relations between agents as explanatory variables (see Figure 12). The influence of each layer +on the interaction counts as the dependent variable is then determined by fitting the Ωij such +that the observed network has the highest likelihood. In other words, the optimal propensities +are proxies for the constraints that shape the network ensemble. As an added benefit of the +method, one can test the statistical significance of the explanatory variables for the observed +interactions [28]. +Figure 12: Illustration of the network regression method [26]. +24/54 + +Frienashiip +nteractionsF. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Potentiality. +With the calibrated propensities, gHypEG allows to calculate how many possible +configurations of the observed network exist, given the constraints for links. This issue becomes +of importance if we later want to quantify adaptivity, i.e., the ability of an organization to attain +different configurations. Then we need to know not only the number, but also the diversity of +possible network configurations. +This information is aggregated in a new measure, potentiality which is based on the normalized +Shannon entropy [187]. Importantly, the calculation is feasible without computational problems. +The larger the potentiality, the more alternatives an organization has to respond to shocks. We +discuss below how this will impact the organization’s resilience. +Significant relations. +“Social” is not “random”, therefore, social relations should significantly +differ from random interactions. To test this, we filter the adjacency matrix with the observed +number of interactions, ˆ +aij, using a significance threshold α and our probability distribution for +the network ensemble. If Pr(Aij ≤ ˆaij) > 1 − α, links are significant [32]. Figure 13 demonstrates +that removing insignificant, i.e., random, links from the network has a considerable impact on +determining, for instance, communities. +(a) +(b) +Figure 13: Community detection of a social network considering (a) all links, (b) only signifi- +cant links. [32] +If the observed network is not expected from the network ensemble, we have to apply an itera- +tive procedure, to refine the probability distribution. In a first step, we measure the significant +deviations. This additional information is used in a second step to update the constraints for +the network ensemble, i.e., to generate a new ensemble. The iterative procedure reveals what +information is relevant to explain the observed network. +25/54 + +68 +G +5 ++ +% +% +36 +25 +31 +73 +15 +61 +2 +14 +30 +55 +86 +24 +88 +76 +39 +10 +81 +67 +84 +27 +54 +32 +35 +28 +0g +26 +0 +51 +18 +6 +G +56 +8 +8 +25 +9 +87 +% +22 +% +22 +0 +% +te +21 +73 +0 +63 +74 +3 +51 +0 +17 +26 +57 +23 +1 +59 +84 +30 +81 +91 +0g +64 +43 +11 +31 +10 +40 +71 +76 +47 +18 +56 +35 +% +心 +3 +3 +2 +3 +m +2F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Signed relations. +Eventually the probability distribution for network ensembles allows to +test whether the number of observed interactions exceeds expectations. This issue is important if +we wish to map interactions to social relations which have positive or negative signs. Empirical +studies have shown that more interactions indicate a positive social relation, e.g., a stronger +friendship [85, 89, 169], whereas less interactions indicate a negative relation which causes e.g., +avoidance behavior [74, 96]. +aij +0.00 +0.05 +0.10 +0.15 +0 +5 +10 +15 +20 +Aij +Pr(Aij) +Figure 14: Determining overrepresented interactions [5]. +As illustrated in Figure 14, we infer the weight and the sign of the social relation between +two agents from ωij = Pr(Aij < ˆaij) − Pr(Aij > ˆaij) [5]. This procedure allows us to obtain from +a multi-edge network of observed interactions a network with signed relations, as shown in +Figure 15. The weighted signs, on the other hand, will enter the formalism to determine the +social impact of agents in a network. +(a) +(b) +Figure 15: Multi-edge network (a) and the resulting network of signed relations (b) [5]. +4.3 +Dynamics of social organizations +So far, we have narrowed down our investigations to social organizations of a particular type +(delimitation), which are modeled as complex adaptive systems (conceptualization). For the rep- +26/54 + +十 ++F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +resentation we have chosen the network approach, which offers a great variety of network types, +but also a statistical description using network ensembles. +This discussion has focused only on the structure, but not on the dynamics of these networks +which will be done in the following. Further, we have not addressed yet the agents and their +properties which will follow in Section 5.1. +Concurrent changes. +Models of complex socio-economic systems often use the concept of +separated time scales. The dynamics on the faster time scale is assumed to reach an equilibrium +state, which allows to describe the dynamics on the slower time scale as a sequence of different +equilibrium states. Similar approaches are used to separate different network dynamics. For +instance changes of the network topology are assumed to be slow, therefore the fast dynamics +running on the network can neglect the changing topology. +As already pointed out, we cannot use such assumptions to model the dynamics of collectives. +Instead, the different processes discussed below should be seen as concurrent. This leads to a +number of issues, such as overlapping or sliding time windows, choice of the appropriate time +scale for aggregation, etc., which are not discussed here, but should be kept in mind. +Entry and exit dynamics. +The most visible changes regard the network topology. For social +organizations we have to consider an entry and exit dynamics of nodes, i.e., newcomers connect +to the network [146, 149], whereas incumbents may leave. This implies also the addition and +deletion of links, as shown in Figure 16 for the case of a multiplex network. Social organizations +often exhibit a life cycle, i.e., a predominant growth of both nodes and links in early stages is +followed by a saturation and a decline caused by many nodes leaving [60, 150, 154]. +... +... +time +... +... +i +j +ts3 +ts2 +ts1 +Figure 16: Coupled growth dynamics in a two-layer network. Intra-layer links are between +nodes of the same color, inter-layer links between nodes of different color [119]. +These processes do not occur at random. Further, they impact the collective as a whole as well +as individual agents. Newcomers may not be able to connect to core nodes initially and thus +connect to the periphery. Their integration into the collective may improve over time, as can be +27/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +measured by their coreness [58, 151, 155]. If core nodes leave, this may trigger cascades of other +nodes leaving as empirical and simulation studies have demonstrated [34, 60]. +Restructuring. +Next to the addition and deletion of nodes and links, the rewiring of links +between nodes plays a major role. Their impact can be measured by tracking changes in the +global and local topological measures discussed below. Structural changes often reflect changes +in the organization, e.g., in responsibilities, hierarchical positions and roles of agents. Figure 17 +illustrates such restructuring processes for a developer collective in which a central developer has +assumed the main responsibilities for task assignments. Already the visual inspection makes clear +that this has lead to considerable problems in the robustness of the collective, which eventually +lead to a collapse and the establishment of more resilient structures. +(a) +(b) +(c) +Figure 17: Topological change of a collaboration network of developers. Aggregated interac- +tions (a) before October 2004, (b) between October 2004 and March 2008, (c) after March +2008. [183]. +Temporal networks. +Whereas the dynamics of networks addressed above changes the topol- +ogy, the dynamics on networks captures interactions between agents. These can be exchange +processes, e.g., load redistribution in case of an agent’s failure, but also communication of in- +formation. These processes are strongly path dependent, i.e., the sequence of interaction matters +and has to respect causal relations. Models of causal paths [97, 142] provide a formal approach. +They build on higher-order networks, where each order captures a causal path of a given length. +In addition to time directedness, temporal networks also reflect the burstiness of activities [142], +i.e., the fact that not every link in a network is active at all times. The temporal component +significantly impacts the centrality measures of individual agents [141], as Figure 18 shows. +28/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Betweenness preference [128] was introduced as an agent-centered measure to quantify its im- +portance in transferring information. +(a) +(b) +Figure 18: Identification of important individuals (a) on the aggregated and (b) on the tempo- +ral network. +External and internal shocks. +The different dynamics described above are continuously +perturbed by internal and external shocks of various size and origin. Internal shocks, for instance, +may cause agents to leave, this way triggering cascades of drop-outs and restructuring. External +shocks, e.g., directives during the pandemics, may change working conditions and collabora- +tion relationships. Because of the volatile dynamics, we cannot clearly separate shocks from the +“normal” dynamics, which both occur on the same time scale. +We note that from our modeling perspective we model shocks, but not the origin of shocks, e.g., +the government that changes the legal regulations. But we need to have models for the impact +of these shocks and for the collective’s response to different kind of shocks. In other words, we +need to estimate the robustness, or the absorptive capacity, of the collective facing a particular +shock, and to estimate the adaptivity of the collective to overcome this shock. Only then we can +calculate the social resilience of the collective, as outlined below. +29/54 + +18F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Conclusions +Different types of networks capture different aspects of relations between individuals. It +depends on the research question and the available data which of these network represen- +tations shall be used. Building up a network ensemble allows us to go beyond the observed +network, to include constraints for the social organization. In particular, we can distinguish +significant from random interactions and infer signed relations from interaction data. These +link characteristics are important to build an agent-based model, in the next step. Our +model has to consider various concurrent dynamics, including growth, entry and exit of +individuals, internal restructuring and external shocks. +5 +What should we do to calculate resilience? +After completing the steps delimitation, conceptualization and representation, we eventually have +to master the last step, operationalization, where we merge the network approach with agent- +based modeling. The overview is presented in Figure 19. +Questions +• How can we turn concepts into measures for robustness and adaptivity? +• How can we characterize agents, using topological information? +• Why do we consider the social impact of agents? How can we quantify it? +• How is resilience composed of robustness and adaptivity? +5.1 +Quantifying agent properties +The major goal of our framework is a micro-perspective on resilience. This is an emerging systemic +property, that means it can neither be reduced to, nor explained by, the dynamics of the agents. +As we demonstrate below, we need to consider the network structure to calculate the robustness +and the adaptivity of the social organization. But the agents’ importance, their social impact, +will provide the right weights in calculating these two measures. +Quantifying agents’ importance. +A bottom-up approach to quantify resilience has to start +from the agents. In each organization, agents have a different importance, ri, that reflects their +hierarchical status, reputation, embedding in the organization, knowledge, etc. To obtain values +for ri is a challenge in itself and depends on the available data. Because there is no general +solution, we resort to some guiding examples. +30/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +System +Structure +Dynamics +Robustness +Adaptivity +Social resilience +Agents +Importance +Interactions +Embedding +Relations +Dyads +Triads +Social impact +Weighted balance +Signed relations +Propensities +Data +Relational +Longitudinal +Multivariate +Figure 19: Operationalization to calculate social resilience. +In the simplest case, importance is defined in the hierarchical structure of a team [148]. There +are also ways to determine hierarchies based on interaction patterns. In the absence of such +information, we may utilize topological information from the reconstructed network (see also +Figure 17). In a directed social network we can use the eigenvector centrality of agents as a +measure of their reputation [16, 152]. For undirected networks, coreness [155] or weighted k-core +centralities [58] can quantify an agent’s embeddedness in a network [151], assuming that more +important agents are closer to the core (see also Figure 6). These measures also estimate the +robustness of an agent’s network position against failure cascades. +For temporal networks different centrality measures can be used [141] (see also Figure 18). Be- +tweenness preference [128] quantifies an agent’s importance in communication processes. Func- +tional roles can be only partially inferred from communication patterns or specific topological +embeddings of agents. Existing algorithms for role detection [78] do not detect organizational +roles, but classify network positions. +Social impact. +What matters in an organization is not just the importance, ri, but also +the support or opposition an agent receives from others. Their influences are combined in an +31/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +individual social impact, Ii. The total impact of an agent is then the sum of its own importance +and the social impact exerted by others, qi = ri + Ii. +Here, we define the social impact as Ii = ∑j wijrj = Ip +i − In +i . The wij denote the weighted and +signed relations between agents, which can be positive, negative or zero. Ip +i is the sum of all +positive contributions, while In +i sums up the negative contributions [84, 99, 122]. Ii can become +negative, reflecting the fact that an agent may not have a high esteem in an organization because +it receives little support, but strong opposition from others. +Infer signed relations. +To calculate the social impact, Ii, we also have to determine the +signed relations wij. For this, we apply the method described above in Section 4.2. It returns for +every pair of agents a weight and a sign to characterize their relationship. +To conclude, our measure of the total importance, qi = ri + Ii, combines different, but rather +complete information about each agent, namely information about its topological embedding and +about its activities because its repeated interactions with other agents determine its relations. +qi aggregates in one value the positive, negative or neutral influences from all counterparties, +weighted by their individual importance. Hence, with qi we have a non-local measure about the +true impact an agent can have in the organization. +5.2 +Quantifying social resilience +In order to obtain a measure for resilience, we have to solve two problems: (i) defining different +proxies to measure robustness and adaptivity based on the available information about agents and +their relations, (ii) determining a functional form for resilience dependent on the two dimensions. +Again, there is not the one way of combining available information into meaningful measures. +Therefore in the following we list a number of candidates to quantify robustness, which we can +choose from. For adaptivity instead we provide only one measure based on the assumption that +we have data available to construct a multi-edge temporal network. +Topological robustness. +As noted above, robustness can only be defined with respect to a +specific shock. A software developer team can be robust against an external shock, e.g., stronger +legal regulations, but not against an internal shock, e.g., the dropout of a leading developer. +Therefore, we must consider different ways for defining the robustness of social organizations. +Topological measures are often easy to calculate and reflect specific aspects. The robustness +against agent removal can be linked to agents’ coreness [29, 58, 117, 151, 155] (see also Figure 6). +It helps understanding cascading effects from removing a specific agent. Centralization [179] +takes the concentration of interactions in a few agents into account, which increases the systemic +32/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +risk if these agents fail [34] (see also Figure 17). Betweenness preference [128] and Eigengap +[112] indicate communication bottlenecks and identify gate keepers. These measures can be used +separately, as demonstrated for centralization [154, 183], or in combination to quantify robustness. +We can further utilize higher-order models of temporal networks to capture robustness. The +Second-Order Algebraic Connectivity, for instance, can be interpreted as a temporal-topological +robustness measure [184]. +Structural robustness. +A different measure of robustness is proposed by the concept of +structural balance as explained in Section 4.1. It decomposes the network into triads and deter- +mines their balance Sijk by multiplying the signs of the signed relations, sign(ωij). This approach +has several shortcomings. First, triads are evaluated independently, i.e., the fact that each agent +is likely part of different triads at the same time is ignored. Secondly, the different weights of +each signed relation, ωij, are not taken into account. Thirdly, the importance ri of the agents +composing the triad is ignored. That implies all triads have the same weight in estimating the +robustness of the organization, which is not justifiable. +Correcting for these shortcomings is an open discussion. As a possible alternative we have pro- +posed a new weighted balance measure Tijk [143, 148] that takes into account not only the +signs and the weights of the signed relations, but also the impact of the agents involved in the +triad. To determine the structural balance of the whole collective, we take the arithmetic mean, +⟨T⟩ = ∑Tijk/Nt, where Nt = ∥Tijk∥ is the total number of triads in the network. +Quantifying adaptivity. +Ideally a maximally resilient system would have maximal robust- +ness, i.e., it could withstand any shock, and maximal adaptivity, i.e., if a shock impacts the +system it will always recover. That means resilience R should increase both with robustness R +and adaptivity A, R(R,A) ∼ R ⋅ A. +Adaptivity does not simply mean “dynamics”. Instead, it refers to the ability of the organization +to attain different states, which we also call potential. But will the system actually attain these +alternative states at random, without a response to a shock? If so, we call this the propensity to +change to indicate that it is independent of the quality of the current state. It turns out that +the propensity to change is a two-edged sword. If a team is in a bad shape, it should be able +to leave such bad state. Then a high propensity to change allows the team to attain other, and +likely better, configurations. On the other hand, if the team has reached a good state, it should +be interested in keeping it. A high propensity to change would be counter productive because the +good state could be easily lost. This means that resilience should increase with the propensity +to change if the system is in a bad state, and decrease in a good state. Figure 20(a) illustrates +the problem. +33/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +low +high +Propensity to change +high +low +Robustness +Fragile state +can not be left +Bad +(1) +Fragile state +can be left +Good +(4) +Robust state +can be kept +Good +(2) +Robust state +can be lost +Bad +(3) +(a) +0 +0.2 +0.4 +0.6 +0.8 +1 0 +0.5 +1 +0 +0.5 +1 +Robustness +Propensity to change +Resilience +0 +0.2 +0.4 +0.6 +0.8 +1 +(b) +Figure 20: Resilience R as a function of robustness ˆR and propensity to change ˆP: (a) Qualita- +tive assessment of different states. (b) Exemplary quantification of R( ˆR, ˆP). [154] +A functional form. +A decomposition of resilience into robustness and adaptivity, R(R,A) ∼ +R ⋅ A, rests on the fact that we can capture the potential to change of the system independently +from its propensity to change, which is in fact not possible. Therefore, we use the propensity to +change ˆP as an empirical proxy for adaptivity. +Our measure of potentiality [187], introduced in Sect. 4.2, allows us to proxy this propensity. It +quantifies the probability distribution of states attainable by a system at a given point in time. +The larger the potentiality, the larger is the number of alternative states attainable by the system, +and the more likely is the system to change towards one of them. The smaller the potentiality, +the smaller the number of states attainable and the smaller the probability the system will move +away from the current state. +These considerations have determined us to propose the following functional form for the re- +silience of social organizations: R( ˆR, ˆP) = ˆR(1 − ˆP) + ˆP(1 − ˆR) [148, 154]. The quantity ˆP is a +convenient transformation of potentiality P. I.e., low values of ˆP (below 0.5) map to a state +with low propensity to change, while large values of ˆP (above 0.5) map to a state with high +propensity to change. The lowest achievable potentiality is mapped to ˆP = 0, while the highest +to ˆP = 1. Similarly, the value of robustness R should be always positive and conveniently scaled +between 0 and 1. This can be achieved for most topology based measures. For structural balance +measures, however, ⟨T⟩ can become negative. To use these measures, we have to map them as +ˆR = 1/(1+eβ⟨T⟩), where β = 0.2 gives a rather smooth mapping. ⟨T⟩ = 0 would then be equivalent +to ˆR = 0.5. Note that the function plotted in Figure 20(b) reflects the arguments summarized +Figure 20(a). +34/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Relation to ecological concepts. +It is worth to get back from here to the discussion about +“potential” and “connectedness” as constituents of ecological resilience in Section 2.3. Potential +shall define the number of possible alternatives states. But so far it was only a conceptual +proposal because of the lack of operationalization. This gap is closed by our concept of adaptivity +which indeed can be calculated and also compared across different systems [187]. Connectedness +refers to the robustness of the system, capturing topological aspects. Again, with our measure of +robustness we are able to calculate and to compare the robustness of different systems. Moreover, +both adaptivity and robustness can be monitored over time, making our resilience measure an +instantaneous early warning signal. +Most interesting is the relation between low connectedness and high resilience, on the one hand, +and high connectedness and low resilience, on the other hand, discussed in Section 2.3 [82]. This +was presented together with the hypothesis about the “adaptive cycle”, which emerges if low +connectivity is met by high potentiality. While this adaptive cycle was considered a “metaphor” +[25] or a “thinking tool”, we are able to demonstrate its existence in data about real world +organizations [154]. +Resilience as a compromise. +Our framework reflects that high resilience requires both, +the maintenance of a valuable organizational structure to withstand shocks, and the ability +to change this structure quickly if needed. Reasons to change can result from internal or from +external problems, for instance from an incapable management or from governmental restrictions. +The resilient organization has to achieve conditions under which it can respond even without +prior knowledge about the shock. Instead of rigidity, it needs fluidity. But instead of fragility, +it also needs stability, dependent on the situation. Hence, the maximum resilience should be a +compromise to balance these different requirements in an efficient manner. +Conclusions +Turning concepts into measures is the hardest part of modeling. There are always different +options to operationalize measures, dependent on available information. Topological mea- +sures alone are not enough to estimate the robustness and adaptivity of social organizations. +Instead, we need to quantify the impact of agents, to correct structural balance. Optimal +resilience is a compromise between robustness and adaptivity. +6 +Network construction and interventions +So far we have translated our concepts for the robustness and adaptivity of social organizations +into measures. As the last step we have to discuss possibilities of obtaining the data needed +35/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +to calculate these measures. If we achieve to have a calibrated generative model of the social +organization, we can address the problem of system design [144]. That means, we can test how +possible intervention strategies impact the organization’s resilience. +Questions +• How do we construct networks from data? +• What type of data is needed to calculate our resilience measure? +• How can we obtain such data from repositories of social organizations? +• Is there a way to validate our generative model? +• How can we control resilience using network interventions? +6.1 +Data acquisition and analysis +We want to emphasize that our methodology inverts the usual supply driven approach found +in computational social science. This starts from the data given, often collected without a clear +purpose and a research question in mind, to subsequently squeeze out interesting features. In +contrast, our demand driven approach has first identified in four steps shown in Figure 4 what +data will be needed to inform our models. Then, utilizing this data we can infer information +about agents and their properties, but also about their interactions with others, as shown in +Figure 19. +Such data cannot directly provide the input for our models and is not sufficient to simply +estimate social resilience. Instead, it has to be pre-processed, before we can construct the networks +that are essential for our framework. These networks are never given, and their generation and +subsequent statistical interpretation bears some of the most overlooked problems in modeling +social organizations. +Extract interactions. +One of our reasons to study software developer teams as prototypes +of social organizations is the availability of vast git repositories. These contain fine-grained +records of all changes made to the software, together with information who changed it, what was +changed and when. We developed a software package, git2net [66, 67], that is able to extract +this information, to create bipartite networks and their projections into an interaction network +between developers (see Figure 21). +We note that, in addition to the co-editing network, i.e., the collaboration network of developers, +we can obtain additional information about the social organization. For instance, analyzing the +amount of code changes can quantify the productivity of developers [139] and analyzing the +sequence of code changes gives insights into their hierarchical structure. +36/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +git2net can be also used to mine other git repositories, e.g., for publications. Additionally we +have developed the rule-based disambiguation tool gambit to solve the persistent problem of +name disambiguation occurring in most real-world user data [68]. +Figure 21: Extracting the collaboration network of developers A, B, C using git2net [66]. +Time window detection. +To obtain interaction networks, a sliding window approach ag- +gregates interactions over a certain time interval. Choosing the right window size is a problem +in itself, because the window size impacts the network density and subsequently all topological +analyses. +Often we have no data about interactions and need to infer them from time series of observed +events. For instance, from co-location data, i.e., observations about two individuals i,j acting +at times ti and tj at a given place, we need to detect the time interval ∆t = ∣ti − tj∣. Only +observations with a ∆t lower than a given threshold ∆tthr will count as interactions [113]. Such +considerations are important to quantify, e.g., the transmission of information within a social +organization. +Analyzing temporal data. +The dynamics of temporal networks crucially depend on ∆t. The +problem, who can potentially influence whom, requires to reconstruct temporal paths of various +lengths [65], on which our networks can be generated. +We have developed different software packages to support the analysis of temporal networks. They +are combined in the toolbox pathpy [140]. It implements, for instance, statistical techniques to +find optimal graphical models for the causal topology. These models balance model complexity +with explanatory power for empirically observed paths in relational time series. As part of pathpy, +MOGen [65] is a multi-order generative model to statistically evaluate paths of various lengths. +It can be used to improve the computation of different temporal centrality measures in case of +insufficient observations. +37/54 + +c +c +B +B +血血 +血血血血血血面 +血血 +B +timeF. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +Infer signed relations. +Signed relations ωij are instrumental to calculate the social impact +and subsequently the robustness of the organization. As described above, our framework uses +interaction data to infer signed relations. This method can be enhanced by taking additional data +sets into account that can provide information about the positive or negative relations between +individuals. +If written or spoken text is available, we can use sentiment analysis to obtain information about +the emotional content [59, 61], to infer social relations. Natural language processing (NLP) pro- +vides an extended tool box to further extract information about opinions, attitudes, or ideological +positions [1, 134]. These can help quantifying the social impact that individuals exert on others. +Information from collaboration platforms. +Another important source of information are +online collaboration platforms, such as slack, zoom, or GitHub. In addition to interaction data +and text messages, they often provide information about attention, e.g., via likes, about declared +trust, recommendations, and activity patterns. +Based on reconstructed collaboration networks, one can analyze the presence of social mechanisms +like reciprocity, homophily, triadic closure [17, 131], or of other motifs [182]. This information +can be used to further characterize the importance of agents and their signed relations and to +estimate their impact on the resilience of the organization. +Network regression. +If the topological information is sufficient to reconstruct an additional +network layer, it can be utilized for the network regression outlined above. To facilitate the +computation, we have developed an R package ghypernet [31]. It implements gHypEG, the +network ensemble considering propensities. In addition to network regressions, the package can +be used to infer significant relations from observed interactions [32]. +Once gHypEG is calibrated, we can also compute our potentiality measure even for large en- +sembles. SciPy [172] provides an efficient implementation for computing the entropy of a given +multinomial distribution. +Calibration and validation. +To find the optimal combination for the different measures +mentioned above is recognized as an open problem. Symbolic regression [44, 163] and other +machine learning (ML) techniques are increasingly used to find solutions. In many cases, ground +truth data is not available. Then, we have to rely on in-sample and out-of-sample predictions to +aggregate different information in a meaningful manner. +This issue becomes relevant if we, for instance, want to improve the importance measures for +agents. If reliable aggregation methods are not available, we have to resort on determining the +38/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +ri values from topological measures, combined with dynamic processes as, e.g., in feedback cen- +tralities, as we will do in the following example. +6.2 +Resilience and control +Improving the resilience of complex systems implies that, to some extent, we are able to influence +such systems in a way that their functionality and their stability is maintained, or even enhanced. +The formal model of social organizations described above allows testing such intervention strate- +gies. +Top-down control. +Generally we distinguish between bottom-up and top-down interventions +[145]. The latter mostly focus on the boundary conditions for organizations, either to prevent +shocks or to enhance their business environment. These can be financial measures during the +Corona crisis, but also legislative measures to ensure fair competition. +In general, to use the top-down approach, one needs to identify global control parameters which +is a challenge on its own. Often they can be derived from the known macroscopic or system +dynamics. As a major conceptual drawback, control parameters usually reflect limitations of +stability, rather than of resilience. +While the top-down approach is discussed in macro-economics and recently in macro-prudential +regulations, we are interested in the bottom-up approach which is more in line with the complex +systems philosophy. +Bottom-up interventions. +Our bottom-up approach to resilience uses interventions targeting +specific agents and their interactions [153]. Structural interventions focus on the interaction struc- +ture, basically changing the adjacency matrix of the network. Functional interventions change, +for instance, the interaction rules to affect timing of interactions [137]. +Dynamical interventions instead influence the internal state of nodes, i.e., the agents. Such mea- +sures can include nudging or mechanism design [145], but the most promising way for us are +network interventions (see Figure 22). They require to first identify the driver nodes, i.e., those +agents that should be targeted, and secondly to decide about the type and the amount of inter- +ventions [106, 185]. Often these interventions change the agents’ utilities ui using control signals. +Indirect influence. +To get access to agents, we can utilize the multi-layer structure of the +organization. For instance, if one layer contains friendship relations and the second one task +assignments, the friendship layer can be used to influence the work relations. +39/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +(a) +(b) +(d) +(c) +u1 +u3 +u2 +u4 +Layer 0 +Layer 1 +Figure 22: Network control in a two-layer network. (b-d) Different couplings between the two +layers, dependent on the peripheral or central position of agents. [185] +If the impacted agent responds appropriately, changes can propagate through the network, this +way influencing agents that were not targeted directly. As the example of Figure 22 shows, we +can target agents at the periphery of the network to impact agents in the core [118, 185, 186]. +Influence on decisions. +Network interventions only control a small number of agents at a +comparably low cost, while utilizing the systemic feedback. But the method requires a model of +the organization to forecast the impact. Further, it assumes that the agent’s utility is known. +For the latter we can have at least reasonable assumptions. +Rational agents want to keep or even increase their impact, qi, by either increasing their impor- +tance, ri, or by decreasing a negative social impact, Ii, they experience. But changing signed +relations or maintaining collaborations is costly. Agents may decide to leave the organization if +their costs exceed their benefits. Conversely, they may decide to stay if their benefits have been +increased. +Changing agents’ utility has the advantage of influencing these decisions. If agents leave or +reorganize their links, this changes the network topology and impacts the dynamics in each +layer. Consequently, both the robustness and the adaptivity of the organization are impacted. +This can lead to counter intuitive effects. For instance, removing some agents may stabilize the +organization [33, 153]. While this is known in human resource management, models are hardly +able to reproduce such behavior. +40/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +Figure 23: Network interventions to prevent the breakdown of a social network, indicated by the +drop of k-coreness. (a) No intervention, (b) peripherial agents targeted, (c) agents close to the +core targeted [33, 153]. +Conclusions +We provide a whole tool box for mining and analyzing data of social organizations. In +particular, interaction data can be obtained from repositories. Other tools allow to calculate +temporal centralities to characterize communication, and to infer propensities for interacting +individuals. Which of the different measures are calculated depends on the available data. +There are various ways to proxy robustness, adaptivity and resilience of social organizations. +Network interventions allow to improve the resilience of organizations. +7 +Conclusions +7.1 +What is resilience? +Structural and dynamic dimensions. +Summarizing our tour through the modeling of +social organizations, some important insights should be noted. First, resilience is a concept that +combines two dimensions, robustness and adaptivity. Robustness, as the structural dimension, +41/54 + +6 +Average k-coreness +2 +0 +10000 +20000 +30000 +40000 +50000 +Network Time6 +Average k-coreness +2 +0 +10000 +20000 +30000 +40000 +50000 +Network Time6 +4 +2 +0 +25000 +50000 +75000 +100000 +Network TimeF. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +captures the ability of a social system to withstand shocks. Adaptivity, as the dynamic dimension, +captures the ability of a social system to recover from shocks. Neither maximal robustness, +nor maximal adaptivity alone are sufficient to warrant resilience for social organizations. Both +dimensions create a tension, because increasing robustness may lower adaptivity and the other +way round. Therefore, a resilient state is a compromise balancing the influence of both dimensions. +This insight is important because, following arguments from engineering, resilience is too often +just treated as a synonym for stability. This leads to the conclusion that maximizing resilience +means maximizing robustness. Such a perspective may hold for designed infrastructure systems, +but not for self-organizing systems such as social organizations. +Resilience measure. +To turn a concept into a measure requires operationalization which +points to a different problem domain [77]. Even if we agree about our resilience concept, there may +be different proposals to operationalize it. They have to solve two problems. Firstly, the functional +form of resilience dependent on robustness and adaptivity should be specified. Secondly, measures +for robustness and adaptivity have to be proposed and subsequently operationalized. +The latter is the real difficulty. What should we measure to quantify robustness or adaptivity? +A system may be robust against some specific shocks but will fail for others. Therefore, the +question cannot be answered without an appropriate formal model of the social organization. In +this paper, we made an operationalization proposal based on networks which can be constructed +from data. In general, these are multi-edge, temporal, multiplex and dynamic networks. From +these networks, topological information can be used to calculate robustness. Using the ensemble +approach, we are further able to calculate adaptivity. +Resilience as an emergent property. +For our modeling framework of social organizations +we have adopted the complex systems perspective, in general, and the complex networks ap- +proach, in particular. This implies to explain resilience as an emerging property of the social +organization. Following the bottom-up approach, we have to focus on the micro level of inter- +acting agents. Measures for resilience need to be derived from this perspective. It requires to +characterize agents in some detail regarding their importance, their signed relations and their +social impact on others. Simple network measures that treat agents as dots to just calculate their +network position are not sufficient to estimate robustness, even less to understand adaptivity as +the dynamic component of resilience. +Resilience as a systemic property has to be constantly maintained, which requires the activity +and the cooperation of the members of the social organization. Conversely, big threats to social +resilience are not coming only from external shocks, but from internal challenges as well. As +our model framework demonstrates, negative signed relations and negative social impact hamper +42/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +robustness and adaptivity. Biased interactions, the lacking integration of newcomers and low +connectivity undermine the conditions under which resilience can be established. +7.2 +The policy dimension +Science versus policy. +Our framework for modeling the resilience of social organizations +helps to understand under which conditions resilience is lost. Our reasoning does not refer to +the loss of “robustness” and “adaptivity” in an abstract manner. It instead relates these losses to +the underlying properties of agents and their dynamic interactions. Only this way we are able to +propose network interventions for improving resilience. +But do these models, while successful from a scientific perspective, benefit policy makers in any +way? Are we able to tell them what to do? To put this challenging question into perspective, we +remind on some preconditions and some findings. +Our models focus on a specific type of social organizations, namely teams of collaborating mem- +bers sharing a common goal. This means that we are not considering societies and, hence, our +models neither aim at, nor are suited for, making suggestions on how to improve the resilience +of societies against political, economic or environmental shocks. Delimitation was the first step +for developing our framework. With these restrictions in mind, our models indeed support some +general insights. +Awareness. +The most important insight is probably about the role of awareness for what +resilience really means. This requires distinguishing it from concepts of robustness, stability, +functionality, or optimality. Resilient systems are not obtained by maximizing or optimizing +specific functions or key figures. A resilient organization has to withstand various kinds of shocks +and to recover from them. That means it needs to be prepared for the unknown, instead of being +specialized to fit the known. +This addresses a policy issue: Instead of improving resilience, organizations have strong incen- +tives to rather improve performance as the most visible indicator of success. This reminds on +the classical conflict between short-term benefits and long-term deterioration and points to the +misallocation of limited resources needed for maintaining resilience. Ideally, a social organiza- +tion should be able to anticipate possible shocks to some degree, and to prepare in advance +for this, also by securing resources. This requires collective awareness, a state of consciousness +that is based on continuously analyzing and recognizing the situation inside and outside the +organization. +Flexibility. +Next to robustness, our models highlight the role of adaptivity. It proxies the num- +ber of options that an organization may have to respond to shocks. Consequently, we measured +43/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. Zingg: +Modeling social resilience: Questions, answers, open problems +To appear in: Advances in Complex Systems, vol. 25, no. 8 (2022) +adaptivity by potentiality. It does not imply that these options are taken, but that they exist +in a current situation. Resilience depends on alternatives. That means, concepts like flexibility +or fluidity become increasingly important. We remind on the concept of adaptive capacity which +already refers to the ability of an organization to adapt either in preparation, or in response to +perturbations. +Quantifying resilience principles. +Ten years ago, Biggs et al. [15] identified seven principles +for building resilient socio-ecological systems: +(1) maintain diversity and redundancy, +(2) manage connectivity, +(3) manage slow variables and feedbacks, +(4) foster complex adaptive systems thinking, +(5) encourage learning, +(6) broaden participation, +(7) promote poly-centric governance systems. +These principles already highlight the importance of a complex systems perspective, the role +of adaptivity and decentralized control. But now we provide a modeling framework for social +resilience where formal models allow to quantify the value of redundancy and connectivity using +multi-edge and multi-layer networks. They show how agents’ diversity, i.e., their heterogeneity, +their social impact and their signed relations impact social resilience. +From a broader perspective, our paper wishes to contribute to a better concept of resilience +management. This requires both an understanding of the system that should be managed and an +active involvement of those who are managing and those being managed. Social organizations +are a prime example for those systems. We are the system elements, the agents, of our own social +organization. We are in the position to change our organization to improve resilience. At the +same time, we are also affected by these changes, as well as by internal and external shocks. +Our modeling framework helps to raise attention for the role of diversity and feedback processes, +the power of decentralized network interventions and collective learning. In the end, however, it +depends on us how much of these insights can be implemented in our social organizations. +Acknowledgements +The authors thank A. Garas, D. Garcia, P. Mavrodiev and S. Schweighofer for early discussions. +44/54 + +F. Schweitzer, G. Andres, G. Casiraghi, C. Gote, +R. Roller, I. Scholtes, G. Vaccario, C. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Giona Casiraghi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Christoph Gote1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ramona Roller1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ingo Scholtes3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Giacomo Vaccario1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Christian Zingg1 1Chair of Systems Design,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Switzerland 2Complexity Science Hub,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Austria 3Department of Informatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' University of Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Switzerland 4Chair of Computer Science XV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Julius-Maximilians-Universität Würzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Germany Abstract Resilience denotes the capacity of a system to withstand shocks and its ability to re- cover from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We develop a framework to quantify the resilience of highly volatile, non- equilibrium social organizations, such as collectives or collaborating teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It consists of four steps: (i) delimitation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', narrowing down the target systems, (ii) conceptualization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', identifying how to approach social organizations, (iii) formal representation using a combi- nation of agent-based and network models, (iv) operationalization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' specifying measures and demonstrating how they enter the calculation of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our framework quantifies two dimensions of resilience, the robustness of social organizations and their adaptivity, and combines them in a novel resilience measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It allows monitoring resilience instantaneously using longitudinal data instead of an ex-post evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 1 Introduction Why do some social organizations succeed to persist and thrive in the presence of crises and shocks, while others fail under the same conditions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They have different levels of resilience that can be most generally described as a system’s capacity to withstand shocks and its ability to recover from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such a description already implies different features: (i) resilience is a systemic property, as opposed to a property of system elements, (ii) resilience is not restricted to a specific system, it rather seems to be a general property of different systems, (iii) resilience is described as a response to a shock, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', it can be only recognized in the presence of shocks, or perturbations, (iv) resilience is not a static property because shocks and recovery imply time dependent pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' ∗Corresponding author, fschweitzer@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='ch 1/54 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='00183v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='SI] 31 Dec 2022 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Resilience has to consider not only the magnitude of shocks, but also different types of shocks the system has to absorb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The same system can be robust to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the impact of an earthquake, but not to the spreading of a disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' More importantly, to be resilient a system also needs to have the ability to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A system may be robust even in the presence of various perturbations, but once it is impacted by a critical shock it will not recover, so it cannot be seen as resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Both the response to a shock and ability to recover are system specific and therefore difficult to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This also hampers a more precise or formal definition of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We should not expect that there is a universal concept of resilience, applicable to various types of systems [25, 116, 177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In fact, resilience concepts diverge across and sometimes even within scientific disciplines [6, 9, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We therefore do not provide a review of existing resilience concepts and refer to the literature already available [9, 53, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, with our paper we want to broadly inspire researchers from different scientific disciplines who already study social organizations by providing new modeling perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The analysis of collaborative teams and collectives is a core topic of social psychology [14] and organizational theory [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, case studies inform about, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', collective decisions, coordination and conflict resolution in teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But the models used are most often descriptive models, not generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Descriptive models include statistical models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', regression models, or database models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', conceptual entity- relationship models that indeed resemble our knowledge graphs (see Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In organizational psychology there are mental models of teams and team members to describe perceived relation- ships or the collective representation of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Descriptive models try to include as much detail as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But generative models try to include as much detail as necessary to generate a macro-social behavior from the micro dynamics of the constitutive individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This method- ological approach is advocated in analytical sociology [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Agent-based and network models belong to the class of generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Stochastic actor- oriented models (SAOM) [162] and exponential random graph models (ERGM) [95, 101, 133] aim at combining agent-based and network approaches [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They also belong to the class of data- driven models, using methods similar to logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' With their focus on link prediction to detect reciprocity, transitivity or homophily they are less suited to study systemic properties of social systems, such as resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Computational issues, in particular scalability, assumptions about utility maximization of actors and problems in model specification prevent a broader range of applications [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But there are better solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In this overview paper we want to sketch a framework how to utilize them for the study of social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This framework provides interfaces for mining larger and more fine grained data about interactions between individuals (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1), an analytically tractable network ensemble to avoid computational issues (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2), statistical methods to 2/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) infer signed relations and significant interactions from observed data (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2) and formal ways to estimate the social impact of agents involved in these relations (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But most of all, this framework allows to calculate robustness and adaptivity instantaneously, to estimate the resilience of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The attention is shifted from the micro configurations, the dyads and triads, towards the macro-properties of social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These are no longer reduced to topological features, but involve a dynamic component to describe the response to shocks, namely the potential for change in an organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' With this the formal modeling of social organizations can be moved to a new level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It will also impact research on resilience which is dominated by two paradigmatic views, engineering and ecological resilience (Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' So far, theoretical research on resilience and empirical research on resilience indicators have been largely segregated [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Many studies restrict their focus on specific systems, in particular engineered systems, ecological, or urban systems [43, 72, 79, 116, 157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There are also studies about the resilience of socio-economic systems and organizations [86, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As we detail below, they are of little help for the problems discussed in this paper, for two reasons: (i) When referring to social systems, most often our modern human society is addressed [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This bears a complexity way too large to be captured in a formal modeling approach and restricts the discussion to a discourse level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (ii) Our research interest are social organizations at smaller scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Organizational resilience studies have pointed out “factors” for improving the resilience of such systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', integration or redundancy [110, 158] or social capital [91, 127, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But they do not instruct us what to do if such factors shall be modeled and quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A major aim of this paper is to provide a framework to overcome this research gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To de- velop a broader foundation, we will specifically address the problems of conceptualizing social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' One main issue is their volatility, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the continuous change of their structure and dynamics that makes it difficult to define stability, to measure the impact of shocks, or to distinguish recovery from change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A second and probably more ambitious aim is to revise the premature anxiety-laden understanding of “shocks” and “breakdowns” that dominate the discus- sion of societal resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To cope with social organizations, we need to shift the focus from the fear to breakdown towards the faith to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 2 What do we know about resilience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In order to investigate the resilience of social organizations, we should first take a look at two of the most prominent resilience concepts in engineering and in ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They may provide already good starting points to formalize robustness and adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If so, we have to test weather such formalization could be utilized to model social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But even if that is not the case we learn from the shortcomings about requirements for social resilience concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This helps to further clarify the underlying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 3/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Questions Is there a difference between robustness, stability, and resilience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What is the relation between adaptivity and recovery?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How is robustness related to existing systemic risk measures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Should a resilient system always return to equilibrium?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 Robustness and adaptivity Constituting dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' While the concrete meaning of resilience may vary across scientific disciplines, there are also conceptual commonalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As pointed out in Figure 1, resilience bears relations to concepts of robustness and adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The former relates to the property of a system to withstand shocks, the latter to its ability to overcome their impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Robustness represents the structural and adaptivity the dynamic dimension of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' However, we need to understand how they constitute resilience as a function defined on these two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience Robustness: Structural Absorptive [54, 174] Buffer [87] Coping [90] Redundant [19] Robust [19] Adaptivity: Dynamic Adaptive [54, 90] Learning [87] Recoverability [54] Self-organization [87] Transformative [90] Figure 1: Examples of resilience factors used in the literature, which we assigned to the struc- tural and the dynamic dimension of resilience without claiming consistency across different systems and shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We argue that both dimensions need to be combined to explain resilience, but most often they have been studied as stand-alone concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, the robustness of inter- connected systems is defined with respect to the failure of nodes or links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Robustness measures then estimate the size of failure cascades after a shock [4, 12, 37, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This approach uses the complex networks perspective that we also utilize in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But it assumes that simple 4/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) topological features, like connectedness, are sufficient to describe the robustness of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This leaves out the dependence on individual properties which becomes a problems when applying the approach to social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In addition to topological robustness, other robustness measures build on dy- namical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It analyzes how small perturbations of an assumed equilibrium state affect a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If the system is able to regain its equilibrium state, it is robust at least against this type of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This approach has several limitations when applying it to social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It requires to know the dynamics of the system, to estimate its response to shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Further, the underlying assumption of different types of equilibrium states can hardly be justified for very volatile social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptivity is often simply understood as dynamics, which neglects the quality of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptive systems do not simply react to a shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their response aims at preserving the system’s functionality, to ensure its persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Recovery after a shock therefore implies a certain directedness which depends on the type of shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Most important, the system needs to have several options to adapt even to unexpected challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, adaptivity should estimate how many options exist in a given situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This points towards the problems of quantification and measurement discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Creative destruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Additionally, simply adopting existing concepts of robustness and adaptivity may lead to misconceptions about social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Quite often, the understanding of these concepts builds on a negative perception of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This ignores the role of “creative destruction” that, according to the economist Joseph Schumpeter, is instrumental for renewing and further developing the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Stable systems do not evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, challenging their stability is one of the driving forces of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This also regards social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The leave of established members is not only a threat, it is also an opportunity for newcomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Creative organizations often respond to shocks with innovations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, the discussion about robustness and adaptivity should not just focus on maintaining the status quo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience means to cope with change in a sustainable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 Infrastructure systems Critical functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' With respect to critical technical infrastructures such as power grids or communication networks, the design of resilient systems has been studied extensively [4, 21, 83, 161, 164, 166].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Here resilience, sometimes also called “resiliency”, refers to a system’s capacity to “maintain an acceptable level of service in the presence of [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' ] challenges” [161].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such challenges 5/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) include software and hardware faults, human mistakes, malicious attacks or large-scale natural disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This concept of resilience is goal-oriented, preserving a system’s functionality after the shock as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A power system has a defined functionality which remains as long as no critical shocks are caused by internal malfunction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', lack of maintenance, or by external disruptions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', an earthquake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If the system has to “absorb” a shock, this functionality is partially or entirely destroyed because the system lost its robustness, to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Time Critical Functionality Plan Absorb Recover Adapt Figure 2: The common perception of resilience for engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We note that functionality is assumed here as a function of an underlying network structure, in this case the power grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Consequently, the focus is on the robustness of this under- lying structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptivity refers to possible changes in the processes running on this structure, and not to the structure itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This can be illustrated by the following example: During the 9/11 attacks in 2001, the Internet infrastructure in downtown Manhattan was largely destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This posed a severe shock to the global Internet because of (i) the termination of transatlantic cables in the basement of the World Trade Center (WTC) and (ii) the failure of the major Internet exchange point NYIIX next to the WTC, which was responsible for 70 % of transatlantic traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Despite these combined failures, the attack caused only minor disruptions and the global routing infrastructure continued to operate normally within a few hours [46, 164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This resilience of the Internet was not obtained by the robustness of the hardware, which was heavily affected in the above example, but by the adaptive capacity of the dynamics on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' However, redundancies built into the underlying network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', a level of robustness, were essential to allowing for the quick rerouting of the communication flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, a minimum level of robustness can be a precondition for the adaptive capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 6/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What distinguishes resilience from robustness is the ability to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Infrastruc- ture systems do not recover by themselves [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They have to be rebuilt by human activity, reestablishing the functionality and often even improving it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Thus, attributing the ability to recover to the infrastructure would reduce the concept of resilience to absurdity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Every bridge that was rebuilt after a collapse would then be “resilient”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In engineering, hence resilience is often used as a synonym for robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For example, Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' [43] define a “resilient reactor” as one achieved by “[using] a tank designed to withstand high pressures and temperatures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' However, once a tank has exploded, it has no ability to recover a functional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Systemic risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Quantifying the impact of shocks requires to model the system’s response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Only such models allow to estimate whether large parts of the system will be destroyed, which is commonly denoted as systemic risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Based on the theory of extreme events [41], one can calculate probabilities for rare, but severe external shocks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', floods or earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Using engineering knowledge about materials [120] and constructions, then allows to propose critical values for system properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the minimum width of walls, or the maximum height of buildings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Systemic risk, from this perspective, is reduced to the risk that an external event with a critical magnitude may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Consequently, robustness is defined only with respect to such rare events, for instance, “the ability of power systems to withstand low-probability high-impact incidents in an efficient manner” [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Loss estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In infrastructure systems, such as transportation networks, it is common to measure robustness by employing loss estimation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These models evaluate the ability of the system elements to withstand a given level of stress or demand without suffering degradation or loss of function [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Quantification allows to control the robustness of such systems to some degree, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', counterbalancing shocks by a central control station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Engineered systems are designed systems, top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their functions are clearly defined, and therefore the impact of a shock can be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Failure cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The focus on external extreme events neglects another major cause for systemic collapse, namely the amplification of a few small failures inside the system into a failure cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If such cascades reach a critical size, they can also destroy the system from inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Robustness in this case has to be defined with respect to (i) the probability that a few system elements fail, and (ii) the mechanisms that can amplify such failures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', by redistributing load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Both may depend, in addition to internal conditions, also on external feedback processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Large scale blackouts in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' have been explained by such failure cascades in the power grid [4, 178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To study failure cascades we need an appropriate representation of the engineered system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For power grids, communication networks or transportation infrastructure, the network approach 7/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) is used most often [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' System elements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', transformers or responders, are represented by nodes, and their physical connections by links in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Many models of failure cascades test the robustness of networks by removing nodes or links either in a random or a targeted manner [21, 170] and measure whether the network breaks down into disconnected components after such attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A modular network is robust if shocks trigger small failure cascades [8, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Targeted attacks, however, usually result in large failures as the network becomes disconnected [10, 156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Amplifying mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Removal tests reduce the problem of systemic risk to mere topo- logical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To understand amplifying mechanisms inside a systems, we need models for the dynamics inside the nodes, but also models for the exchange of quantities between nodes [22– 24, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That means we need a framework that couples the dynamics of the network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the failure of nodes or links, with the dynamics on the network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the rerouting of communication or the redistribution of load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Without such a framework, we cannot understand the robustness of the system, even less its adaptivity, and hence, its resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Extreme value theory [35] can at best estimate the probability that large-scale failure cascades happen, whereas network models can provide an explanation why they happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience, as a systemic property, cannot be reduced to the robustness against extreme shocks, it has to be derived from a broader perspective that helps to understand why and when small events can be turned into big disasters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3 Ecological systems The classic view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Historically, the notion of resilience first appeared in ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ecosystems constantly change under dynamic processes of renewal and reorganization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, resilience concepts do not primarily focus on robustness as a static property of the system, but rather on the adaptivity as the dynamic component of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For Holling [81], resilience is based on the “ability of a system to return to an equilibrium state after a temporary disturbance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Because shocks and perturbations are unavoidable, emphasis is on the survival, or the persistence, of the ecological system, regardless of the impact of a shock [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such a notion of resilience essentially builds on the theory of dynamical systems and its concept of “stability”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In different scientific areas, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', nonlinear dynamics, control theory, physico-chemical reaction kinetics, or biological pattern formation, a system is said to be stable if it returns to the equilibrium state after a shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Approaches to assess resilience in biological and engineered systems are based on this idea [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Most systems are in fact metastable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', they are stable in the presence of small perturbations, but become unstable if these perturbations exceed a certain critical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Then, instead of returning to the previous equilibrium state, the system moves away from it, possibly to another equilibrium state, as Figure 3 indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This new equilibrium state corresponds to a “different” system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', to a system that has evolved and adapted to the new situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This aspect has been studied as robust adaption, combining the notions of robustness and adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There are analogies to concepts of phase transitions in physics and chemistry and regime shifts in social and biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' shock low recovery rate State Potential Figure 3: Mechanical analogy for a metastable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This figure is adapted from Fig- ure 1 in [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The meaning of recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In the resilience concept of ecological systems recovery may have different connotations dependent on whether a system returns to a previously attained stable state or to a different one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Even a previous state would be gradually different because evolving systems would never reach identical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In more general terms, resilience can be implicitly defined by “the amount of disturbance that an ecosystem could withstand without changing self-organized processes and structures (defined as alternative stable states)” [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To capture differences in the quality of a resilient ecosystem, the additional factors of ecological “vulnerability” and “sensitivity” have been proposed [40, 115, 135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They consider that species are more sensitive to perturbations in certain time periods or under special conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This complements the notion of “stability” of ecosystems or “robustness” of engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What matters for ecological resilience is the time scale of recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If it would take forever to return to a previous state, the system is not resilient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, “the inverse of 9/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) the length of time required for an ecosystem to return to near-normal” [70] is often used as a resilience measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But resilience itself can also be timescale specific [25, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' On a daily timescale, the system may return to a near-normal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But on a time scale of decades, the system may lose this ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, resilience cannot be reduced to some concept of “stability”, the dimension of adaptivity is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilient ecosystems maintain a critical balance between stability and instability, the former providing a level of robustness against small perturbations, the latter allowing them to adapt to changing environmental conditions [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Connectedness and potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To further grasp the generic mechanisms underlying ecolog- ical resilience, two system properties have been proposed: potential and connectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' “Potential sets limits to what is possible - it determines the number of alternative options for the future” [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, potential resembles our notion of adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Connectedness, on the other hand, is related to robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' “Connectedness is assumed to increase over time, leading to high internal control and limited potential to cope with disturbances” [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is an important remark as it points out to the fact that robustness above a certain critical value may have a negative impact on resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' “When connectedness is low, resilience is high because the system can vary over a wide range of states and respond to disturbances in many different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' When connectedness, however, is high, ecosystem resilience is low because the system is more tightly organized and has fewer options for responding to disturbances” [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We will return to this argument when discussing our own concept of social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It was argued that resilience changes in a cyclic manner because an increase in connectedness may lead to a breakdown of the system: “Naturally such over-connected systems crash into a release period, where they have the potential to reorganize, thereby coping with dis- turbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This development is believed to be cyclic” [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We note that the ability to reorganize is given only if during “release phases” of low connectivity the potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the adaptivity, is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is in line with our arguments for the recovery of social organizations discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' So far, adaptive cycles have not been found empirically: “Because of its very general nature, the concept of the adaptive cycle should be considered a metaphor [25] or thinking tool rather than a testable scientific theory” [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We may add here that with our approach we move the adaptive cycle from a metaphor to a testable concept, which is also accessible to formal modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Couplings between adaptive cycles on different temporal and spatial scales may lead to a nested hierarchy, called panarchy [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 10/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Losing resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead of specifying properties for resilient states, we could address the complementary question, namely about the properties that indicate a loss of resilience or about states that need to be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This reveals critical conditions for stabilizing feedback cycles, critical magnitudes for perturbations or critical levels of diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To identify factors that lead to the loss of resilience, non-parametric regression models or machine learning tools, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', symbolic regression, can be used to analyze data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Early warning signals for losing resilience can be obtained from time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A slower recovery rate from perturbations, known as critical slowing down [104], right before a tipping point is a possible indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Also an increase in the auto-correlation of systemic variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', order parameters, indicates the vicinity of a regime shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Coupling to social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Insights into ecological resilience are limited if the dynamics of ecological systems is predominantly driven by the coupling to the human sphere [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' On the global scale social and ecological systems are, in fact, coupled inseparably, mainly because of hu- man interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In recent years, this has triggered integrative research studying the resilience of social-ecological systems [51, 175, 176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It relates ecological resilience with the resilience of societies responding to environmental challenges that originate from the ecological systems into which they are embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Conclusions Different scientific disciplines have their own understanding of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We should not expect to find a universal resilience measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The key question is [25]: Resilience of what to what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Answering this question requires to have a model of the respective system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience concepts for social organizations may benefit from ecological concepts, because issues of time scales, different equilibria and adaptive cycles are already addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 3 Why are social systems different?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Existing concepts of resilience from engineering or ecology do not seem to provide the best basis for social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Although formal approaches exist, we cannot simply reuse them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Before developing suitable alternatives, we may first clarify what makes social systems different from other types of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This leads us to more fundamental questions about defining systems and models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What seems to be a detour at this point will later allow us to better ground our notion of social resilience from a methodological perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 11/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Questions What types of social systems do we want to investigate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What characterizes social organizations and collectives?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What is the focus of organizational resilience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What modeling consequences entails the complex systems approach?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 Social organizations No model of society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The need to understand social resilience is often motivated by the many crises that our societies face, today [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their vulnerabilities have been widely recognized, ranging from pandemics to political polarization, from climate change to budget crises, from infrastructure breakdown to poverty migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Consequently, the resilience of societies cannot be decoupled from the resilience of ecosystems, political systems, infrastructure systems, financial and economic systems, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' While these connections cannot be denied, they raise a methodological question that, unfortunately, is not addressed with the same emphasis: How should we model all of these interdependencies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Four steps towards a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We aim at a quantitative understanding of social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, we specify the resilience problem in a tractable manner in four steps that are sum- marized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The first step is delimitation: Which types of systems should be specifically investigated, and which ones not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This question is discussed in the remainder of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2, we further distinguish our problem from existing concepts of organizational resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The second step is conceptualization: Which approaches should we use to describe social systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Which of the many possible features will we focus on?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Only this clarification will enable the third step, representation: To build a model means to represent the system and its elements in a formal manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 we introduce different network concepts as candidates to represent properties of social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In Section 5, eventually, we address the fourth step, operationalization: How do we specify measures such that they can be calculated?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Again, we introduce different solutions to choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Finally, in Section 6, we comment on the data required for the fourth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Systems of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For methodological reasons we distinguish social resilience from the resilience of societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The latter is not the target of our formal modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Society is a system of systems, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To understand the resilience of societies requires modeling the interaction of these systems [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 12/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Step 1: Delimination Type of system: Social organization Example: Collaborative teams Focus on relations between individuals Step 2: Conceptualization Complex adaptive system Resilience as an emergent property Self-organization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' no time scale separation Step 3: Representation Combine agent-based and network models Multi-edge networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' statistical ensembles Signed relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' propensities of interaction Step 4: Operationalization Social impact and importance of agents Structural/topological robustness Resilience: Robustness & adaptivity Data Collection Extract interaction data from repositories Network regression: Propensities Analyzing temporal data: Causal relations Figure 4: Our framework to quantify social resilience Models for systems of systems exist only in rudimentary form [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Particular emphasis was given to the coupling between socio-economic systems and ecological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' More recently, also the coupling between climate systems and socio-economic systems and/or ecological systems is captured by formal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Most of the systemic relations that constitute a “society” are not formalized at all [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Collectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We have to restrict our investigation of social resilience to clearly defined social entities rather than “social systems” in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our focus are social organizations, or collectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' With the term social organization we refer to formal or informal groups of interrelated individuals who pursue a collective goal and who are embedded into an environment [80, 124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For illustrative purposes, our running examples are project teams, in particular teams of software developers [80, 139, 182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These teams face numerous shocks during their development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Compet- ing products, technical evolution, organizational problems, lack of motivation or resources put up challenges and let them fail quite often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their common goal, namely to develop a software for a certain scope, is important to distinguish this type of social system from a collection of a hundred persons who, for example, use the subway without being interrelated or contributing 13/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) to a common good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Collective goals in most cases go beyond the mere survival [47, 124], which would characterize ecological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Beyond individual resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The term “social” specifically refers to human individuals in the following, although many animal societies also have a remarkable degree of social or- ganization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A broad range of psychological studies focuses on the resilience of an individual [7, 45, 94, 171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Given that we want to understand resilience as a systemic property, psychologi- cal resilience would require us to model the individual as a system [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That could be possible, but is not aligned with our aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, we are interested in collectives of many individuals, for which we use the term social organization synonymously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We assume collectives of the order of 102 individuals, small enough that the impact of a few individuals can still matter, but large enough to distinguish individual and collective in a meaningful manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This implies that resilience, as a systemic property, is neither identical to, nor the mere combination of the psychological resilience of its members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 Organizational resilience Different from engineering, in psychology and organizational science the concept of resilience focuses on the dynamic component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', adaptivity as the system’s ability to cope with shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But in social organizations resilience does not require to return to a previous state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, resilience is generally seen as “the ability of groups or communities to cope with external stresses and disturbances as a result of social, political and environmental change” [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We note that in such a definition shocks primarily result from other systems an organization is embedded in, rather than from internal processes, which is the main focus of our concept of social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Community resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Examples for this outside orientation are studies of citizen commu- nities in urban or rural areas [2, 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their response to natural hazards or disasters [19, 38] or to climate change [111] is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Community resilience generally refers to the ability of communities to cope and adjust to stresses and to engage community resources to overcome adversity [121, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' An organization should not only persist after a disturbance, but also manage to strengthen its capability for future adjustments [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The ability to transform challenges into advantages is known as transformational resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Whether a new state is resilient may depend on specific positive outcomes that need to be achieved [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, resilience comprises more than just persistence: “The capacity of actors to access capitals in order to – not only cope with and adjust to adverse conditions (that is, reactive capacity) – but also search for and create options (that is, proactive capacity), and thus develop increased competence (that is, positive outcomes) in dealing with a threat” [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 14/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Resilience factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It is an open question why and how some organizations manage to thrive and enhance core capabilities when faced with a crisis, while others fail [173].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Recent research highlights three factors that influence the resilience of a social system: (i) its vulnerability (or susceptibility) to disruptions, (ii) its level of anticipation, foresight, or situational awareness for such vulnerabilities, and (iii) its adaptive capacity, flexibility or fluidity which allows to mitigate vulnerabilities or respond to disruptions [62, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To better understand the role of these factors, most empirical resilience studies have followed a “hindsight approach”, focusing on organizations which have recovered from a shock and trans- formed crises into advantages [55, 103, 132, 180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For such social organizations, Sutcliffe and Vogus [165] define organizational resilience as the ability to maintain a “positive adjustment under challenging conditions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptive capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A system’s capacity to adapt in a constantly changing environment is also referred to as adaptive capacity [56, 160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' From a social science perspective, the adaptive capacity is expressed in a number of different ways, for instance in terms of the ability to learn and store knowledge, the ability to anticipate disruptive events, the level of creativity in problem solving, or the dynamics of organizational structures [52, 160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Some of these aspects have been assessed by means of survey research designs, such as the learning capability [36], situational awareness, creativity [114], or the fluidity of structures [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Missing macro-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Most notions of resilience proposed above reveal their limitations when it comes to quantification [100, 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Two problems need to be solved, (i) to define a measure for resilience that considers also the dynamics of the system, and (ii) to measure the defined variables against available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Many studies of social resilience, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', in disaster management [38, 88, 109, 121, 129, 136, 168], monitor resources for basic needs or survey social well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But we lack macro-variables to describe social organizations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', to measure their adaptive capacity and their elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such variables exist in economics, for instance productivity and efficiency measures, but also for ecological systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', biomass production or recovery rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In engineering functional resilience can be computed through the integral below the function of performance [20, 129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In absence of these variables, tools to derive early-warning signals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the critical slowing down or the increase of auto-correlations mentioned above, cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, one of our aims is to provide such macro-variables for robustness and adaptivity and to show how they can be monitored over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These variables will help separating the ability to resist a shock from the capacity to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 15/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3 Conceptualization In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 we have already processed the first step towards a model of social resilience, namely delimitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The second step, conceptualization, now requires us to specify how to approach highly dynamic social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This particularly regards the modeling framework that shall foster our understanding of the micro-processes to explain social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Different conceptual frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ecological systems are often modeled using concepts from system dynamics [147], where species are described by densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Interactions between dif- ferent species in a food web are then formally expressed by coupled differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This approach does not focus on individuals, but mostly this is also not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Models of engineered systems often use concepts from control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This allows to steer system elements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', transformers, from a central perspective, but requires to have precise models of such elements and their relations to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is often the case because engineered systems are designed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Both of these modeling approaches cannot be applied to social organizations the way we see them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They are much more volatile, more adaptive in response to shocks and, most importantly, have no defined reference state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, in the following we specify what concepts we will use to describe their structure and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We start from the insight that social organizations are complex adaptive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They comprise a larger number of interacting system elements, commonly denoted as agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Taking the complex system perspective implies that systemic properties, such as resilience, need to be understood as emerging from the interaction of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, we have to develop a bottom-up perspective for social resilience, starting from the micro, or agent, level rather than from the macro, or systemic, level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is in line with the method- ological principles of analytical sociology [75], which aims at explaining macro-social phenomena from the micro-processes that generate them [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Agent-based and network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To formalize both the dynamics of agents and their relations, we combine agent-based modeling with temporal multi-layer network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The agents, as the nodes of the network, are characterized by different properties, such as status, roles, knowledge, opinions, which depend on other agents and can change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Furthermore, agents are heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They can be of different types and even within one type their properties are not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, agents’ function and efficiency in solving tasks vary across agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We therefore have to model agents explicitly, to overcome approaches solely based on topological features to describe the functioning of a social system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 16/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Agents’ interactions and their social relations are captured in different network layers, which evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This requires us to also model interactions explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In particular, we have to distinguish random from meaningful interactions and to find ways to infer roles and social relations from interaction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This paves the ground for a statistical approach based on net- work ensembles, which also provides the interface for data-driven modeling, which we discuss in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Finite systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our approach explicitly addresses the finite number of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Focusing on larger systems would have the advantage that we could calculate simple statistical measures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', averages, to overcome details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For the type of social organizations discussed here details matter and are therefore explicitly addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We need to consider individuals and discrete events instead of continuous variables characterizing a whole system, such as densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Work teams, online chat groups, or school classes differ from large social systems not only in their interaction structures or perceived goals, they also differ in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Emergent phenomena of social systems, such as coherence or cooperation, depend on size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Large systems necessarily behave differently from smaller ones because regime shifts or phase transitions can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore our models for social resilience are not expected to describe very large social systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', political parties or urban populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In small systems, such as collectives, stochastic influences can have a larger relative impact on the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Further, path dependent processes in the evolution of these systems cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Local effects, such as neighborhood relations become important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, known limit cases of formal modeling, such as the mean-field approach in which all agents interact in a similar manner, cannot be readily applied to collectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, we need to build agent-based models that reflect agents’ heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Self-organized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' An important difference to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', technical systems, is the level of adaptivity in social organizations which cannot be simply reduced to “dynamics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, emerg- ing structures in social systems feed back on the interaction of agents and cause further change, often denoted as second-order emergence [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is related to co-evolution and learning, which occurs on the individual and on the organizational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The outcome of these dynamics can hardly be predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Social organizations cannot be com- pletely controlled and agents cannot be forced to behave in a predictable manner when facing changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead of central control distributed influences and self-organization play a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In general, agents respond to changes both in intended and in unintended ways [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This makes the response of social organizations to internal or external shocks so difficult to model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 17/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Time Critical Functionality Plan Absorb Recover Adapt Figure 5: Problems defining a resilient state for volatile organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' No separation of time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Social organizations are very volatile systems which makes it almost impossible to define a reference state, as Figure 5 illustrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' More importantly, we have to account for the fact that the absorption of shocks and the subsequent recovery cannot be clearly separated as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, changes of robustness and adaptivity follow instantaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is a noted difference to ecological systems where the time scale of adaptivity is usually much larger and an out-of-equilibrium state can be clearly separated from the equilibrium and the relaxation time scale is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Conclusions Our notion of social resilience focuses on social organizations and teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To develop a formal model we adopt the viewpoint of complex adaptive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Modeling resilient societies would require a system dynamics approach, instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Existing concepts of organizational resilience mostly take a management perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We aim instead to model resilience bottom up, as an emerging property of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our framework will combine agent-based and network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 4 How shall we model social organizations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We continue to go from the general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', delimitation and conceptualization, to the particular, now addressing the problem of system representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Once we agreed upon the complex systems approach with its agent-based and network models, the biggest hurdle is to turn these concepts into formal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead of presenting just one solution, we have to prepare for a broader perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The following descriptions should therefore be seen as alternatives for choosing formal approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2, when we introduce network ensembles, we want to highlight possible options of utilizing ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 18/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Questions Why is system representation a central problem for modeling?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What are the differences between the various network representations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Why do we need a network ensemble?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How shall it be used?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What types of dynamics do we consider for social organizations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 Network representation Various options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There is not the one way to construct a network, not only because of different topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There are different types of networks, as we outline below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Which network type is most suited to represent the organization depends on the context and the available information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' One could argue that we should indeed start our discussion with the latter, to explain what data we got.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is denoted as the supply driven approach in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We instead follow the demand driven approach, which requires us to first identify what data we will need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, before collecting data from an empirical system, a suitable formal system representation has to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Only then the question about the minimal set of data needed should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Link properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Networks are one way of representing complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The nodes of the network are the agents, and links aij between nodes i and j represent their relations or interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 6 shows one example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The network approach focuses on the topological structure, which can be conveniently summarized in an adjacency matrix A with the entries aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Links between agents are usually directed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', agent i assigns a task to agent j and aij ≠ aji, repeated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', there are multiple links between the same pair of agents, aij ≥ 1, and time bound, aij(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', they have to respect causal ordering or bursts of activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Network inference builds on the assumption that the topological structure encodes information about agents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Utilizing this information could reduce the model complexity because it allows for operationalizing the structure and dynamics of social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But studying the network topology would reveal hidden information about individuals and collectives only to some degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, the network approach has to be extended by explicit models of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Links versus signed relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The network reconstruction is most often based on inter- action data to determine links, aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Interactions may be frequent and short-lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What matters for the resilience of organizations are rather the social relations ωij between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These are 19/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) 72 69 58 95 52 71 6 73 42 79 49 4 53 107 91 92 76 46 32 59 93 40 83 81 39 100 66 70 68 89 18 104 30 65 44 101 103 88 27 56 20 63 57 7 28 77 21 85 80 67 108 23 24 50 48 55 96 61 78 25 9 51 94 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', they have positive or negative signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It takes time to establish social relations and they usually change on a longer time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Compared to interaction data, data about signed relations is rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore we need methods to infer signed relations from interaction data, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Signed relations crucially impact the robustness of a social network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The theory of struc- tural balance [64, 73] considers triads involving three agents (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A network is assumed to be robust, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', stable, if it contains balanced triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To determine the balanced state, the classical approach only takes the signs of the signed relations into account, Sijk = sign(wij) sign(wik) sign(wkj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If Sijk = 1, triads are balanced, if Sijk = −1, they are unbalanced and have the tendency to change into balanced triads as Figure 7 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The line index [73] mea- sures the minimal amount of signs that need to be changed to turn all unbalanced into balanced triads and can therefore serve as a measure of structural robustness, which is explained later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Bipartite networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' One of the main challenges in modeling social organizations comes from the vast heterogeneity not only in the agents’ properties, but also in their interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The notion of a link, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', a direct interaction, is already an abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Taking the developer example, collaboration means that two developers work on the same code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This would be most appropriately represented as a bipartite network between different entities, the developers and the pieces of code (see Figure 8 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The collaboration network then is a projection, where developers have a direct link if they have changed the same piece of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A second network results from the projection on the code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' two pieces of code are connected if they were changed by the same developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 20/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Figure 7: Unbalanced triad and the three ways to obtain a balanced triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Multi-edge networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If two developers collaborate on more than one piece of code, nodes in each network projection can be connected by more than one link, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', we have a multi-edge network (see Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' From this network we are able to construct an important network ensemble, as we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 P1 P2 P3 P4 P5 P6 P7 P8 (a) D1 D2 D3 D4 D5 D6 D7 D8 D9 D10 (b) P1 P2 P3 P4 P5 P6 P7 P8 (c) Figure 8: (a) Bipartite network of developers (P) and software code (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (b) Projected collabo- ration network of developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (c) Projected network of code changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Knowledge graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Following these considerations, the starting point for representing or- ganizations by means of networks is not the social network between agents, which is already a reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead we have to start from a relational graph, also known as knowledge graph, that visualizes the various ways of connecting individuals, as shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 21/54 2 + + Paradise State .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 k + 2 + + + 1 k k 1 Polarized State + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 kF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) User Repository File Organization Commit Addition Deletion Issue fork star watch own own belong to follow link to make contain contain modify modify consist of consist of open comment on close link to belong to belong to Figure 9: Relational graph of software development activities on GitHub Multi-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' From a knowledge graph we construct different projections, each of which creates its own network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These networks are combined in a multilayer network, as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In each layer the nodes and their interactions are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If the nodes are the same in each layer, but the links represent different types of interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', friendship, work relations) this is known as a multiplex network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, we have now intra-layer links within each layer and inter-layer links between layers [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The multilayer network is accessible to mathematical investigations, by representing the topo- logical structure as tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This allows to apply methods of spectral analysis [3, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A noted limitation of networks is the decomposition of any type of interactions into bilateral interactions between two agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, in a group of five agents this proce- dure results in ten links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To overcome this limitation in modeling group interactions, we resort to hypergraphs [13, 125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is a special type of higher-order networks, in which higher-order nodes contain groups of simultaneously interacting agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Links between higher-order nodes then capture group interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Similar to multi-layer networks, higher-order networks can have levels of increasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The first order would be then the standard network, the second order level contains groups of two agents, the third order groups of three agents, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This way hypergraphs allow to model 22/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Figure 10: Multilayer network with intra-layer and inter-layer links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' interactions between groups of different sizes by means of inter-layer links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For the formation and the dissolution of groups, however, more refined dynamic models are needed [50, 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 Network ensembles Probabilistic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A network representation of the collective constructed from avail- able data will be only one possible realization and not necessarily a very typical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ideally, we would need a probability distribution that assigns to all possible networks a probability to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such a network ensemble is largely determined by the constraints of agents to form links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 11 shows sample networks from such an ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 11: Six networks sampled from a network ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They look similar, but differ in their details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If no link constraints are taken into account but only the total number of nodes, n, and links, m, we would arrive at a very large ensemble of random networks that all have the same n and m and the same probability to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The network constructed from data will be part of this ensemble, but it is statistically indistinguishable from the other networks, most of which will look 23/54 VF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, we need to incorporate more information to restrict the network ensemble, and to increase the probability for our reconstructed network in comparison to others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Generalized Hypergeometric Ensemble of Graphs (gHypEG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To model multi-edge networks characterized by heterogenous constraints, we have proposed gHypEG [30], a broad class of analytically tractable statistical ensembles of finite, directed, and multi-edge networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It introduces dyadic link propensities Ωij, which capture the preference of nodes to form links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Precisely, the ratio Ωij/Ωik is the odds to draw a link (i,j) rather than a link (i,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The propen- sities reflect social mechanisms such as homophily or reciprocity [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Furthermore, gHypEG can incorporate formal assignments to classes or communities [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To do so, it employs propensities ΩB kl for links between nodes i,j that are in different “blocks”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', communities, k,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' gHypEG has the benefit of being defined by closed form probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Thanks to this, we are able to calculate the weights for all incorporated features by means of efficient numerical Maximum Likelihood Estimation (MLE), without the need of expensive Markov Chain Monte Carlo (MCMC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Network regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The challenge to obtain the propensities Ωij can be mastered by means of a multiplex network regression [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Each network layer l encodes different types of known relations between agents as explanatory variables (see Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The influence of each layer on the interaction counts as the dependent variable is then determined by fitting the Ωij such that the observed network has the highest likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In other words, the optimal propensities are proxies for the constraints that shape the network ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As an added benefit of the method, one can test the statistical significance of the explanatory variables for the observed interactions [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 12: Illustration of the network regression method [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 24/54 Frienashiip nteractionsF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Potentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' With the calibrated propensities, gHypEG allows to calculate how many possible configurations of the observed network exist, given the constraints for links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This issue becomes of importance if we later want to quantify adaptivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the ability of an organization to attain different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Then we need to know not only the number, but also the diversity of possible network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This information is aggregated in a new measure, potentiality which is based on the normalized Shannon entropy [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Importantly, the calculation is feasible without computational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The larger the potentiality, the more alternatives an organization has to respond to shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We discuss below how this will impact the organization’s resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Significant relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' “Social” is not “random”, therefore, social relations should significantly differ from random interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To test this, we filter the adjacency matrix with the observed number of interactions, ˆ aij, using a significance threshold α and our probability distribution for the network ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If Pr(Aij ≤ ˆaij) > 1 − α, links are significant [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 13 demonstrates that removing insignificant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', random, links from the network has a considerable impact on determining, for instance, communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (a) (b) Figure 13: Community detection of a social network considering (a) all links, (b) only signifi- cant links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' [32] If the observed network is not expected from the network ensemble, we have to apply an itera- tive procedure, to refine the probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In a first step, we measure the significant deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This additional information is used in a second step to update the constraints for the network ensemble, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', to generate a new ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The iterative procedure reveals what information is relevant to explain the observed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25/54 68 G 5 + % % 36 25 31 73 15 61 2 14 30 55 86 24 88 76 39 10 81 67 84 27 54 32 35 28 0g 26 0 51 18 6 G 56 8 8 25 9 87 % 22 % 22 0 % te 21 73 0 63 74 3 51 0 17 26 57 23 1 59 84 30 81 91 0g 64 43 11 31 10 40 71 76 47 18 56 35 % 心 3 3 2 3 m 2F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Signed relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Eventually the probability distribution for network ensembles allows to test whether the number of observed interactions exceeds expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This issue is important if we wish to map interactions to social relations which have positive or negative signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Empirical studies have shown that more interactions indicate a positive social relation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', a stronger friendship [85, 89, 169], whereas less interactions indicate a negative relation which causes e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', avoidance behavior [74, 96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' aij 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='15 0 5 10 15 20 Aij Pr(Aij) Figure 14: Determining overrepresented interactions [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As illustrated in Figure 14, we infer the weight and the sign of the social relation between two agents from ωij = Pr(Aij < ˆaij) − Pr(Aij > ˆaij) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This procedure allows us to obtain from a multi-edge network of observed interactions a network with signed relations, as shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The weighted signs, on the other hand, will enter the formalism to determine the social impact of agents in a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (a) (b) Figure 15: Multi-edge network (a) and the resulting network of signed relations (b) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3 Dynamics of social organizations So far, we have narrowed down our investigations to social organizations of a particular type (delimitation), which are modeled as complex adaptive systems (conceptualization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For the rep- 26/54 十 +F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) resentation we have chosen the network approach, which offers a great variety of network types, but also a statistical description using network ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This discussion has focused only on the structure, but not on the dynamics of these networks which will be done in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Further, we have not addressed yet the agents and their properties which will follow in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Concurrent changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Models of complex socio-economic systems often use the concept of separated time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The dynamics on the faster time scale is assumed to reach an equilibrium state, which allows to describe the dynamics on the slower time scale as a sequence of different equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Similar approaches are used to separate different network dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance changes of the network topology are assumed to be slow, therefore the fast dynamics running on the network can neglect the changing topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As already pointed out, we cannot use such assumptions to model the dynamics of collectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, the different processes discussed below should be seen as concurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This leads to a number of issues, such as overlapping or sliding time windows, choice of the appropriate time scale for aggregation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', which are not discussed here, but should be kept in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Entry and exit dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The most visible changes regard the network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For social organizations we have to consider an entry and exit dynamics of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', newcomers connect to the network [146, 149], whereas incumbents may leave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This implies also the addition and deletion of links, as shown in Figure 16 for the case of a multiplex network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Social organizations often exhibit a life cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', a predominant growth of both nodes and links in early stages is followed by a saturation and a decline caused by many nodes leaving [60, 150, 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' time .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' i j ts3 ts2 ts1 Figure 16: Coupled growth dynamics in a two-layer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Intra-layer links are between nodes of the same color, inter-layer links between nodes of different color [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These processes do not occur at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Further, they impact the collective as a whole as well as individual agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Newcomers may not be able to connect to core nodes initially and thus connect to the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their integration into the collective may improve over time, as can be 27/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) measured by their coreness [58, 151, 155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If core nodes leave, this may trigger cascades of other nodes leaving as empirical and simulation studies have demonstrated [34, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Restructuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Next to the addition and deletion of nodes and links, the rewiring of links between nodes plays a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their impact can be measured by tracking changes in the global and local topological measures discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Structural changes often reflect changes in the organization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', in responsibilities, hierarchical positions and roles of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 17 illustrates such restructuring processes for a developer collective in which a central developer has assumed the main responsibilities for task assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Already the visual inspection makes clear that this has lead to considerable problems in the robustness of the collective, which eventually lead to a collapse and the establishment of more resilient structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (a) (b) (c) Figure 17: Topological change of a collaboration network of developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Aggregated interac- tions (a) before October 2004, (b) between October 2004 and March 2008, (c) after March 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' [183].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Whereas the dynamics of networks addressed above changes the topol- ogy, the dynamics on networks captures interactions between agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These can be exchange processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', load redistribution in case of an agent’s failure, but also communication of in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These processes are strongly path dependent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the sequence of interaction matters and has to respect causal relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Models of causal paths [97, 142] provide a formal approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They build on higher-order networks, where each order captures a causal path of a given length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In addition to time directedness, temporal networks also reflect the burstiness of activities [142], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the fact that not every link in a network is active at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The temporal component significantly impacts the centrality measures of individual agents [141], as Figure 18 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 28/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Betweenness preference [128] was introduced as an agent-centered measure to quantify its im- portance in transferring information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (a) (b) Figure 18: Identification of important individuals (a) on the aggregated and (b) on the tempo- ral network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' External and internal shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The different dynamics described above are continuously perturbed by internal and external shocks of various size and origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Internal shocks, for instance, may cause agents to leave, this way triggering cascades of drop-outs and restructuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' External shocks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', directives during the pandemics, may change working conditions and collabora- tion relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Because of the volatile dynamics, we cannot clearly separate shocks from the “normal” dynamics, which both occur on the same time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We note that from our modeling perspective we model shocks, but not the origin of shocks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the government that changes the legal regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But we need to have models for the impact of these shocks and for the collective’s response to different kind of shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In other words, we need to estimate the robustness, or the absorptive capacity, of the collective facing a particular shock, and to estimate the adaptivity of the collective to overcome this shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Only then we can calculate the social resilience of the collective, as outlined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 29/54 18F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Conclusions Different types of networks capture different aspects of relations between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It depends on the research question and the available data which of these network represen- tations shall be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Building up a network ensemble allows us to go beyond the observed network, to include constraints for the social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In particular, we can distinguish significant from random interactions and infer signed relations from interaction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These link characteristics are important to build an agent-based model, in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our model has to consider various concurrent dynamics, including growth, entry and exit of individuals, internal restructuring and external shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 5 What should we do to calculate resilience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' After completing the steps delimitation, conceptualization and representation, we eventually have to master the last step, operationalization, where we merge the network approach with agent- based modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The overview is presented in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Questions How can we turn concepts into measures for robustness and adaptivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How can we characterize agents, using topological information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Why do we consider the social impact of agents?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How can we quantify it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How is resilience composed of robustness and adaptivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 Quantifying agent properties The major goal of our framework is a micro-perspective on resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This is an emerging systemic property, that means it can neither be reduced to, nor explained by, the dynamics of the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As we demonstrate below, we need to consider the network structure to calculate the robustness and the adaptivity of the social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But the agents’ importance, their social impact, will provide the right weights in calculating these two measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Quantifying agents’ importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A bottom-up approach to quantify resilience has to start from the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In each organization, agents have a different importance, ri, that reflects their hierarchical status, reputation, embedding in the organization, knowledge, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To obtain values for ri is a challenge in itself and depends on the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Because there is no general solution, we resort to some guiding examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 30/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) System Structure Dynamics Robustness Adaptivity Social resilience Agents Importance Interactions Embedding Relations Dyads Triads Social impact Weighted balance Signed relations Propensities Data Relational Longitudinal Multivariate Figure 19: Operationalization to calculate social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In the simplest case, importance is defined in the hierarchical structure of a team [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There are also ways to determine hierarchies based on interaction patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In the absence of such information, we may utilize topological information from the reconstructed network (see also Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In a directed social network we can use the eigenvector centrality of agents as a measure of their reputation [16, 152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For undirected networks, coreness [155] or weighted k-core centralities [58] can quantify an agent’s embeddedness in a network [151], assuming that more important agents are closer to the core (see also Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These measures also estimate the robustness of an agent’s network position against failure cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For temporal networks different centrality measures can be used [141] (see also Figure 18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Be- tweenness preference [128] quantifies an agent’s importance in communication processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Func- tional roles can be only partially inferred from communication patterns or specific topological embeddings of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Existing algorithms for role detection [78] do not detect organizational roles, but classify network positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Social impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What matters in an organization is not just the importance, ri, but also the support or opposition an agent receives from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Their influences are combined in an 31/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) individual social impact, Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The total impact of an agent is then the sum of its own importance and the social impact exerted by others, qi = ri + Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Here, we define the social impact as Ii = ∑j wijrj = Ip i − In i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The wij denote the weighted and signed relations between agents, which can be positive, negative or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ip i is the sum of all positive contributions, while In i sums up the negative contributions [84, 99, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ii can become negative, reflecting the fact that an agent may not have a high esteem in an organization because it receives little support, but strong opposition from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Infer signed relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To calculate the social impact, Ii, we also have to determine the signed relations wij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For this, we apply the method described above in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It returns for every pair of agents a weight and a sign to characterize their relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To conclude, our measure of the total importance, qi = ri + Ii, combines different, but rather complete information about each agent, namely information about its topological embedding and about its activities because its repeated interactions with other agents determine its relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' qi aggregates in one value the positive, negative or neutral influences from all counterparties, weighted by their individual importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, with qi we have a non-local measure about the true impact an agent can have in the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 Quantifying social resilience In order to obtain a measure for resilience, we have to solve two problems: (i) defining different proxies to measure robustness and adaptivity based on the available information about agents and their relations, (ii) determining a functional form for resilience dependent on the two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Again, there is not the one way of combining available information into meaningful measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore in the following we list a number of candidates to quantify robustness, which we can choose from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For adaptivity instead we provide only one measure based on the assumption that we have data available to construct a multi-edge temporal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Topological robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As noted above, robustness can only be defined with respect to a specific shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A software developer team can be robust against an external shock, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', stronger legal regulations, but not against an internal shock, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the dropout of a leading developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, we must consider different ways for defining the robustness of social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Topological measures are often easy to calculate and reflect specific aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The robustness against agent removal can be linked to agents’ coreness [29, 58, 117, 151, 155] (see also Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It helps understanding cascading effects from removing a specific agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Centralization [179] takes the concentration of interactions in a few agents into account, which increases the systemic 32/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) risk if these agents fail [34] (see also Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Betweenness preference [128] and Eigengap [112] indicate communication bottlenecks and identify gate keepers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These measures can be used separately, as demonstrated for centralization [154, 183], or in combination to quantify robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We can further utilize higher-order models of temporal networks to capture robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The Second-Order Algebraic Connectivity, for instance, can be interpreted as a temporal-topological robustness measure [184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Structural robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A different measure of robustness is proposed by the concept of structural balance as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It decomposes the network into triads and deter- mines their balance Sijk by multiplying the signs of the signed relations, sign(ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This approach has several shortcomings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' First, triads are evaluated independently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the fact that each agent is likely part of different triads at the same time is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Secondly, the different weights of each signed relation, ωij, are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Thirdly, the importance ri of the agents composing the triad is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That implies all triads have the same weight in estimating the robustness of the organization, which is not justifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Correcting for these shortcomings is an open discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As a possible alternative we have pro- posed a new weighted balance measure Tijk [143, 148] that takes into account not only the signs and the weights of the signed relations, but also the impact of the agents involved in the triad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To determine the structural balance of the whole collective, we take the arithmetic mean, ⟨T⟩ = ∑Tijk/Nt, where Nt = ∥Tijk∥ is the total number of triads in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Quantifying adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ideally a maximally resilient system would have maximal robust- ness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', it could withstand any shock, and maximal adaptivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', if a shock impacts the system it will always recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That means resilience R should increase both with robustness R and adaptivity A, R(R,A) ∼ R ⋅ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptivity does not simply mean “dynamics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, it refers to the ability of the organization to attain different states, which we also call potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But will the system actually attain these alternative states at random, without a response to a shock?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If so, we call this the propensity to change to indicate that it is independent of the quality of the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It turns out that the propensity to change is a two-edged sword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If a team is in a bad shape, it should be able to leave such bad state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Then a high propensity to change allows the team to attain other, and likely better, configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' On the other hand, if the team has reached a good state, it should be interested in keeping it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A high propensity to change would be counter productive because the good state could be easily lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This means that resilience should increase with the propensity to change if the system is in a bad state, and decrease in a good state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 20(a) illustrates the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 33/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) low high Propensity to change high low Robustness Fragile state can not be left Bad (1) Fragile state can be left Good (4) Robust state can be kept Good (2) Robust state can be lost Bad (3) (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='5 1 Robustness Propensity to change Resilience 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='8 1 (b) Figure 20: Resilience R as a function of robustness ˆR and propensity to change ˆP: (a) Qualita- tive assessment of different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (b) Exemplary quantification of R( ˆR, ˆP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' [154] A functional form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A decomposition of resilience into robustness and adaptivity, R(R,A) ∼ R ⋅ A, rests on the fact that we can capture the potential to change of the system independently from its propensity to change, which is in fact not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, we use the propensity to change ˆP as an empirical proxy for adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our measure of potentiality [187], introduced in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2, allows us to proxy this propensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It quantifies the probability distribution of states attainable by a system at a given point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The larger the potentiality, the larger is the number of alternative states attainable by the system, and the more likely is the system to change towards one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The smaller the potentiality, the smaller the number of states attainable and the smaller the probability the system will move away from the current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These considerations have determined us to propose the following functional form for the re- silience of social organizations: R( ˆR, ˆP) = ˆR(1 − ˆP) + ˆP(1 − ˆR) [148, 154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The quantity ˆP is a convenient transformation of potentiality P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', low values of ˆP (below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='5) map to a state with low propensity to change, while large values of ˆP (above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='5) map to a state with high propensity to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The lowest achievable potentiality is mapped to ˆP = 0, while the highest to ˆP = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Similarly, the value of robustness R should be always positive and conveniently scaled between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This can be achieved for most topology based measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For structural balance measures, however, ⟨T⟩ can become negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To use these measures, we have to map them as ˆR = 1/(1+eβ⟨T⟩), where β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 gives a rather smooth mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' ⟨T⟩ = 0 would then be equivalent to ˆR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Note that the function plotted in Figure 20(b) reflects the arguments summarized Figure 20(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 34/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Relation to ecological concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It is worth to get back from here to the discussion about “potential” and “connectedness” as constituents of ecological resilience in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Potential shall define the number of possible alternatives states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But so far it was only a conceptual proposal because of the lack of operationalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This gap is closed by our concept of adaptivity which indeed can be calculated and also compared across different systems [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Connectedness refers to the robustness of the system, capturing topological aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Again, with our measure of robustness we are able to calculate and to compare the robustness of different systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Moreover, both adaptivity and robustness can be monitored over time, making our resilience measure an instantaneous early warning signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Most interesting is the relation between low connectedness and high resilience, on the one hand, and high connectedness and low resilience, on the other hand, discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='3 [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This was presented together with the hypothesis about the “adaptive cycle”, which emerges if low connectivity is met by high potentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' While this adaptive cycle was considered a “metaphor” [25] or a “thinking tool”, we are able to demonstrate its existence in data about real world organizations [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience as a compromise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our framework reflects that high resilience requires both, the maintenance of a valuable organizational structure to withstand shocks, and the ability to change this structure quickly if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Reasons to change can result from internal or from external problems, for instance from an incapable management or from governmental restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The resilient organization has to achieve conditions under which it can respond even without prior knowledge about the shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead of rigidity, it needs fluidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But instead of fragility, it also needs stability, dependent on the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Hence, the maximum resilience should be a compromise to balance these different requirements in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Conclusions Turning concepts into measures is the hardest part of modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There are always different options to operationalize measures, dependent on available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Topological mea- sures alone are not enough to estimate the robustness and adaptivity of social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, we need to quantify the impact of agents, to correct structural balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Optimal resilience is a compromise between robustness and adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 6 Network construction and interventions So far we have translated our concepts for the robustness and adaptivity of social organizations into measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As the last step we have to discuss possibilities of obtaining the data needed 35/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) to calculate these measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If we achieve to have a calibrated generative model of the social organization, we can address the problem of system design [144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That means, we can test how possible intervention strategies impact the organization’s resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Questions How do we construct networks from data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What type of data is needed to calculate our resilience measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How can we obtain such data from repositories of social organizations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Is there a way to validate our generative model?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' How can we control resilience using network interventions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 Data acquisition and analysis We want to emphasize that our methodology inverts the usual supply driven approach found in computational social science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This starts from the data given, often collected without a clear purpose and a research question in mind, to subsequently squeeze out interesting features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In contrast, our demand driven approach has first identified in four steps shown in Figure 4 what data will be needed to inform our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Then, utilizing this data we can infer information about agents and their properties, but also about their interactions with others, as shown in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such data cannot directly provide the input for our models and is not sufficient to simply estimate social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Instead, it has to be pre-processed, before we can construct the networks that are essential for our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These networks are never given, and their generation and subsequent statistical interpretation bears some of the most overlooked problems in modeling social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Extract interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' One of our reasons to study software developer teams as prototypes of social organizations is the availability of vast git repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These contain fine-grained records of all changes made to the software, together with information who changed it, what was changed and when.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We developed a software package, git2net [66, 67], that is able to extract this information, to create bipartite networks and their projections into an interaction network between developers (see Figure 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We note that, in addition to the co-editing network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the collaboration network of developers, we can obtain additional information about the social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, analyzing the amount of code changes can quantify the productivity of developers [139] and analyzing the sequence of code changes gives insights into their hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 36/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) git2net can be also used to mine other git repositories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', for publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Additionally we have developed the rule-based disambiguation tool gambit to solve the persistent problem of name disambiguation occurring in most real-world user data [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Figure 21: Extracting the collaboration network of developers A, B, C using git2net [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Time window detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To obtain interaction networks, a sliding window approach ag- gregates interactions over a certain time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Choosing the right window size is a problem in itself, because the window size impacts the network density and subsequently all topological analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Often we have no data about interactions and need to infer them from time series of observed events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, from co-location data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', observations about two individuals i,j acting at times ti and tj at a given place, we need to detect the time interval ∆t = ∣ti − tj∣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Only observations with a ∆t lower than a given threshold ∆tthr will count as interactions [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such considerations are important to quantify, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the transmission of information within a social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Analyzing temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The dynamics of temporal networks crucially depend on ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The problem, who can potentially influence whom, requires to reconstruct temporal paths of various lengths [65], on which our networks can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We have developed different software packages to support the analysis of temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They are combined in the toolbox pathpy [140].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It implements, for instance, statistical techniques to find optimal graphical models for the causal topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These models balance model complexity with explanatory power for empirically observed paths in relational time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As part of pathpy, MOGen [65] is a multi-order generative model to statistically evaluate paths of various lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It can be used to improve the computation of different temporal centrality measures in case of insufficient observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 37/54 c c B B 血血 血血血血血血面 血血 B timeF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) Infer signed relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Signed relations ωij are instrumental to calculate the social impact and subsequently the robustness of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As described above, our framework uses interaction data to infer signed relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This method can be enhanced by taking additional data sets into account that can provide information about the positive or negative relations between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If written or spoken text is available, we can use sentiment analysis to obtain information about the emotional content [59, 61], to infer social relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Natural language processing (NLP) pro- vides an extended tool box to further extract information about opinions, attitudes, or ideological positions [1, 134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These can help quantifying the social impact that individuals exert on others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Information from collaboration platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Another important source of information are online collaboration platforms, such as slack, zoom, or GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In addition to interaction data and text messages, they often provide information about attention, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', via likes, about declared trust, recommendations, and activity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Based on reconstructed collaboration networks, one can analyze the presence of social mechanisms like reciprocity, homophily, triadic closure [17, 131], or of other motifs [182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This information can be used to further characterize the importance of agents and their signed relations and to estimate their impact on the resilience of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Network regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If the topological information is sufficient to reconstruct an additional network layer, it can be utilized for the network regression outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To facilitate the computation, we have developed an R package ghypernet [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It implements gHypEG, the network ensemble considering propensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In addition to network regressions, the package can be used to infer significant relations from observed interactions [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Once gHypEG is calibrated, we can also compute our potentiality measure even for large en- sembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' SciPy [172] provides an efficient implementation for computing the entropy of a given multinomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Calibration and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To find the optimal combination for the different measures mentioned above is recognized as an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Symbolic regression [44, 163] and other machine learning (ML) techniques are increasingly used to find solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In many cases, ground truth data is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Then, we have to rely on in-sample and out-of-sample predictions to aggregate different information in a meaningful manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This issue becomes relevant if we, for instance, want to improve the importance measures for agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If reliable aggregation methods are not available, we have to resort on determining the 38/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) ri values from topological measures, combined with dynamic processes as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', in feedback cen- tralities, as we will do in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 Resilience and control Improving the resilience of complex systems implies that, to some extent, we are able to influence such systems in a way that their functionality and their stability is maintained, or even enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The formal model of social organizations described above allows testing such intervention strate- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Top-down control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Generally we distinguish between bottom-up and top-down interventions [145].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The latter mostly focus on the boundary conditions for organizations, either to prevent shocks or to enhance their business environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These can be financial measures during the Corona crisis, but also legislative measures to ensure fair competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In general, to use the top-down approach, one needs to identify global control parameters which is a challenge on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Often they can be derived from the known macroscopic or system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As a major conceptual drawback, control parameters usually reflect limitations of stability, rather than of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' While the top-down approach is discussed in macro-economics and recently in macro-prudential regulations, we are interested in the bottom-up approach which is more in line with the complex systems philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Bottom-up interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our bottom-up approach to resilience uses interventions targeting specific agents and their interactions [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Structural interventions focus on the interaction struc- ture, basically changing the adjacency matrix of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Functional interventions change, for instance, the interaction rules to affect timing of interactions [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Dynamical interventions instead influence the internal state of nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such mea- sures can include nudging or mechanism design [145], but the most promising way for us are network interventions (see Figure 22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They require to first identify the driver nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', those agents that should be targeted, and secondly to decide about the type and the amount of inter- ventions [106, 185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Often these interventions change the agents’ utilities ui using control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Indirect influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To get access to agents, we can utilize the multi-layer structure of the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, if one layer contains friendship relations and the second one task assignments, the friendship layer can be used to influence the work relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 39/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) (a) (b) (d) (c) u1 u3 u2 u4 Layer 0 Layer 1 Figure 22: Network control in a two-layer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (b-d) Different couplings between the two layers, dependent on the peripheral or central position of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' [185] If the impacted agent responds appropriately, changes can propagate through the network, this way influencing agents that were not targeted directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As the example of Figure 22 shows, we can target agents at the periphery of the network to impact agents in the core [118, 185, 186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Influence on decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Network interventions only control a small number of agents at a comparably low cost, while utilizing the systemic feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But the method requires a model of the organization to forecast the impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Further, it assumes that the agent’s utility is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For the latter we can have at least reasonable assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Rational agents want to keep or even increase their impact, qi, by either increasing their impor- tance, ri, or by decreasing a negative social impact, Ii, they experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But changing signed relations or maintaining collaborations is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Agents may decide to leave the organization if their costs exceed their benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Conversely, they may decide to stay if their benefits have been increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Changing agents’ utility has the advantage of influencing these decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' If agents leave or reorganize their links, this changes the network topology and impacts the dynamics in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Consequently, both the robustness and the adaptivity of the organization are impacted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This can lead to counter intuitive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For instance, removing some agents may stabilize the organization [33, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' While this is known in human resource management, models are hardly able to reproduce such behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 40/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 23: Network interventions to prevent the breakdown of a social network, indicated by the drop of k-coreness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' (a) No intervention, (b) peripherial agents targeted, (c) agents close to the core targeted [33, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Conclusions We provide a whole tool box for mining and analyzing data of social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In particular, interaction data can be obtained from repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Other tools allow to calculate temporal centralities to characterize communication, and to infer propensities for interacting individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Which of the different measures are calculated depends on the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' There are various ways to proxy robustness, adaptivity and resilience of social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Network interventions allow to improve the resilience of organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 7 Conclusions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='1 What is resilience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Structural and dynamic dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Summarizing our tour through the modeling of social organizations, some important insights should be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' First, resilience is a concept that combines two dimensions, robustness and adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Robustness, as the structural dimension, 41/54 6 Average k-coreness 2 0 10000 20000 30000 40000 50000 Network Time6 Average k-coreness 2 0 10000 20000 30000 40000 50000 Network Time6 4 2 0 25000 50000 75000 100000 Network TimeF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) captures the ability of a social system to withstand shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Adaptivity, as the dynamic dimension, captures the ability of a social system to recover from shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Neither maximal robustness, nor maximal adaptivity alone are sufficient to warrant resilience for social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Both dimensions create a tension, because increasing robustness may lower adaptivity and the other way round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, a resilient state is a compromise balancing the influence of both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This insight is important because, following arguments from engineering, resilience is too often just treated as a synonym for stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This leads to the conclusion that maximizing resilience means maximizing robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Such a perspective may hold for designed infrastructure systems, but not for self-organizing systems such as social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To turn a concept into a measure requires operationalization which points to a different problem domain [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Even if we agree about our resilience concept, there may be different proposals to operationalize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They have to solve two problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Firstly, the functional form of resilience dependent on robustness and adaptivity should be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Secondly, measures for robustness and adaptivity have to be proposed and subsequently operationalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The latter is the real difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' What should we measure to quantify robustness or adaptivity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A system may be robust against some specific shocks but will fail for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Therefore, the question cannot be answered without an appropriate formal model of the social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In this paper, we made an operationalization proposal based on networks which can be constructed from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In general, these are multi-edge, temporal, multiplex and dynamic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' From these networks, topological information can be used to calculate robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Using the ensemble approach, we are further able to calculate adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience as an emergent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' For our modeling framework of social organizations we have adopted the complex systems perspective, in general, and the complex networks ap- proach, in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This implies to explain resilience as an emerging property of the social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Following the bottom-up approach, we have to focus on the micro level of inter- acting agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Measures for resilience need to be derived from this perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It requires to characterize agents in some detail regarding their importance, their signed relations and their social impact on others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Simple network measures that treat agents as dots to just calculate their network position are not sufficient to estimate robustness, even less to understand adaptivity as the dynamic component of resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience as a systemic property has to be constantly maintained, which requires the activity and the cooperation of the members of the social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Conversely, big threats to social resilience are not coming only from external shocks, but from internal challenges as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' As our model framework demonstrates, negative signed relations and negative social impact hamper 42/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) robustness and adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Biased interactions, the lacking integration of newcomers and low connectivity undermine the conditions under which resilience can be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='2 The policy dimension Science versus policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our framework for modeling the resilience of social organizations helps to understand under which conditions resilience is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our reasoning does not refer to the loss of “robustness” and “adaptivity” in an abstract manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It instead relates these losses to the underlying properties of agents and their dynamic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Only this way we are able to propose network interventions for improving resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But do these models, while successful from a scientific perspective, benefit policy makers in any way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Are we able to tell them what to do?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' To put this challenging question into perspective, we remind on some preconditions and some findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our models focus on a specific type of social organizations, namely teams of collaborating mem- bers sharing a common goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This means that we are not considering societies and, hence, our models neither aim at, nor are suited for, making suggestions on how to improve the resilience of societies against political, economic or environmental shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Delimitation was the first step for developing our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' With these restrictions in mind, our models indeed support some general insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' The most important insight is probably about the role of awareness for what resilience really means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This requires distinguishing it from concepts of robustness, stability, functionality, or optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilient systems are not obtained by maximizing or optimizing specific functions or key figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' A resilient organization has to withstand various kinds of shocks and to recover from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That means it needs to be prepared for the unknown, instead of being specialized to fit the known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This addresses a policy issue: Instead of improving resilience, organizations have strong incen- tives to rather improve performance as the most visible indicator of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This reminds on the classical conflict between short-term benefits and long-term deterioration and points to the misallocation of limited resources needed for maintaining resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ideally, a social organiza- tion should be able to anticipate possible shocks to some degree, and to prepare in advance for this, also by securing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This requires collective awareness, a state of consciousness that is based on continuously analyzing and recognizing the situation inside and outside the organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Next to robustness, our models highlight the role of adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It proxies the num- ber of options that an organization may have to respond to shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Consequently, we measured 43/54 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweitzer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Andres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Casiraghi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Gote, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Roller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Scholtes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Vaccario, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Zingg: Modeling social resilience: Questions, answers, open problems To appear in: Advances in Complex Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 8 (2022) adaptivity by potentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' It does not imply that these options are taken, but that they exist in a current situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Resilience depends on alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' That means, concepts like flexibility or fluidity become increasingly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We remind on the concept of adaptive capacity which already refers to the ability of an organization to adapt either in preparation, or in response to perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Quantifying resilience principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Ten years ago, Biggs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' [15] identified seven principles for building resilient socio-ecological systems: (1) maintain diversity and redundancy, (2) manage connectivity, (3) manage slow variables and feedbacks, (4) foster complex adaptive systems thinking, (5) encourage learning, (6) broaden participation, (7) promote poly-centric governance systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' These principles already highlight the importance of a complex systems perspective, the role of adaptivity and decentralized control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' But now we provide a modeling framework for social resilience where formal models allow to quantify the value of redundancy and connectivity using multi-edge and multi-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' They show how agents’ diversity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=', their heterogeneity, their social impact and their signed relations impact social resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' From a broader perspective, our paper wishes to contribute to a better concept of resilience management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' This requires both an understanding of the system that should be managed and an active involvement of those who are managing and those being managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Social organizations are a prime example for those systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We are the system elements, the agents, of our own social organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' We are in the position to change our organization to improve resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' At the same time, we are also affected by these changes, as well as by internal and external shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Our modeling framework helps to raise attention for the role of diversity and feedback processes, the power of decentralized network interventions and collective learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' In the end, however, it depends on us how much of these insights can be implemented in our social organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Acknowledgements The authors thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Garas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Garcia, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Mavrodiev and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' Schweighofer for early 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Entropy 21, 901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} +page_content=' 54/54' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctAyT4oBgHgl3EQfXfdE/content/2301.00183v1.pdf'} diff --git a/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf b/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..754932b38321447ea393e2dc1eb0cc88d3e5885a --- /dev/null +++ b/fdE4T4oBgHgl3EQfqg0X/content/2301.05200v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:939ccb9d849954cf2c4e71ee36cefc43fcde939555a42afd0a07398873fdd5c9 +size 1137239 diff --git a/fdE4T4oBgHgl3EQfqg0X/vector_store/index.pkl b/fdE4T4oBgHgl3EQfqg0X/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..453b0df7ade17d1ccd4aee394240d5fdef9ae609 --- /dev/null +++ b/fdE4T4oBgHgl3EQfqg0X/vector_store/index.pkl @@ -0,0 +1,3 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+firstname.lastname@lfb.rwth-aachen.de +Abstract +The VVC codec is applied to the task of multispectral image (MSI) compression using adap- +tive and scalable coding structures. In a “plain” VVC approach, concepts from picture-to- +picture temporal prediction are employed for decorrelation along the MSI’s spectral dimen- +sion. The popular principle component analysis (PCA) for spectral decorrelation is further +evaluated in combination with VVC intra-coding for spatial decorrelation. This approach +is referred to as PCA-VVC. A novel adaptive MSI compression algorithm, named HPCLS, +is introduced, that uses PCA and inter-prediction for spectral and VVC intra-coding for +spatial decorrelation. Further, a novel adaptive scalable approach is proposed, that pro- +vides a separately decodable spectrally scaled preview of the MSI in the compressed file. +Information contained in the preview is exploited in order to reduce the overall file size. All +schemes are evaluated on images from the ARAD HS data set containing outdoor scenes +with a high variety in brightness and color. We found that “Plain” VVC is outperformed by +both PCA-VVC and HPCLS. HPCLS shows advantageous rate-distortion (RD) behavior +compared to PCA-VVC for reconstruction quality above 51 dB PSNR. The performance of +the scalable approach is compared to the combination of an independent RGB preview and +one of HPCLS or PCA-VVC. The scalable approach shows significant benefit especially at +higher preview qualities. +1 +Introduction +Multispectral (MS) images capture a scene’s spectro-spatial information in a set of 2D +intensity images called spectral bands, where each band corresponds to a wavelength +range within the electromagnetic spectrum. Due to it’s remote sensing capability and +non-intrusive character, MS imaging is used in numerous areas of application such +as medicine, food processing and satellite imaging [1][2][3]. The higher the spectral +resolution, however, the higher the number of bands in the 3D MS image (MSI), +which in turn generates large image files. This calls for MSI compression techniques, +that, next to the spatial correlation, are able to efficiently exploit the high correlation +along the spectral dimension, in order to lower transmission and storage costs. +MSI compression is an active field of research using a variety of techniques, which +can be categorized into lossless and lossy compression. In lossless compression, only +redundant information is discarded in order to reduce the amount of data. For lossy +compression, also non-redundant information is discarded based on an importance cri- +terion determined by the specific application. In the state of the art, most approaches +arXiv:2301.04117v1 [eess.IV] 10 Jan 2023 + +use methods from transform coding, predictive coding, tensor decomposition and ma- +chine learning. +The linear Principle Component Analysis [4] (PCA) is a popular choice among +transforms used for spectral decorrelation. [5]-[9] use the PCA for spectral decorre- +lation in combination with a Discrete Wavelet Transform (DWT) for spatial decor- +relation, which is also used for spatial compression in the JPEG2000 [10] standard. +In [11], the DWT is performed on both spatial and spectral direction, resulting in a +3D DWT, and [12] uses a 3D Discrete Cosine Transform [13] (DCT) followed by a +support vector machine regression on the DCT coefficients for further compression. +Several new codecs have been released since the launch of the JPEG2000 standard, +which motivated Kwan et al. [14] to test the performance of J2K, X264, X265, and +Daala in combination with PCA on MSI compression. +Karami et al. [15] combine a 2D DWT with a tensor decomposition of the 3D +concatenation of wavelet coefficients, while in [16] Zhang et al. propose a pure tensor +decomposition approach. +In [17]-[22], linear DPCM prediction is used for compression of MSIs, where the +predictor weights are usually determined by the least squares (LS) method. Intra- +band prediction uses pixel values from within the same band for prediction, while +inter-band prediction uses pixel values from previously coded bands to predict the +current band. Toivanen et al. [18] and Zhang and Liu [19] experiment with correlation- +based band reordering in order to improve inter-band prediction performance. Li et +al. [20] use correlation-based K-means clustering to obtain a set of reference bands to +predict the entire MSI. In [21], Abrardo et al. perform separate inter-band prediction +on spatially non-overlapping rectangular image blocks, while Karaca and G¨ull¨u [22] +propose performing inter-band prediction on a superpixel basis, where superpixels are +obtained using the simple linear iterative clustering algorithm [23]. +Inter-band prediction using a non-linear multilayer propagation network of pre- +determined structure is proposed in [24], where only the parameters of the network +trained in the encoding stage have to be transmitted to the decoder side together with +the prediction error. Deng et al. in [25] employ an autoencoder network that directly +learns a mapping from the MSI to a latent space representation of fewer degrees of +freedom, that constitutes the compressed version of the MSI. +In this paper, Versatile Video Coding (VVC) [26, 27], the most recent video codec +released by the JVET, is applied to the problem of MSI compression. In a “plain” +VVC approach, VVC inter-prediction is employed for spectral decorrelation using a +Group of Picture (GOP) structure has been adapted to treat the spectral dimension +of the MSI image as the temporal dimension. This is used as the baseline to compare +the other compression algorithms. VVC intra-coding is tested in combination with +the popular PCA for spectral decorrelation, referred to as PCA-VVC. Further, two +novel adaptive compression algorithms are introduced, that also use VVC intra-coding +for spatial decorrelation. Hybrid Principle Component Least Squares (HPCLS) uses +linear inter-band prediction to predict the entire MSI from a set of reference bands, +that are obtained via PCA. The prediction error and the reference bands are further +compressed via VVC intra-coding. A fourth compression algorithm is motivated by + +the desire of having a spectrally scaled preview of the MSI’s contents included in the +compressed MSI file. The RGB Least Squares (RGBLS) algorithm provides such a +preview in the form of a RGB representation of the MSI included in the compressed +image file. The algorithm makes use of the information contained in the RGB base +layer by predicting all MSI bands from the RGB values. +Similar to HPCLS, the +RGB and the prediction error image are coded using PCA and VVC. Experimental +results obtained for the non-scalable PCA-VVC and HPCLS are used to asses the +performance of the spectrally scaled solution. +The remainder of this paper is structured as follows: In section 2, the data set +used for evaluation is briefly introduced, followed by a description of the implemented +compression algorithms in section 3. Section 4 features the experimental setup and +evaluation results, which are then discussed in section 5. Section 6 draws a conclusion +of the findings from this study and proposes possible future directions of research. +2 +Data Set +The ARAD HS data set published for the NTIRE2020 challenge on spectral recon- +struction [28] is used in this work. The data set comprises 470 images of outdoor +scenes with a high variety in brightness and color. The 10 images that were used as +validation data set in the challenge were selected for evaluation of the compression +algorithms implemented in this paper. +Exemplary RGB versions of images taken +from the data set are shown in figure 1. Each MSI comprises 31 482×512 sized band +images at 10 nm increments in the visible 400-700 nm range. Each image is cropped +to yield four 256×256×31 sized images to fit to the VVC coding tree unit size of +128×128, resulting in a total of 40 evaluation images. +Figure 1: Example images from the ARAD HS data set +3 +Methods +The compression algorithms introduced in the following are evaluated by their achieved +reconstruction quality, measured in Peak Signal to Noise Ratio (PSNR), at a given +compressed file size, measured in bits. The PSNR is defined as follows: +PSNR = 10 · log10 +� +I2 +max/MSE +� +[dB] , +(1) +with MSE representing the Mean Squared Error and the maximum pixel value of the +image before compression Imax. + +FRTheF-3.1 +VVC Configuration +In this study, the VVC reference software [29] is used and the MSIs were quantized +to 10 bit precision before VVC coding. The concept could be directly transferred to +higher precision (e.g. 12, 14, or 16 bit) if indicated by the application. For VVC +intra-only coding of MSI bands, the default VTM intra encoder configuration is used. +By using VVC inter-coding for inter-band prediction of MSIs, concepts from temporal +prediction are applied to the task of spectral decorrelation. For encoding the MSI, +we adapt the default VTM random access configuration. This configuration uses a +Group Of Pictures structure of 32 pictures (GOP-32) and a dyadic hierarchy with +bi-prediction for encoding. For adaptation to the MSI data set used here, we drop +the first and last pictures on the lowest hierarchy level of the structure, resulting in +a GOP-30 structure. By coding the first and last band both as key pictures (highest +hierarchy level), this structure can be readily applied to the 31 bands of the MSI. +Note that for simplicity, the bands are kept in sequential order here. The same QP +was used for all bands but the key pictures, since opposed to pictures of a video +sequence, accurate reconstruction is assumed to be of equal importance across all +bands. Key pictures are encoded with a QP offset of -3. This configuration will +be referred to as ”plain” VVC in the following. In an initial experiment, it showed +superior compression performance compared to other simple GOP structures. More +sophisticated GOP structures adapted to the spectral characteristics of MSIs are +subject of further study. +3.2 +PCA-VVC +PCA-VVC describes a combination of a PCA for spectral and VVC intra for spatial +decorrelation. The PCA is applied to the spectral pixel vectors of an MS image, +resulting in a concatenation of B 2D principle component (PC) images, where B is +the number of bands in the MSI. For encoding, only the first nc PCs representing the +highest energy in the data are kept, resulting in a lossy reconstruction. The PCA +basis vectors have to be transmitted together with the PC images for reconstruction +on the decoder side. Before transmission, the PCA basis vectors are quantized using +a uniform quantizer with a step size of ∆v=2−13. +3.3 +HPCLS +The coding scheme of HPCLS is visualized in Fig. 2a. To obtain the reference bands +used for inter-band prediction, a PCA retaining the first nc PCs is applied to the +MSI as described in section 3.2. For this PCA, nc will correspond to the number of +reference bands nref. The reference bands are further VVC intra coded. For every +spatially non-overlapping square MSI block of size Sp×Sp and for every band, predic- +tor weights wi are calculated via LS, that predict the MSI from the decoded reference +bands ˜Ic. To account for quantization of the wi at the decoder side, the prediction +error is calculated between the original MSI and the prediction using quantized wi. +The resulting 3D prediction error image is then further compressed by a PCA-VVC. + +Determine and quantize +predictor weights +Predict bands +Calculate prediction +error + +VVC intra encode +Original image bands +VVC intra encode +VVC intra decode +PCA( +) +bitstream +PCA( +) +(a) +Determine and quantize +predictor weights +Predict bands +Calculate prediction +error + +VVC intra encode +Original image bands +VVC intra encode +VVC intra decode +PCA( +) +PCA( +) +VVC encode +bitstream +Determine and quantize +predictor weights +Predict RGB +Calculate prediction error + +(b) +Figure 2: (a) Block diagram of HPCLS. (b) Block diagram of HPCLS-RGB. +The compressed file comprises the prediction error image and PCA basis vectors, the +reference bands and the predictor weights. +3.4 +HPCLS-RGB +The HPCLS-RGB algorithm is the scalable extension of HPCLS. A second set of +weights performs a linear prediction of the RGB preview, which is calculated from +the MSI by applying the CIE Standard Observer spectral sensitivities as done in [28]. +The resulting prediction error of the RGB preview eRGB is then directly encoded +using VVC in a GOP-2 configuration and added to the bitstream together with the +quantized predictor weights wRGBq. Fig. 2b shows a schema of the algorithm. +4 +Experiments +The compression algorithms introduced in section 3 are tested for different parameter +configurations on the evaluation data set described in section 2. In order to nar- +row down the parameter search space, all algorithms were first run in a test set up +investigating a bigger variation of parameter settings. In the test set up, the VVC +codec was substituted by a 2D DCT for spatial decorrelation, and compressed file +sizes were entirely estimated through entropy calculations. The same was done for +the predictor weights and PCA basis vectors. Repeating these runs using VVC for a +careful selection of parameter settings revealed comparable rate distortion behavior. +Parameter preset choices in the following sections were thus based on findings from +the test runs. +To obtain the points belonging to one rate-distortion (RD) curve for HPCLS and +HPCLS-RGB, the quantization parameter (QP) used for VVC intra-coding of the +PCs of the prediction error image is increased by 5 within [5, 50]. RD curves for +PCA-VVC are generated using the same QP values for intra-coding of the MSI’s +PCs. The results are further averaged over the entire 40 image evaluation set to yield +the final plots. + +0.0 +100.0 +200.0 +300.0 +400.0 +500.0 +600.0 +kbits +30 +35 +40 +45 +50 +55 +60 +PSNR in dB +HPCLS +convex hull +Figure 3: HPCLS for varying qref with nref=1 and nc=3 fixed. Lower qref shift the +RD curve towards the to right. The convex hull is established by the best-performing +rate points. +4.1 +Non-scalable Adaptive Compression +HPCLS is run for nref∈{1, 2, 3}, varying QP for compression of the reference bands +qref∈ [5, 50] and varying number of PCs for compression of the prediction error nc ∈ +[1, 6]. PCA-VVC is run for different nc∈ [1, 10]. The prediction block size is preset to +Sp=64. The PCA basis vectors v and the HPCLS and HPCLS-RGB predictor weights +are quantized with a step size of ∆v=2−13 and ∆w=2−12, respectively1. Figure 3 shows +the RD plots for an HPCLS example run where only qref was varied, plotted in blue. +In order to investigate the best performance a encoder might achieve, the convex hull +of RD points generated by varying the parameter settings of each scheme is reported. +The process is illustrated in Fig. 4. The design of the parameter adaptation is subject +of further study. In this manner, the convex hull RD curves of all investigated PCA- +VVC and HPCLS configurations are plotted against the curve belonging to “plain” +VVC coding of the entire MSI in Fig. 4. Besides the modified GOP structures, the +VTM configurations are used as-is. +4.2 +Scalable Adaptive Compression +The HPCLS-RGB algorithm is run for the same parameter settings as done for HP- +CLS while additionally varying the QP used to encode the RGB preview. By this, +the performance of compression method can be evaluated for different RGB preview +qualities. As done for HPCLS, the predictor weight quantization step size and the +prediction block size are preset to ∆w=2−12 and Sp=64, respectively. For HPCLS- +RGB to be a feasible option for scalable compression, it should provide superior RD +performance when compared to simply adding an RGB preview to the compressed +files produced by the non-scalable approaches. The latter technique is referred to +as “simulcast” in the following. Fig. 5 compares the scalable HPCLS-RGB to the +simulcast version of HPCLS and PCA-VVC for different preview qualities measured +in PSNR. The curve of non-scalable coding by HPCLS/PCA-VVC is added to illus- +trate the coding cost for the scalability feature. Note that the convex hull of HPCLS +1As of now, the wi and PCA basis vectors are fixed length coded, and coding cost could very +likely be reduced, e.g. by an entropy coding stage. + +0.0 +100.0 +200.0 +300.0 +400.0 +500.0 +kbits +30 +35 +40 +45 +50 +55 +60 +65 +PSNR in dB +HPCLS +PCA-VVC +VVC +Figure 4: Comparison of HPCLS, PCA-VVC and “plain” VVC. +0.0 +100.0 +200.0 +300.0 +400.0 +500.0 +kbits +30 +35 +40 +45 +50 +55 +60 +65 +PSNR in dB +PSNR ≥ 40 +PSNR ≥ 45 +single layer HPCLS/PCA-VVC +0.0 +100.0 +200.0 +300.0 +400.0 +500.0 +kbits +30 +35 +40 +45 +50 +55 +60 +65 +PSNR in dB +PSNR ≥ 35 +single layer HPCLS/PCA-VVC +Figure 5: Comparison of simulcast HPCLS/PCA-VVC (dotted) and HPCLS-RGB +(solid) using adaptive and scalable coding structures for three different qualities of +the RGB preview (35, 40 and 45 dB PSNR), plotted against single layer HPCLS/PCA- +VVC, that does not provide a preview. +and PCA-VVC have been combined to a single best performing HPCLS/PCA-VVC +curve here. +5 +Discussion +The results plotted in Fig. 4 reveal that the proposed adaptive HPCLS algorithm +outperforms the PCA-VVC approach for reconstruction qualities above 51 dB PSNR +on the evaluated image set. +Note that this crossing point might be significantly +shifted to lower PSNRs by developing an entropy encoding scheme for the HPCLS +predictor weights. This makes HPCLS a viable option, since MSIs are usually used at +very high qualities. Both, PCA-VVC and HPCLS outperform “plain” VVC coding +for reconstruction qualities between 30 and 60 dB. At 50 dB, both PCA-VVC and +HPCLS achieve a compression ratio of roughly 1:100. +The compression performance of the proposed scalable HPCLS-RGB compared to +the non-scalable simulcast versions of HPCLS and PCA-VVC depends on the desired +quality of the RGB preview and the reconstructed MSI. As can be seen from Fig. 5, +HPCLS-RGB outperforms the simulcast configuration significantly. + +6 +Conclusions +In this work, the VVC codec was applied to the task of MSI compression using +adaptive and scalable coding structures. VVC inter prediction was found to be out- +performed by the PCA for spectral decorrelation of the MSIs. With HPCLS, a novel +adaptive MSI compression algorithm combining concepts from transform and predic- +tive coding was introduced, that outperforms the combination of PCA for spectral +and VVC intra-coding for spatial compression for reconstruction qualities above 51 dB +PSNR. Further, an adaptive salable approach was developed, that provides a sepa- +rately decodable RGB preview of the MSI’s contents. The RD performance of the +scalable approach is superior to simply adding the RGB preview to the compressed +files produced by the non-scalable approaches. +In future work on scalable MSI compression, the design of the preview may be +revisited in order to achieve better exploitation of its spectral information for predic- +tion. This could be achieved e.g. by replacing the RGB preview with the output of a +colour transform which is more optimized for the compression task. 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Available: https://vcgit.hhi.fraunhofer. +de/jvet/VVCSoftware VTM/-/tree/VTM-15.0 + diff --git a/gNE2T4oBgHgl3EQfyAjI/content/tmp_files/load_file.txt b/gNE2T4oBgHgl3EQfyAjI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..40d3aab0bc0035e85ab7e97162a8320ef1bcd0e7 --- /dev/null +++ b/gNE2T4oBgHgl3EQfyAjI/content/tmp_files/load_file.txt @@ -0,0 +1,468 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf,len=467 +page_content='Adaptive and Scalable Compression of Multispectral Images using VVC Philipp Seltsam, Priyanka Das, Mathias Wien Institute of Imaging and Computer Vision RWTH Aachen University 52056 Aachen, Germany firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='lastname@lfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='de Abstract The VVC codec is applied to the task of multispectral image (MSI) compression using adap- tive and scalable coding structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In a “plain” VVC approach, concepts from picture-to- picture temporal prediction are employed for decorrelation along the MSI’s spectral dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The popular principle component analysis (PCA) for spectral decorrelation is further evaluated in combination with VVC intra-coding for spatial decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This approach is referred to as PCA-VVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' A novel adaptive MSI compression algorithm, named HPCLS, is introduced, that uses PCA and inter-prediction for spectral and VVC intra-coding for spatial decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Further, a novel adaptive scalable approach is proposed, that pro- vides a separately decodable spectrally scaled preview of the MSI in the compressed file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Information contained in the preview is exploited in order to reduce the overall file size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' All schemes are evaluated on images from the ARAD HS data set containing outdoor scenes with a high variety in brightness and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' We found that “Plain” VVC is outperformed by both PCA-VVC and HPCLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' HPCLS shows advantageous rate-distortion (RD) behavior compared to PCA-VVC for reconstruction quality above 51 dB PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The performance of the scalable approach is compared to the combination of an independent RGB preview and one of HPCLS or PCA-VVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The scalable approach shows significant benefit especially at higher preview qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 1 Introduction Multispectral (MS) images capture a scene’s spectro-spatial information in a set of 2D intensity images called spectral bands, where each band corresponds to a wavelength range within the electromagnetic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Due to it’s remote sensing capability and non-intrusive character, MS imaging is used in numerous areas of application such as medicine, food processing and satellite imaging [1][2][3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The higher the spectral resolution, however, the higher the number of bands in the 3D MS image (MSI), which in turn generates large image files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This calls for MSI compression techniques, that, next to the spatial correlation, are able to efficiently exploit the high correlation along the spectral dimension, in order to lower transmission and storage costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' MSI compression is an active field of research using a variety of techniques, which can be categorized into lossless and lossy compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In lossless compression, only redundant information is discarded in order to reduce the amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For lossy compression, also non-redundant information is discarded based on an importance cri- terion determined by the specific application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In the state of the art, most approaches arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='04117v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='IV] 10 Jan 2023 use methods from transform coding, predictive coding, tensor decomposition and ma- chine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The linear Principle Component Analysis [4] (PCA) is a popular choice among transforms used for spectral decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' [5]-[9] use the PCA for spectral decorre- lation in combination with a Discrete Wavelet Transform (DWT) for spatial decor- relation, which is also used for spatial compression in the JPEG2000 [10] standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In [11], the DWT is performed on both spatial and spectral direction, resulting in a 3D DWT, and [12] uses a 3D Discrete Cosine Transform [13] (DCT) followed by a support vector machine regression on the DCT coefficients for further compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Several new codecs have been released since the launch of the JPEG2000 standard, which motivated Kwan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' [14] to test the performance of J2K, X264, X265, and Daala in combination with PCA on MSI compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Karami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' [15] combine a 2D DWT with a tensor decomposition of the 3D concatenation of wavelet coefficients, while in [16] Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' propose a pure tensor decomposition approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In [17]-[22], linear DPCM prediction is used for compression of MSIs, where the predictor weights are usually determined by the least squares (LS) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Intra- band prediction uses pixel values from within the same band for prediction, while inter-band prediction uses pixel values from previously coded bands to predict the current band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Toivanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' [18] and Zhang and Liu [19] experiment with correlation- based band reordering in order to improve inter-band prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' [20] use correlation-based K-means clustering to obtain a set of reference bands to predict the entire MSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In [21], Abrardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' perform separate inter-band prediction on spatially non-overlapping rectangular image blocks, while Karaca and G¨ull¨u [22] propose performing inter-band prediction on a superpixel basis, where superpixels are obtained using the simple linear iterative clustering algorithm [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Inter-band prediction using a non-linear multilayer propagation network of pre- determined structure is proposed in [24], where only the parameters of the network trained in the encoding stage have to be transmitted to the decoder side together with the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' in [25] employ an autoencoder network that directly learns a mapping from the MSI to a latent space representation of fewer degrees of freedom, that constitutes the compressed version of the MSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In this paper, Versatile Video Coding (VVC) [26, 27], the most recent video codec released by the JVET, is applied to the problem of MSI compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In a “plain” VVC approach, VVC inter-prediction is employed for spectral decorrelation using a Group of Picture (GOP) structure has been adapted to treat the spectral dimension of the MSI image as the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This is used as the baseline to compare the other compression algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' VVC intra-coding is tested in combination with the popular PCA for spectral decorrelation, referred to as PCA-VVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Further, two novel adaptive compression algorithms are introduced, that also use VVC intra-coding for spatial decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Hybrid Principle Component Least Squares (HPCLS) uses linear inter-band prediction to predict the entire MSI from a set of reference bands, that are obtained via PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The prediction error and the reference bands are further compressed via VVC intra-coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' A fourth compression algorithm is motivated by the desire of having a spectrally scaled preview of the MSI’s contents included in the compressed MSI file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The RGB Least Squares (RGBLS) algorithm provides such a preview in the form of a RGB representation of the MSI included in the compressed image file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The algorithm makes use of the information contained in the RGB base layer by predicting all MSI bands from the RGB values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Similar to HPCLS, the RGB and the prediction error image are coded using PCA and VVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Experimental results obtained for the non-scalable PCA-VVC and HPCLS are used to asses the performance of the spectrally scaled solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The remainder of this paper is structured as follows: In section 2, the data set used for evaluation is briefly introduced, followed by a description of the implemented compression algorithms in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Section 4 features the experimental setup and evaluation results, which are then discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Section 6 draws a conclusion of the findings from this study and proposes possible future directions of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 2 Data Set The ARAD HS data set published for the NTIRE2020 challenge on spectral recon- struction [28] is used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The data set comprises 470 images of outdoor scenes with a high variety in brightness and color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The 10 images that were used as validation data set in the challenge were selected for evaluation of the compression algorithms implemented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Exemplary RGB versions of images taken from the data set are shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Each MSI comprises 31 482×512 sized band images at 10 nm increments in the visible 400-700 nm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Each image is cropped to yield four 256×256×31 sized images to fit to the VVC coding tree unit size of 128×128, resulting in a total of 40 evaluation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Figure 1: Example images from the ARAD HS data set 3 Methods The compression algorithms introduced in the following are evaluated by their achieved reconstruction quality, measured in Peak Signal to Noise Ratio (PSNR), at a given compressed file size, measured in bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The PSNR is defined as follows: PSNR = 10 · log10 � I2 max/MSE � [dB] , (1) with MSE representing the Mean Squared Error and the maximum pixel value of the image before compression Imax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' FRTheF-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='1 VVC Configuration In this study, the VVC reference software [29] is used and the MSIs were quantized to 10 bit precision before VVC coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The concept could be directly transferred to higher precision (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 12, 14, or 16 bit) if indicated by the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For VVC intra-only coding of MSI bands, the default VTM intra encoder configuration is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' By using VVC inter-coding for inter-band prediction of MSIs, concepts from temporal prediction are applied to the task of spectral decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For encoding the MSI, we adapt the default VTM random access configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This configuration uses a Group Of Pictures structure of 32 pictures (GOP-32) and a dyadic hierarchy with bi-prediction for encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For adaptation to the MSI data set used here, we drop the first and last pictures on the lowest hierarchy level of the structure, resulting in a GOP-30 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' By coding the first and last band both as key pictures (highest hierarchy level), this structure can be readily applied to the 31 bands of the MSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Note that for simplicity, the bands are kept in sequential order here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The same QP was used for all bands but the key pictures, since opposed to pictures of a video sequence, accurate reconstruction is assumed to be of equal importance across all bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Key pictures are encoded with a QP offset of -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This configuration will be referred to as ”plain” VVC in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In an initial experiment, it showed superior compression performance compared to other simple GOP structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' More sophisticated GOP structures adapted to the spectral characteristics of MSIs are subject of further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='2 PCA-VVC PCA-VVC describes a combination of a PCA for spectral and VVC intra for spatial decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The PCA is applied to the spectral pixel vectors of an MS image, resulting in a concatenation of B 2D principle component (PC) images, where B is the number of bands in the MSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For encoding, only the first nc PCs representing the highest energy in the data are kept, resulting in a lossy reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The PCA basis vectors have to be transmitted together with the PC images for reconstruction on the decoder side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Before transmission, the PCA basis vectors are quantized using a uniform quantizer with a step size of ∆v=2−13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='3 HPCLS The coding scheme of HPCLS is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' To obtain the reference bands used for inter-band prediction, a PCA retaining the first nc PCs is applied to the MSI as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For this PCA, nc will correspond to the number of reference bands nref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The reference bands are further VVC intra coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For every spatially non-overlapping square MSI block of size Sp×Sp and for every band, predic- tor weights wi are calculated via LS, that predict the MSI from the decoded reference bands ˜Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' To account for quantization of the wi at the decoder side, the prediction error is calculated between the original MSI and the prediction using quantized wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The resulting 3D prediction error image is then further compressed by a PCA-VVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Determine and quantize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='predictor weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Predict bands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Calculate prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC intra encode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Original image bands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC intra encode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC intra decode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='PCA( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='bitstream ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='PCA( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Determine and quantize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='predictor weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Predict bands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Calculate prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC intra encode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Original image bands ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC intra encode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC intra decode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='PCA( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='PCA( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='VVC encode ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='bitstream ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Determine and quantize ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='predictor weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Predict RGB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Calculate prediction error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='Figure 2: (a) Block diagram of HPCLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' (b) Block diagram of HPCLS-RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The compressed file comprises the prediction error image and PCA basis vectors, the reference bands and the predictor weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='4 HPCLS-RGB The HPCLS-RGB algorithm is the scalable extension of HPCLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' A second set of weights performs a linear prediction of the RGB preview, which is calculated from the MSI by applying the CIE Standard Observer spectral sensitivities as done in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The resulting prediction error of the RGB preview eRGB is then directly encoded using VVC in a GOP-2 configuration and added to the bitstream together with the quantized predictor weights wRGBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 2b shows a schema of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 4 Experiments The compression algorithms introduced in section 3 are tested for different parameter configurations on the evaluation data set described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In order to nar- row down the parameter search space, all algorithms were first run in a test set up investigating a bigger variation of parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In the test set up, the VVC codec was substituted by a 2D DCT for spatial decorrelation, and compressed file sizes were entirely estimated through entropy calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The same was done for the predictor weights and PCA basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Repeating these runs using VVC for a careful selection of parameter settings revealed comparable rate distortion behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Parameter preset choices in the following sections were thus based on findings from the test runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' To obtain the points belonging to one rate-distortion (RD) curve for HPCLS and HPCLS-RGB, the quantization parameter (QP) used for VVC intra-coding of the PCs of the prediction error image is increased by 5 within [5, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' RD curves for PCA-VVC are generated using the same QP values for intra-coding of the MSI’s PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The results are further averaged over the entire 40 image evaluation set to yield the final plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 kbits 30 35 40 45 50 55 60 PSNR in dB HPCLS convex hull Figure 3: HPCLS for varying qref with nref=1 and nc=3 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Lower qref shift the RD curve towards the to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The convex hull is established by the best-performing rate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='1 Non-scalable Adaptive Compression HPCLS is run for nref∈{1, 2, 3}, varying QP for compression of the reference bands qref∈ [5, 50] and varying number of PCs for compression of the prediction error nc ∈ [1, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' PCA-VVC is run for different nc∈ [1, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The prediction block size is preset to Sp=64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The PCA basis vectors v and the HPCLS and HPCLS-RGB predictor weights are quantized with a step size of ∆v=2−13 and ∆w=2−12, respectively1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Figure 3 shows the RD plots for an HPCLS example run where only qref was varied, plotted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In order to investigate the best performance a encoder might achieve, the convex hull of RD points generated by varying the parameter settings of each scheme is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The design of the parameter adaptation is subject of further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In this manner, the convex hull RD curves of all investigated PCA- VVC and HPCLS configurations are plotted against the curve belonging to “plain” VVC coding of the entire MSI in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Besides the modified GOP structures, the VTM configurations are used as-is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='2 Scalable Adaptive Compression The HPCLS-RGB algorithm is run for the same parameter settings as done for HP- CLS while additionally varying the QP used to encode the RGB preview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' By this, the performance of compression method can be evaluated for different RGB preview qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' As done for HPCLS, the predictor weight quantization step size and the prediction block size are preset to ∆w=2−12 and Sp=64, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For HPCLS- RGB to be a feasible option for scalable compression, it should provide superior RD performance when compared to simply adding an RGB preview to the compressed files produced by the non-scalable approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The latter technique is referred to as “simulcast” in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 5 compares the scalable HPCLS-RGB to the simulcast version of HPCLS and PCA-VVC for different preview qualities measured in PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The curve of non-scalable coding by HPCLS/PCA-VVC is added to illus- trate the coding cost for the scalability feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Note that the convex hull of HPCLS 1As of now, the wi and PCA basis vectors are fixed length coded, and coding cost could very likely be reduced, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' by an entropy coding stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 kbits 30 35 40 45 50 55 60 65 PSNR in dB HPCLS PCA-VVC VVC Figure 4: Comparison of HPCLS, PCA-VVC and “plain” VVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 kbits 30 35 40 45 50 55 60 65 PSNR in dB PSNR ≥ 40 PSNR ≥ 45 single layer HPCLS/PCA-VVC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='0 kbits 30 35 40 45 50 55 60 65 PSNR in dB PSNR ≥ 35 single layer HPCLS/PCA-VVC Figure 5: Comparison of simulcast HPCLS/PCA-VVC (dotted) and HPCLS-RGB (solid) using adaptive and scalable coding structures for three different qualities of the RGB preview (35, 40 and 45 dB PSNR), plotted against single layer HPCLS/PCA- VVC, that does not provide a preview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' and PCA-VVC have been combined to a single best performing HPCLS/PCA-VVC curve here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 5 Discussion The results plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 4 reveal that the proposed adaptive HPCLS algorithm outperforms the PCA-VVC approach for reconstruction qualities above 51 dB PSNR on the evaluated image set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Note that this crossing point might be significantly shifted to lower PSNRs by developing an entropy encoding scheme for the HPCLS predictor weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This makes HPCLS a viable option, since MSIs are usually used at very high qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Both, PCA-VVC and HPCLS outperform “plain” VVC coding for reconstruction qualities between 30 and 60 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' At 50 dB, both PCA-VVC and HPCLS achieve a compression ratio of roughly 1:100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The compression performance of the proposed scalable HPCLS-RGB compared to the non-scalable simulcast versions of HPCLS and PCA-VVC depends on the desired quality of the RGB preview and the reconstructed MSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 5, HPCLS-RGB outperforms the simulcast configuration significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' 6 Conclusions In this work, the VVC codec was applied to the task of MSI compression using adaptive and scalable coding structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' VVC inter prediction was found to be out- performed by the PCA for spectral decorrelation of the MSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' With HPCLS, a novel adaptive MSI compression algorithm combining concepts from transform and predic- tive coding was introduced, that outperforms the combination of PCA for spectral and VVC intra-coding for spatial compression for reconstruction qualities above 51 dB PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' Further, an adaptive salable approach was developed, that provides a sepa- rately decodable RGB preview of the MSI’s contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The RD performance of the scalable approach is superior to simply adding the RGB preview to the compressed files produced by the non-scalable approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' In future work on scalable MSI compression, the design of the preview may be revisited in order to achieve better exploitation of its spectral information for predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' This could be achieved e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' by replacing the RGB preview with the output of a colour transform which is more optimized for the compression task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' For HPCLS-RGB and HPCLS, the linear predictor might be substituted by a small multilayer percep- tron or convolutional network, that is trained and transmitted for each individual image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE2T4oBgHgl3EQfyAjI/content/2301.04117v1.pdf'} +page_content=' The proposed compression algorithms will further be evaluated on other MS data sets.' metadata={'source': 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Reep,3 Vladimir Airapetian,4, 5 Shin Toriumi,6 Keith C. Gendreau,7 and +Zaven Arzoumanian7 +1CRESST II and X-ray Astrophysics Laboratory NASA/GSFC, Greenbelt, MD 20771, USA +2Department of Physics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA +3Space Science Division, US Naval Research Laboratory, Washington, DC 20375, USA +4American University, 4400 Massachusetts Avenue NW, Washington, DC 20016 USA +5NASA/GSFC/SEEC, Greenbelt, MD, 20771, USA +6Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa, +252-5210, Japan +7X-ray Astrophysics Laboratory, NASA/GSFC, Greenbelt, MD 20771, USA +(Received 2022 July 18; Accepted 2022 December 21) +ABSTRACT +The Neutron star Interior Composition ExploreR (NICER) X-ray observatory observed two powerful +X-ray flares equivalent to superflares from the nearby young solar-like star, κ1 Ceti, in 2019. NICER +follows each flare from the onset through the early decay, collecting over 30 cnts s−1 near the peak, +enabling a detailed spectral variation study of the flare rise. The flare in September varies quickly in +∼800 sec, while the flare in December has a few times longer timescale. In both flares, the hard band +(2−4 keV) light curves show typical stellar X-ray flare variations with a rapid rise and slow decay, +while the soft X-ray light curves, especially of the September flare, have prolonged flat peaks. The +time-resolved spectra require two temperature plasma components at kT ∼0.3−1 keV and ∼2−4 keV. +Both components vary similarly, but the cool component lags by ∼200 sec with a 4−6 times smaller +emission measure (EM) compared to the hot component. A comparison with hydrodynamic flare loop +simulations indicates that the cool component originates from X-ray plasma near the magnetic loop +footpoints, which mainly cools via thermal conduction. The time lag represents the travel time of the +evaporated gas through the entire flare loop. The cool component has several times smaller EM than +its simulated counterpart, suggesting a suppression of conductive cooling possibly by the expansion of +the loop cross-sectional area or turbulent fluctuations. The cool component’s time lag and small EM +ratio provide important constraints on the flare loop geometry. +Keywords: Main sequence stars (1000), Solar analogs (1941), Stellar flares (1603), Stellar x-ray flares +(1637) +1. INTRODUCTION +Solar and stellar flares are the most energetic events +on low-mass stars (Haisch et al. 1991; G¨udel & Naz´e +2009). They represent the rapid conversion of magnetic +energy of active regions into kinetic and thermal ener- +gies, radiating from radio to gamma-rays and ejecting +high-energy nuclei and electrons. Powerful solar flares +have disrupted the Earth’s magnetosphere and human +Corresponding author: Kenji Hamaguchi +kenji.hamaguchi@umbc.edu +activity, yet flares of young Sun-like stars can far sur- +pass their solar counterparts in energy and frequency, +with their enhanced magnetic dynamos driven by rapid +rotations and deep convections. Their intense radiation +could impact the exoplanetary environment and habit- +ability (e.g., Airapetian et al. 2020). +These flares, even with substantial energy variations, +share similar behavior and characteristics and arise +from the universal magnetic reconnection mechanism. +Magnetic reconnection efficiently accelerates particles to +high energies (≳10 keV), which bombards the footpoints +of the loops with high-energy particles and heats the +chromosphere; the evaporated gas fills the magnetic loop +arXiv:2301.01377v1 [astro-ph.HE] 3 Jan 2023 + +2 +Hamaguchi et al. +and gradually cools down via radiation. The evaporated +gas at ≈107 K radiates primarily in soft X-rays between +0.1−10 keV (≈1−100 ˚A). Therefore, soft X-ray obser- +vations are crucial in understanding the flare geometry +and heating mechanisms. +During a typical flare, soft X-ray emission rises quickly +as the evaporated gas fills the magnetic loop and de- +cays quasi-exponentially as it gradually cools down ra- +diatively. Earlier studies have focused on the peak and +decay phase of flares (White et al. 1986; van den Oord +& Mewe 1989; Reale & Micela 1998; Tsuboi et al. 1998; +Favata et al. 2000; Sasaki et al. 2021). They suggested +that powerful flares tend to decay slowly and originate +from larger flare loops, which exceed the stellar radius in +extreme cases. Direct solar flare imagings, stellar flare +occultation observations, or theoretical models support +this idea, but the models can significantly overestimate +the flare size due to continuous heating, multiple loop +structures or subsequent flares during the decay phase +(e.g., Toriumi et al. 2017; Schmitt & Favata 1999; G¨udel +et al. 2004; Reep & Toriumi 2017). +The rising phase holds crucial information on the flare +geometry and heating mechanism (e.g., Reale 2007) as it +goes through initial heating, evaporation, and loop fill- +ing. However, the rising phase is often shorter than the +decaying phase (e.g., Reep & Knizhnik 2019; Getman +et al. 2021), and so has been mostly limited to duration +or crude hardness ratio studies in the soft X-ray band. +An exception is an XMM-Newton observation of Prox- +ima Centauri, which caught a bright flare from the onset +to the middle of the decay, recording ≳100 cnts s−1 near +the peak (G¨udel et al. 2002, 2004; Reale et al. 2004). +The X-ray hardness ratio reached its maximum in the +middle of the rise and started to decline near the flux +peak. The timing of maximum hardness coincides with +the U band (3000−3900˚A) flux peak measured with the +onboard Optical Monitor, suggesting a connection be- +tween the coronal and chromospheric heating. +The NICER (Neutron star Interior Composition Ex- +ploreR) X-ray observatory onboard the International +Space Station (ISS) (Gendreau & Arzoumanian 2017) +observed two bright X-ray flares from the nearby solar- +like star κ(kappa)1 Ceti (a.k.a. HD 20630, HIP 15457, +d =9.16 pc, mass: 1.04 M⊙, radius: 0.95±0.10 R⊙, ef- +fective temperature: 5665 K, Ribas et al. 2010; Rucin- +ski et al. 2004) during a monitoring program for the +Sellers Exoplanet Environments Collaboration (SEEC)1 +in 2019. This star shows intense magnetic activity due +to its fast stellar rotation (P =9.2 days), emitting two +1 https://seec.gsfc.nasa.gov +orders of magnitudes higher coronal X-rays and chro- +mospheric UV light than the Sun. +In 1986, the star +showed a signature of a superflare event in the He I D3 +(λ5875.6 ˚A) optical line, with an estimated total flare en- +ergy of E ≈2×1034 ergs (Schaefer et al. 2000). Still, the +the radiation from tranition region and coronal plasma +satisfies a solar magnetic flux scaling law similar to other +Sun-like stars (Toriumi & Airapetian 2022). These char- +acteristics suggest that κ1 Ceti is a young solar analog +at 0.4−0.8 Gyrs old with an enhanced solar-type coro- +nal and chromospheric heating rates (Airapetian et al. +2021). Its global-scale magnetic shear may cause super- +flares that eject huge masses of coronal material (Lynch +et al. 2019). +The NICER X-ray observatory primarily aims at +studying rapidly rotating neutron stars with very high +timing resolution, but its superb soft X-ray collecting +power, wide dynamic range, high throughput and mod- +erate background, decent energy resolution, tolerance +to optical light, and rapid maneuvering capability make +it a powerful tool for observing nearby solar-type stars +with sporadic bright X-ray flares. This manuscript de- +scribes analysis of NICER observations of the two pow- +erful X-ray flares from κ1 Ceti and performs hydrody- +namic simulations of single loop flares to interpret the +observations. +The result provides how X-ray plasmas +develop during the flare rising phase in detail. +2. OBSERVATION +The NICER X-ray Timing Instrument (XTI) is an ar- +ray of aligned 56 X-ray modules, each of which consists +of an X-ray concentrator (XRC, Okajima et al. 2016) +and a silicon drift detector (SDD, Prigozhin et al. 2016). +Each XRC concentrates X-rays within a ∼3′ radius field +of view to the paired SDD, which detects each photon +with accuracy at ∼84 ns. The XTI as a whole has one +of the largest collecting areas among X-ray instruments +between 0.2−12 keV (∼1900 cm−2 at 1.5 keV). We use +50 XTI modules as the remaining six (ID: 11, 14, 20, 22, +34, 60) are inactive or noisy. +NICER can continuously observe a target up to +∼2.5 ksec in every ISS orbit (∼5.5 ksec). However tar- +get visibility can be limited by Earth or ISS structure +occultation, or proximity to high particle regions such as +the South Atlantic Anomaly. NICER can quickly slew +the telescope and so observes multiple targets in each +ISS orbit to maximize the observing efficiency. This ca- +pability enables NICER to visit a target frequently, but +it can also cause scheduling conflicts with other timely +targets. +NICER started monitoring κ1 Ceti on 2019 Septem- +ber 16; it has observed the star for ∼180 ksec be- + +X-ray Flares from κ1 Ceti +3 +60000 +40000 +20000 +0 +0 +5 +10 +15 +20 +25 +Rate (cnts s +1) +Onset: 2019-09-17 09:36:52 +4000 +2000 +0 +2000 +4000 +6000 +Time (sec) +0 +5 +10 +15 +20 +25 +Rate (cnts s +1) +Onset: 2019-12-10 19:13:50 +Figure 1. Background subtracted light curves of κ1 Ceti between 0.6−1.2 keV on 2019 September 16−17 (top) and 2019 +December 10 (bottom). +NICER opportunely caught the rising phase of a bright X-ray flare during each observation. +The +grey and yellow lines show the background level estimated with the NICER tool nibackgen3C50 and an average count of each +Bayesian block derived with the astropy tool bayesian blocks. The time on the top shows the light curve origin, an estimated +flare onset. +The color bars present quiescent (magenta), preflare (red), flare rise & peak (blue) and flare decay end (cyan) +intervals for the spectral analysis. Each data bin is 50 sec. +tween 2019−2021. +During these monitoring observa- +tions, NICER detected two prominent X-ray flares on +2019 September 17 and December 10. +Earlier X-ray +imaging observations did not detect any X-ray sources at +significant X-ray brightness within 3′ from κ1 Ceti (e.g., +Telleschi et al. 2005), indicating that the flares originate +from κ1 Ceti. +We analyze the NICER observations ID: 2300020101, +2300020102, +and +2300020114. +We +reprocess +the +datasets with the NICER calibration ver. CALDB +XTI(20210707), using nicerl2 in HEASoft ver. 6.29c +and NICERDAS ver. V008c. We evaluate particle back- +ground using nibackgen3C50 ver. v7b with the parame- +ters dtmin=10.0, dtmax=60.0, hbgcut=0.1, s0cut=30.0 +(Remillard et al. 2021). We use python ver. 3.7, numpy +ver. 1.20.3, scipy ver. 1.1.0 and astropy ver. 3.1. +3. OBSERVATION RESULTS +3.1. Light Curves +The first flare occurred on 2019 September 17, dur- +ing the last snapshot of the one-day observation of κ1 +Ceti from September 16 (190917, Figure 1 top). +The +snapshot only lasts for ∼800 sec, but it covers the rise +and beginning of the decay of the flare as the flare +varies very quickly. The Bayesian block analysis tool, +bayesian blocks in the astropy package (Scargle et al. +2013) — nonparametric statistical analysis to detect sig- +nificant flux change in time-series data — does not sug- +gest that the 0.6−1.2 keV count rate changes from the +previous snapshot to the first 100 sec of this snapshot. +This result suggests that the flare begins around the +boundary of the last snapshot’s second and third time +bins, 2019 September 17 at 9h 36m 52s UT. + +4 +Hamaguchi et al. +Figure 2. 1st−5th rows: band-sliced light curves of κ1 Ceti around the first flare on 2019 September 17 (left) and the second flare +on December 10 (right). These light curves have the same horizontal scales, so the variation timescales are directly comparable. +The soft band light curves on the upper panels have delayed peaks compared with the 2−4 keV light curve. The grey line shows +the instrumental background level. Each light curve has 50 sec bins. Bottom row: time-series of the hardness ratio defined by +(H − S)/(H + S) where H and S are the net count rates between 1.2−4 keV and 0.3−1.2 keV, respectively. The color bars +present preflare (red), flare rise & peak (blue) and flare decay end (cyan) intervals for the spectral analysis. The first bin of the +September flare is below −1 as nibackgen3C50 overestimates the hard band background. In each flare, the hardness ratio peaks +before the total (0.3−4 keV) band count rate maximum. + +X-ray Flares from κ1 Ceti +5 +Figure 2 left shows band-sliced light curves of the last +snapshot. The 2−4 keV light curve is typical of solar +and stellar X-ray flares (Benz & G¨udel 2010, and refer- +ences therein), showing a sharp rise in ∼200 sec and a +steady decline by a factor of 3 in ∼500 sec after the peak. +However, the softer band light curves rise more slowly, +peak later, and decay more gradually. The light curves +below 1.2 keV may not even show a decay during the +snapshot. The soft band light curves significantly devi- +ate from the hard band light curves. The hardness ratio +in the bottom panel rises quickly, peaks before the total +(0.3−4 keV) light curve, and declines gradually. This +behavior is similar to the giant flare seen from Proxima +Centauri with XMM-Newton (G¨udel et al. 2004). +The second flare occurred during the second snapshot +on 2019 December 10 (191210, Figure 1 bottom). This +observation (for ∼1.6 ksec) is longer than for the first +flare, but it similarly covers the rise and beginning of the +decay as the second flare develops more slowly. NICER +misses the middle of the decay, but the third snapshot +covers the end of the decay as the light curve connects +smoothly from the second snapshot. On the other hand, +the first snapshot shows a slight elevation in the middle, +but both intervals appear to be in a quiescent state with- +out significant variation. The light curve before the first +flare also shows similar count rate variations (Figure 1 +top). A Bayesian block analysis shows that the 0.6−1.2 +keV count rate stays at ∼6.6 cnts s−1 from the latter +half of the first snapshot to the first ∼150 sec of the +second snapshot, suggesting that the flare begins during +the second snapshot at around 19h 14m 40s UT. Mean- +while, the last Bayesian block of the third snapshot has +almost the same count rate (∼6.8 cnts s−1), suggesting +that the quiescent emission does not vary significantly +during the flare. +Besides the slow variation, the second flare has sim- +ilar energy-dependent variations to the first flare. The +2−4 keV light curve reaches its peak before other en- +ergy bands. +The softer band peaks are delayed: the +0.3−0.6 keV light curve does not show a clear peak dur- +ing the second snapshot. This delay continues to the +third snapshot. The softer band light curves gradually +decline, while the 2−4 keV light curve is almost flat. +The hardness ratio reaches a maximum early during the +rise, but otherwise the variation is similar to the first +flare. +3.2. Time Resolved Spectra +To understand the energy-dependent time variations, +we analyze the time-resolved spectra. We first produce +a quiescent spectrum from the snapshot directly pre- +ceding each flare (the magenta bar in Figure 1). The +snapshot of the September flare shows no significant +variation in count rates, while that in December does +show a small but significant count rate increase in the +middle at −4.7 ksec (Figure 1 bottom). Therefore, we +produce two spectra separated at the time with a sta- +tistically significant count rate change (change point) in +the Bayesian analysis (Figure 3). The quiescent spectra +show a prominent hump between 0.7−0.9 keV, consis- +tent with emission lines of Fe XVII-XX, Ne IX-X, and +OVIII ions seen in the XMM-Newton RGS spectra of +κ1 Ceti in 2002 (Telleschi et al. 2005). The spectrum +has a steep hard slope, with negligible emission above +∼2 keV, but no absorption cut-off in the soft band down +to 0.3 keV. We, thus, apply a thermal plasma emission +model (apec) without absorption (Table 1). +Assum- +ing the best-fit elemental abundances of the XMM/RGS +spectrum (Appendix A), we find that a 2-temperature +(2T) model with kT = 0.27 & 0.62 keV provides an +acceptable fit to the quiescent spectrum for the Septem- +ber flare. The quiescent spectra of the December flare +require a 3T model for the former interval and an addi- +tional 1T component for the latter interval to account +for the flux increase. We use this 4T model as the fixed +quiescent component for the December flare spectra. +We produce time-resolved spectra during the flares, +every 50 sec with a minimum 25 counts per bin for the +September flare to track the fast variation and every 100 +sec with a minimum of 50 counts per bin for the Decem- +ber flare to get good photon statistics (Figures 4, 5). For +each flare, we make one spectrum before the flare onset +in the same snapshot (the red bars in Figure 2 between +−95—−35 sec for the first flare and −80—0 sec for the +second flare), which we call the preflare spectrum. In +the third snapshot of the December flare, due to a de- +creased count rate, we increase the time interval of each +bin to 400−600 sec. We also apply longer time intervals +for the third snapshot of the December flare near the +decay end. The preflare spectrum at the top left panel +of each figure matches well the corresponding quiescent +spectrum in the solid yellow line, consistent with the +Bayesian block analysis of the flare onset timings. +During the first flare, the flux in the entire energy +band increases for the first 300 sec. The 0.7−0.9 keV +hump becomes broader to the high energy side as the +flux at ∼1.1 keV, probably originating from Fe XXII- +XXIV or Ne X emission lines, is enhanced. After that, +the hard band emission gradually declines, while the flux +at ∼0.9 keV, probably produced by Fe XVII-XX lines, +strengthens. The second flare evolves more slowly but +similarly to the first flare. +The whole band increases +until ∼800 sec, and then the hump at ∼0.9 keV begins +to strengthen. In the third snapshot, the flux declines + +6 +Hamaguchi et al. +Figure 3. Quiescent spectra of the first flare for an exposure of 870 sec from −16921 sec to −16051 sec (left) and the second +flare for 789 sec from −4700 sec to −3911 sec (right). The solid yellow lines are the best-fit models, and the dotted lines are +the individual components. The upper right corner of each panel shows the best-fit parameters (the units are keV for kT and +cm−3 for EM). The best-fit models reproduce the observed spectra well. +Table 1. Best-fit Values of the Quiescent Spectra +Component +190917 +191210 +kT +EM +kT +EM +(keV) +(1051 cm−3) +(keV) +(1051 cm−3) +1 +0.27+0.03 +−0.03 +2.4+0.4 +−0.4 +0.34+0.02 +−0.03 +3.9+0.4 +−0.6 +2 +0.62+0.04 +−0.03 +1.9+0.3 +−0.4 +0.73+0.06 +−0.06 +1.9+0.5 +−0.4 +3 +1.98+0.87 +−0.38 +1.2+0.2 +−0.5 +4 +1.04+0.29 +−0.20 +0.5+1.1 +−0.2 +∆χ2/d.o.f +55.44/64 +162.06/175 +Note— The errors show 90% confidence ranges. +The 4th com- +ponent is required for the latter interval spectrum of the 191210 +observation. +nearly to the preflare level with some residual in the soft +and hard bands in the prior ∼1 ksec. +3.3. kT and EM Variations during the Flares +The time-resolved spectra do not suggest any signif- +icant variation of the quiescent component during the +flares. We, therefore, reproduce each time-resolved spec- +trum by a model with variable flare components plus +the fixed quiescent component. The hard slopes of most +spectra require a kT ∼3 keV plasma. However, colli- +sional equilibrium plasma at that temperature does not +emit Fe XVII-XX emission lines at ∼0.9 keV, which are +enhanced near the flare peaks. Non-equilibrium ioniza- +tion plasmas emit these lines at τ ∼2×1010 s cm−3, but +they do not reproduce the emission lines at ∼1.1 keV +observed during the rising phase. This result suggests +that the flare spectra need at least two plasma compo- +nents. +We, thus, apply a 2T apec plasma model for +the flare emission, with elemental abundances fixed to +the best-fit XMM/RGS values. We find that a model +with kT ∼2−4 keV and 0.3−1 keV components repro- +duces each spectrum well and that these temperatures +vary monotonically over time. +However, the spectral +parameters are poorly constrained near the flare onset +and end due to weak flare emission. We, therefore, fit +all spectra simultaneously, assuming that each compo- +nent’s plasma temperature varies linearly with time, and +find reasonable results with χ2/d.o.f. at 685.96/616 for +the first flare and 1118.41/919 for the second flare. Fig- +ures 4 & 5 plot the best-fit model for individual spectra +and Figure 6 shows the best-fit kT and EM values. The +third and fourth columns in Table 2 show the best-fit +kT slopes. + +X-ray Flares from κ1 Ceti +7 +preflare +Figure 4. Time-resolved spectra of κ1 Ceti during the first flare (190917). The red/blue line depicts the best-fit cool/hot +component of the flare spectrum, and the yellow line does the fixed quiescent component. The solid black line is their sum. +The hot component soars in the rising phase, dominating most energy bands, while the cool component is more significant at +∼0.9 keV with Fe L emission lines after ∼350 sec. The top right of each panel shows each spectrum’s time interval in seconds. +The cool component EMs are mainly determined by +fits of the Fe L emission line complex to the ∼0.9 keV +excess. +We examine whether the line intensity con- +straints in the applied model are adequate for this anal- +ysis. First, the model fixes the elemental abundances +during the flares at the best-fit XMM/RGS quiescent +spectrum values. However, some solar or stellar X-ray +flares show apparent elemental abundance changes from +the pre or post-flare states (e.g., Osten et al. 2000; Au- +dard et al. 2001; Mondal et al. 2021). The best-fit spec- +tral models also fit well the ∼1.1 keV bump with the Fe +L and Ne K lines in the hot component. Since the hard +spectral slope determines the hot component’s EM, the +hot component’s Fe and Ne abundances are consistent +with the assumed abundances, i.e., the coronal abun- +dances are not observed to significantly change during +the κ1 Ceti flares. +Second, suppose the cool compo- +nent did not reach equilibrium at ionization timescales +of ≲1010 s cm−3 as opposed to the model assumption. +In that case, the plasma should emit weaker Fe L lines +than the equilibrium case and require a larger EM to +account for the observed 1.1 keV bump. However, no +observed spectra show strong emission below ∼0.7 keV +expected from low ionized oxygen and carbon emission +lines from such non-equilibrium plasmas. In addition, +preflare loops probably have densities over ∼1011 cm−3 +(e.g., Osten et al. 2006), suggesting that the Fe L line +complex develops within ≈0.1 sec. These results suggest +that the cool component EM measurements are robust. +The hot component explains most of the initial flux +rise in each flare. The component in the second flare +is slightly hotter and cools down significantly slower + +8 +Hamaguchi et al. +preflare +Figure 5. Time-sliced spectra of κ1 Ceti during the second flare (191210). The flare spectral components behave similarly to +those of the first flare. The cool component exceeds the hot component at ∼0.9 keV after ∼1 ksec but not so much as during +the first flare. +than the first flare, but the component stays kT ≳2 keV +throughout both observations. On the other hand, the +cool component develops more slowly than the hot com- +ponent. The plasma temperature does not vary strongly +at ∼1 keV around the flare peak, but it declines to +∼0.3 keV by the end of the second flare. +EM time series in Figure 6 bottom panels confirms the +similarity of the two flares: i) the hot EM varies with a +linear rise and a slow decay, ii) the cool EM varies simi- +larly to the hot EM but with a delay. To quantitatively +evaluate their variations, we fit the EM time series with + +X-ray Flares from κ1 Ceti +9 +the following conventional formula for stellar flares. +EM(t) = 0 +t < tonset += EMpeak +t − tonset +∆trise +tonset ≤ t < tpeak +(1) += EMpeak exp(−t − tpeak +τdecay +) +tpeak ≤ t +where tonset, tpeak, EMpeak and τdecay are free param- +eters and ∆trise = tpeak − tonset. +For the fittings, we +use curve fit in the scipy package. We fix ∆trise and +τdecay of the 190917 flare’s cool component at the best- +fit values of the hot component as the cool component +does not show a clear EM peak. Table 2 shows the best- +fit result. The hot component’s tonset is close to zero, +again consistent with the Bayesian blocks measurement +of the flare onset in each flare. +In contrast, the cool +component’s tonset is significantly delayed from the hot +component’s tonset (hereafter ∆tdelay = tonset(cool) − +tonset(hot)). +In the second flare, the cool component +has similar ∆trise to, but a factor of two longer τdecay +than, the hot component. The second flare has longer +durations in ∆tdelay, ∆trise, and τdecay than the first +flare. +These behaviors explain the energy-dependent varia- +tions of the light curves. The 2−4 keV band light curve +is dominated by the hot component’s behavior, showing +a conventional stellar flare variation. The softer bands +add the cool component’s behavior with a conventional +flare variation but a time delay compared to the hot +component. The 0.6−1.2 keV light curve deviates most +with the strong 0.9 keV hump from the cool component. +The deviation is stronger in the first flare with a larger +relative time delay (∆tdelay/∆trise) and a larger EMpeak +ratio (EMpeak(cool)/EMpeak(hot)). +4. HYDRODYNAMIC SIMULATIONS OF SINGLE +LOOP FLARES +The hot component constitutes the major part of the +flare emission. +As discussed in numerous studies, it +should originate from radiatively-cooling plasma inside +the flare magnetic loops. Then, what is the cool compo- +nent? Flare spectral fits often require two temperatures +or more (e.g., Sasaki et al. 2021; Paudel et al. 2021), but +the nature of the cool component is poorly known. Our +NICER study provides this component’s time variation +through the flare rise. We run hydrodynamic simula- +tions of single magnetic loop flares to understand the +cool component. +We employ a field-aligned hydrodynamic code, the +HYDrodynamics and RADiation code (HYDRAD2; +2 https://github.com/rice-solar-physics/HYDRAD +Bradshaw & Mason 2003), used to study heating in the +solar corona and solar flares. The code solves the equa- +tions for the conservation of mass, momentum, and en- +ergy for plasma confined to a magnetic flux tube (Brad- +shaw & Cargill 2013). +The loops are assumed to be +heated by non-thermal electrons, accelerated by mag- +netic reconnection near the loop’s apex. As the electrons +propagate, they deposit their energy through Coulomb +collisions with the ambient plasma. +The majority of +the heat is deposited in the upper chromosphere, caus- +ing a rapid increase in temperature and pressure. +It +then drives an expansion of material (chromospheric +evaporation), carrying hot and dense plasma into the +corona. The assumed form of the heating function that +we use was derived by Emslie (1978), with modification +for non-uniform ionization in the chromosphere (Hawley +& Fisher 1994). As the loop evolves, the plasma cools +through thermal conduction and radiation, which we +calculated using the atomic database CHIANTI (Dere +et al. 1997), version 10 (Del Zanna et al. 2021). We use +the elemental abundances derived from the XMM/RGS +spectra (Table A1), but our preliminary study using +solar elemental abundances provides a similar result. +Our simulations assume a magnetic loop with a uniform +cross-section and injected particles with a power-law en- +ergy distribution for 200 sec with an energy flux peak- +ing at 1011.5 ergs cm−2 s−1 at 100 sec. Since the two +NICER flares have different flare decay timescales and +plausibly different magnetic loop lengths (e.g., Toriumi +et al. 2017; Reep & Toriumi 2017), the simulations con- +sider loop lengths at 5, 10, 15, 20, and 25×109 cm. The +derived EM normalization can be adjusted by changing +the cross-section of the magnetic loops (equivalently, the +total volume of the loops). +The left panel of Figure 7 shows the EM evolution of +the 10×109 cm cm flare loop simulation. +The EM is +dominated by the hottest plasma emission that peaks +near ∼600 sec. This component represents a radiative- +cooling, evaporated plasma that fills the magnetic loop, +corresponding to the hot component of the observing +flares. Since the evaporated plasma cools down grad- +ually under thermal equilibrium, a single temperature +bucket dominates near its peak and drops to zero quickly +once the plasma cools. +A secondary component is a +group of low-temperature buckets that rises and falls +similarly to the main component at one-third the EM of +the evaporated plasma component. Each temperature +bucket stays in this group until the evaporated plasma +cools down to its temperature range. +This secondary +component represents plasmas at transition regions near +the magnetic footpoints. Because the conductive heat +flows from the looptop to the footpoints, the plasma + +10 +Hamaguchi et al. +190917 +191210 +Figure 6. Best-fit kT (top) and EM (bottom) values of the time-resolved flare spectra by the 2T apec models (left: 190917, +right: 191210). The red/blue color shows the cool/hot plasma component. In the kT plots, the solid lines and filled areas are +the best-fit kT linear models and the 90% confidence areas. In the EM plots, the data points show the best-fit values and their +90% confidence ranges of the combined spectral fits. The solid lines show the best-fit linear rise plus exponential decay model +to these EM measurements. In each flare, the cool component rises similarly to the hot component but with a time delay. +Table 2. Flare Parameters +Flare +Comp. +kT(t = 0) +kTslope +tonset +∆trise +τdecay +EMpeak +LXpeak +EX +Ebol +(keV) +(keV/ksec) +(sec) +(sec) +(sec) +(1052 cm−3) +(1029 ergs s−1) +(1032 ergs) +(1033 ergs) +190917 +cool +1.00+0.11 +−0.13 +−0.18+0.25 +−0.24 +156 (20) +218 (fix) +572 (fix) +0.71 (0.061) +1.2 +0.79 +2.0/3.2 +hot +2.86+0.51 +−0.46 +−1.6+0.96 +−0.95 +14 (7) +218 (15) +572 (86) +2.6 (0.13) +3.2 +2.2 +191210 +cool +1.02+0.03 +−0.04 +−0.13+0.02 +−0.01 +227 (35) +923 (99) +2591 (324) +0.45 (0.032) +0.75 +2.3 +6.6/4.0 +hot +3.65+0.24 +−0.25 +−0.32+0.16 +−0.09 +33 (37) +1003 (59) +1135 (111) +2.8 (0.12) +3.9 +6.4 +Note— kT(t = 0), kTslope: Best-fit kT linear time variation model of the combined spectral fits. The errors show 90% confidence ranges. +tonset, ∆trise, τdecay, EMpeak: Best-fit linear rise plus exponential decay model of the EM time series. The parentheses show 1σ confidence +ranges. LXpeak: Peak X-ray luminosity between 0.3−10 keV. EX: Total X-ray flare energy between 0.3−10 keV. Ebol: Total bolometric flare +energy. The left values use a relation to the GOES band (1.55-12.4 keV) flare-radiated energy for active stars (Table 2 in Osten & Wolk +2015). The right values use a relation to the GOES band solar flare peak flux (equation (13) in Aschwanden et al. 2017). +has a strong temperature gradient and responds to the +evaporated plasma’s variation. The other loop length +simulations show similar EM variations with different +time scales. +In the first 500 sec, the log T ≥7.3 (K) buckets only +reflect the evaporated plasma component, while the log +T <7.0 (K) buckets reflect the footpoint plasma com- +ponent. We, therefore, define two temperature ranges, +log T =7.3−8.0 (K) and 6.6−6.9 (K), and sum up EMs +within each range to understand their behaviors near the +rising phase (Figure 7 right). +First, the EM[6.6−6.9] +time series of various loop lengths vary similarly for +∼200 sec from the beginning. +This EM base origi- +nates from the initial heating of the plasma in the up- +per chromosphere by the injected particles, and peaks at +∼100 sec in response to the assumed particle injection +flux. The EM[7.3−8.0] does not show this component +clearly, but the slow rise in the first ∼50 sec originates +from the initial plasma heating. +All EM[6.6−6.9] plots except the 5×109 cm simu- +lation show the footpoint components’ onsets as clear +kinks (e.g., at ∼150 sec for the 10×109 cm simulation). +We measure the timing of each kink from a two linear +fit (tonset(foot) in Table 3). The onset ranges between +∼80−300 sec, and longer loops have later onsets. The +evaporated component does not have a clear onset sig- +nature, so we measure the onset timing from a fit to the +first 200 sec of EM[7.3−8.0] by a linear function starting +at tonset(eva). The onset ranges between 17−64 sec and +does not appear to correlate with the loop length. The + +X-ray Flares from κ1 Ceti +11 +log T (K) +5 +10 +15 +20 +25 +5 +20 +15 +25 +10 +7.3-8.0 +6.6-6.9 +log T (K) +Figure 7. Left: Whole loop EM variations of the 10×109 cm loop simulation. The EMs are divided by logarithmic temperature +buckets and binned every 30 seconds. The plot shows two EM components — the evaporated plasma that envelopes individual +temperature peaks standing up from the hot side, getting a maximum at ∼600 sec, and the footpoint plasma, a group of +low-temperature buckets that rises and falls similarly at one-third of the evaporated plasma component. Right: Whole loop EM +variations for the first 500 sec, summed over the temperature ranges log T =7.3−8.0 K (blue) and 6.6−6.9 K (red). The number +that labels each line is the loop length in 109 cm. The evaporated component starts to rise at ∼50 sec, while the footpoint +component is significantly delayed from the evaporated component, more with longer loops. The overlapping triangular base in +the red plot up to ∼250 sec originates from the initial heating. +Table 3. HYDRAD Simulation Result +Loop Length +tonset(eva) +tonset(foot) +∆tdelay +∆trise(eva) +τdecay(eva) +EMpeak ratio +kTpeak(cool/hot) +(109 cm) +(sec) +(sec) +(sec) +(sec) +(sec) +(keV/keV) +5 +63 +77 +15 +330 +760 +1.21 +0.32/1.40 +10 +64 +153 +89 +491 +1207 +1.04 +0.31/1.67 +15 +40 +205 +164 +496 +2268 +0.78 +0.32/1.66 +20 +25 +248 +223 +639 +3013 +0.65 +0.34/1.57 +25 +17 +294 +276 +752 +3601 +0.60 +0.35/1.57 +190917 +142 +218 +572 +0.27 +0.96/2.50 +191210 +194 +1003 +1135 +0.16 +0.88/3.31 +Note—In each simulation, the time origin is the particle injection start. +tonset(eva)/tonset(foot): onset time of +the evaporated/footpoint component derived from one/two linear fits to the EM[7.3−8.0]/EM[6.6−6.9] time series. +∆tdelay = tonset(foot) − tonset(eva). ∆trise(eva)/τdecay(eva): rise/decay time of the evaporated component derived +from a fit to the peak EMs of individual temperature buckets by a linear rise plus exponential decay model. EMpeak +ratio, kTpeak(cool/hot): EMpeak ratio and plasma temperatures at the cool/hot EM peaks, derived from fits to the +synthetic flare spectra with 100 sec bins by a 2T apec model. + +12 +Hamaguchi et al. +time lags ∆tdelay (=tonset(foot) − tonset(eva)) clearly in- +crease with longer loops. +Figure 8 shows why the footpoint component is de- +layed. +In the plots, the evaporated component is lo- +cated in the middle part where temperature and den- +sity quickly increase after the particle injection. +The +footpoint component is located near both ends, whose +temperature and density do not increase until the shocks +produced by the evaporated gas’s collision at the looptop +propagate down to the footpoints. Figure 9 displays the +hydrogen and electron density product and the electron +temperature, magnifying the left end on a logarithmic +scale. Both the log T =6.6−6.9 K depth and density +product increase between 140 sec and 220 sec. The time +delay corresponds to the travel time of the flare loop by +the evaporated flows and the collisional shocks, which is +approximately the sound crossing time. It is, therefore, +roughly proportional to the loop length. +We also measure the decay timescales of the simu- +lated flares. As described above, the evaporated plasma +is in a single temperature bucket at each temperature +peak. +We, thus, take the peak EM of each tempera- +ture bracket and fit them by a linear plus exponential +decay model from equation (1). The fits reproduce the +EM variations well except around the peak (Figure 10 +for the 10×109 cm loop simulation). Longer loop flares +have longer decay timescales (τdecay(eva) in Table 3), as +suggested in earlier studies (e.g., van den Oord & Mewe +1989; Toriumi et al. 2017). +We make synthetic NICER spectra of the simulated +EM distributions to compare the spectral properties. +For each simulation, we produce spectral models with +100 sec bins, assuming an apec plasma model for each +temperature bucket. +We normalize them to a peak +0.3−2 keV flux at 2.2×10−11 ergs cm−2 s−1 to match +the observed two NICER flares. We then generate a syn- +thetic spectrum for each spectral model with the xspec +fakeit tool, by convolving the model with the NICER +on-axis responses, nixtiref20170601v003.rmf and nixti- +aveonaxis20170601v005.arf. We increase photon statis- +tics by a factor of 10 to reduce statistical uncertainty, +equivalent to a 1 ksec exposure. +We, then, bin each +synthesized spectrum to have ≥50 counts per bin, and +fit each spectrum by a 2T apec model. Figure 11 left +shows a synthetic spectrum of the 10×109 cm loop be- +tween 600−700 sec, adding the quiescent component of +the September flare for a comparison. Table 3 shows +the EMpeak ratios and plasma temperatures at the EM +peaks. The peak plasma temperatures ∼0.3−0.35 keV +and 1.4−1.67 keV are similar among the simulations +and significantly lower than the observed values. The +EMpeak ratio is the highest with the 5×109 cm loop sim- +ulation at 1.21 and smaller with longer loop simulations. +This result is naturally understood since the footpoint +plasma volume does not change with the loop length. +The numerical simulations demonstrate that the foot- +point component is delayed to rise by 100−300 sec from +the evaporated component. This result indicates that +the cool component in the NICER spectra originates +from the footpoint plasma. The simulations also suggest +that longer flare loops have longer delays of the footpoint +component rise and smaller EM peak ratios, as well as +longer decay timescales. All these properties are consis- +tent with the properties of the two NICER flares, sug- +gesting that the December flare originates from a longer +flare loop than the September flare. +5. DISCUSSION +The κ1 Ceti flares in 2019 are an order of magnitude +more powerful than the most powerful solar flare ever +seen, the Carrington Event in 1859 (LX ∼1028 ergs s−1, +Cliver & Dietrich 2013; Sakurai 2022). Their X-ray lu- +minosities are near the upper end of the flare luminosity +ranges of solar-type G stars (LX ≲1030 ergs s−1, Schaefer +et al. 2000; Tsuboi et al. 2016). Their bolometric flare- +radiated energies 3−8×1033 ergs, evaluated from two +independent empirical relations to the X-ray radiations +among solar and active stellar flares (Aschwanden et al. +2017; Osten & Wolk 2015, see Table 2), qualify them +as superflares (>1033 ergs, e.g., Maehara et al. 2012) +and are comparable to the κ1 Ceti superflare recorded +in 1986 (∼2×1034 ergs, Schaefer et al. 2000). Nonethe- +less, their X-ray luminosities and released X-ray ener- +gies are modest among active or young stellar flares. +(LX ≲1032−33 ergs s−1, e.g., Benz & G¨udel 2010, and +references therein). The other X-ray characteristics — +the hot plasma temperature, hard band light curve, and +hardness ratio variation — are similar to solar and stel- +lar X-ray flares (e.g., Pye et al. 2015). We conclude that +the κ1 Ceti flares in 2019 are conventional magnetic re- +connection events. +The κ1 Ceti flare spectra require an additional cool +(kT ≲1 keV) temperature component. Although such +a component has not received much attention, well ex- +posed stellar X-ray flare spectra usually require one or +more components with kT ∼0.3−1 keV (e.g., GT Mus: +Sasaki et al. 2021, EV Lac: +Paudel et al. 2021). +The high-resolution XMM/RGS spectra of the Prox- +ima Centauri flare suggested that the flare EM distri- +bution was broad with a peak at ∼30MK and a low- +temperature tail during the rise and steadily moved to +low-temperatures as the flare developed (G¨udel et al. +2004; Reale et al. 2004). +In solar flares, the low- +temperature EM (<16.5 MK) peaks later than the high- + +X-ray Flares from κ1 Ceti +13 +Figure 8. Density (top) and temperature (bottom) spatial distribution of the 10 × 109 cm loop simulation at 40, 60, 140 and +220 seconds from the particle injection start. The horizontal axis shows the distance from a footpoint along the loop: the loop +top is at 5×109 cm and the two footpoints are at 0, 10×109 cm. From left, i) at 40 sec, the particle injection heats the footpoint +chromosphere, and the evaporated gas soars into the magnetic loop. The evaporated spectral component starts to increase. ii) +at 60 sec, the upward evaporation flows collide at the loop top, heating the gas further. iii) at 140 sec, the shock propagate +down the other leg, smoothing the corona’s density. iv) at 220 sec, by the time the shock reaches the footpoints, the loop has +enough high density that thermal conduction becomes extremely efficient, and the footpoint spectral component emerges. The +red and blue lines are for electrons and hydrogen, respectively. The black and white double arrows point to the locations of the +evaporation or shock fronts from either side. +1022 +1023 +1024 +1025 +1026 +Density Product (cm +6) +40 s +60 s +140 s +220 s +10 +1 +100 +Position (109 cm) +105 +106 +107 +108 +Electron Temperature (K) +Figure 9. +Electron and hydrogen density product (top) +and electron temperature (bottom) distributions of the 10 × +109 cm loop simulation near a footpoint region at t =40 sec +(dotted), 60 sec (dash-dotted), 140 sec (dashed), and 220 sec +(solid). The horizontal axis is the same as Figure 8 but on a +logarithmic scale. The filled grey area in the bottom panel +shows the log T =6.6−6.9 K bucket. The boxes in the top +panel show the one-dimensional volumes and density product +ranges of this temperature bucket. Both the density product +and the volume significantly increase between t =140 sec and +220 sec. +temperature EM (McTiernan et al. 1999), perhaps sug- +gesting the presence of a similar cool component. The +cool component is probably ubiquitous in solar and +stellar flares and represents an average of the low- +temperature tail in the EM distribution. +The cool component’s EMs of the κ1 Ceti flares in- +crease steadily during the rising phase, but the footpoint +plasma’s EMs in the HYDRAD simulations rapidly in- +crease to half the maximum at the onsets. The Septem- +ber flare may be statistics limited due to its quick rise, +but the December flare clearly shows that the cool com- +ponent’s EM steadily increases with a possible step-wise +increase in the middle of the rise. This may suggest that +each flare is an assembly of multiple loops, which is well +known from e.g. spatially-resolved UV and optical imag- +ing of solar flares (e.g., Aschwanden & Alexander 2001). +Multiple loop models can reproduce energy-dependent +X-ray time variations of solar flares (Reep & Toriumi +2017). +If the observed flares are multiple loop events, we +should ideally convolve the HYDRAD simulations with +single loop event rates. +Very hard X-rays (>20 keV) +or microwave emission can trace the flux variation of +the injected non-thermal reconnection particles (e.g., +Benz 2017), but we do not have simultaneous data in +these bands, unfortunately. +Earlier flare observations +suggest that these emissions drop before the soft X-ray +peaks (Lin et al. 2003; Asai et al. 2004; Veronig et al. +2005), which are ∼200 sec in the September flare and +∼1 ksec in the December flare. A convolution in each +timescale may change the EMpeak ratio and the flare de- + +14 +Hamaguchi et al. +( ) +( ) +( ) +Figure 10. EM value (blue) at each temperature bucket peak and the corresponding plasma temperature (grey) in the 10×109 cm +loop simulation. The solid green line is the best-fit model of the EM values by a linear plus exponential decay model. The +model reproduces the EM variation well. +Figure 11. Left: Synthetic NICER spectrum of the 10×109 cm simulation at the flare peak (600−700 sec). The flare spectrum is +normalized to have the 0.3−2 keV flux at 2.2×10−11 ergs cm−2 s−1 and combined with the quiescent spectrum of the September +flare. Right: The same spectrum but with EMs below the evaporated plasma temperature (log T <7.3) reduced to 10%. The +left spectrum has strong emission at ∼0.8 keV, while the right spectrum is close to the observed flare peak spectra. The upper +right corner of each panel shows the best-fit parameters of a 2T apec model (the units are keV for kT and cm−3 for EM). Each +plot uses the same color scheme as of Figures 4 and 5. +cay timescale. It should not change the cool component +delay timescales. +We compare the derived κ1 Ceti flare parameters, +∆tdelay, τdecay, and the EMpeak ratio with the simula- +tion (Table 3). These values vary monotonically with +the loop length in the simulation, so we estimate the +loop length for each parameter by linearly interpolating +or extrapolating the two neighboring values (Table 4). +We also list three other estimates from the literature. +The first estimate is an empirical relation of the ribbon +distance with the decay timescale in solar flares (Tori- +umi et al. 2017, equation 4). We approximate the decay +timescale of the 1−8˚A energy flux with τdecay of the hot +component in Table 2 and assume a semi-circular flare +loop shape to derive the loop length. The second es- +timate is a quasi-statistic cooling model for a constant +radiative and conductive timescale ratio (van den Oord +& Mewe 1989; Tsuboi et al. 2000, equation A5). A prob- +lem with this estimate is that flares never truly cool stat- +ically (e.g., Cargill et al. 1995). The third estimate is +a magnetic reconnection model, assuming that the gas +pressure of a flare loop is comparable to the magnetic +pressure (Shibata & Yokoyama 2002, SY02). A caveat is +that the model requires the unmeasurable preflare pro- +ton density. All estimates but the EMpeak ratio are con- +sistent with ≈1010 cm loop lengths. All estimates but + +X-ray Flares from κ1 Ceti +15 +Table 4. Flare Loop Length Estimate +Flare +HYDRAD +Sun +QS +SY02 +∆tdelay +EMpeak ratio +τdecay +190917 +13.5 +60.3 +2.9 +6.2 +8.9 +12.9 +191210 +17.6 +72.3 +9.2 +13.6 +22.5 +8.7 +Note— Unit in 109 cm. HYDRAD: linear interpolation or extrap- +olation of the nearest two values of the HYDRAD simulation in +Table 3. Sun: Solar flare ribbon distance relation in equation 4 of +Toriumi et al. (2017). The derived dribbon values are multiplied +by π/2. QS: Quasi-Static cooling model in Tsuboi et al. (2000), +equation A5. SY02: Equation 7b in Shibata & Yokoyama (2002) +for the preflare proton density at 1011 cm−3. +SY02 suggest that the December flare has a longer flare +loop than the September flare. +The derived loop length of ≈1010 cm is near the upper +end but still within the range of solar flare loops (e.g., +Toriumi et al. 2017). Since κ1 Ceti has about the same +stellar radius as the Sun, we can safely assume that the +observed κ1 Ceti flares have similar magnetic field ge- +ometries to moderately large solar flare loops. However, +the peak EMs ∼3×1052 cm−3 are about two orders of +magnitudes larger than the EMs of solar flares with sim- +ilar loop lengths. One solution is that the κ1 Ceti flares +have an order of magnitude higher flare plasma density. +Such high-density plasma radiatively cools with an order +of magnitude shorter timescales, but the κ1 Ceti’s flare +decay timescales are consistent with the solar flare’s de- +cay time relation (Table 4). The other solution is that +the κ1 Ceti flares have two orders of magnitude larger +widths and/or have thicker magnetic loops. +The EMpeak ratio derives inconsistent loop lengths, +possibly because the HYDRAD simulation systemati- +cally overestimates the footpoint component. The foot- +point component is comprised of all temperature buckets +below the evaporated plasma temperature (see Figure 7 +left). +We therefore reduce the footpoint component’s +EMs of Figure 11 left simulation — the EMs below the +evaporated plasma temperature, log T =7.3 (K) — to +10%, as a trial. +Then, the synthetic spectrum looks +more similar to the observed spectra near the flare peaks +(Figure 11 right), and the best-fit 2T apec model has a +smaller EMpeak ratio at ∼0.18. As observed, this model +also derives a higher hot component temperature at 2.0 +keV. +The +footpoint +plasma +at +the +height +of +∼1.2−1.6×108 cm is in the transition region (Figures 8, +9). The line of sight should have more intervening ma- +terial than the evaporated plasma in the corona. Still, +attenuating ∼0.9 keV X-rays by ∼80% requires a hydro- +gen column density at NH ∼1022 cm−2, corresponding +to a physical depth of ∼1010−11 cm for the density of +the transition region (n ∼1011−12 cm−3). Flare loops +need to be viewed almost edge-on to have this depth, +but realizing such geometries for both loops are less +likely. Therefore, the observed flares should have less +footpoint plasma to the evaporated plasma than the +HYDRAD simulations. The RHESSI observatory found +in the hard X-ray band (>10 keV) that solar flares have +several times higher electron rates at the looptop than at +the footpoints during the impulsive phase3, implying the +electrons accumulate in the looptop (Sim˜oes & Kontar +2013). The κ1 Ceti flares may also have a mechanism to +suppress electron transportation to the footpoints and +to reduce thermal conduction. Such a mechanism may +also explain the slower cooling of the evaporated plasma +compared to the HYDRAD simulations. +One possible mechanism to suppress electron trans- +port is that the flare magnetic loop expands toward +the looptop, trapping charged particles in a magnetic +mirror. Solar coronal loops, whether quiescent or flar- +ing, do not necessarily show an expansion of the loop +width along their lengths (Klimchuk et al. 1992; Klim- +chuk 2000; Klimchuk & DeForest 2020, and references +therein). However, the magnetic field strength falls off +with height in the corona, implying that there should be +an expansion of the cross-sectional area of loops (e.g., +Dud´ık et al. 2014), and models are unable to reproduce +both hot and cool emission simultaneously without an +area expansion (Warren et al. 2010; Reep et al. 2022a). +The loop expansion reduces the thermal conductivity +near the footpoints. +A preliminary 10×109 cm loop +simulation with the expansion geometry in Reep et al. +(2022b) does not produce a small EMpeak ratio, but the +cool component EM peaks significantly later than the +constant loop simulation. The other possible mechanism +is that the flare loops have turbulent magnetic fluctua- +tions, which would increase the frequency of Coulomb +collisions, suppressing the energy transport and reduc- +ing the thermal conductivity (e.g., Bian et al. 2016; +Allred et al. 2022). This mechanism increases coronal +temperatures compared to those in a model with colli- +sionally dominated transport. +6. CONCLUSION +NICER observed two moderately strong X-ray flares +from κ1 Ceti, a nearby young solar analog, in 2019. +3 During the impulsive phase, the magnetic reconnection acceler- +ates charged particles, which emit hard non-thermal X-rays (e.g., +Benz 2017). This phase occurs mostly before the cool component +rises. + +16 +Hamaguchi et al. +NICER’s excellent soft X-ray sensitivity, good energy +resolution, and large collecting area bring rare details of +bright X-ray flares from the onsets through the peaks. +Both flares show conventional stellar flare variations +above 2 keV with a rapid rise and decay, having sim- +ilar X-ray fluxes at ∼2.2×10−11 ergs cm−2 s−1 between +0.3−2 keV and high plasma temperatures at ∼3 keV +near the peaks. +Their bolometric energies estimated +from the X-ray radiated energies, ∼3−9×1032 ergs, are +comparable to superflares. The flare on September 17 +varies in several hundred seconds in X-rays, with an in- +teresting flat soft X-ray flux peak. The flare on Decem- +ber 10 varies 2−4 times more slowly, showing a similar +but less extreme variation in the soft band. +The time-resolved spectra show that, in the rising +phase, the hard band slope increases first, and a hump +at ∼0.9 keV, originating from the Fe L line complex, fol- +lows. Most spectra require two temperature optically- +thin thermal plasma components at kT ∼1 keV and +∼3 keV on top of the quiescent component. The hot +component mainly reproduces the hard band slope, and +the cool component does the 0.9 keV hump. Both com- +ponents’ EMs rise linearly on similar timescales, but the +cool component is delayed by 100−200 sec. The Septem- +ber flare has a longer delay time relative to the flare rise +duration and a more substantial cool component than +the December flare, producing a heavily rounded flare +peak. +The HYDRAD field-aligned numerical simulations +demonstrate that the cooler footpoint plasmas start to +increase a few hundred seconds after the hot evaporated +plasmas increase — longer flare loops have longer time +delays and weaker cool components. This result indi- +cates that the cool components in the κ1 Ceti flares orig- +inate primarily from the footpoint plasma and that the +September flare stems from a shorter flare loop than the +December flare. The estimated loop lengths of ≈1010 cm +are large but still within the range of solar flare loops. +Since the κ1 Ceti flares have more than two orders of +magnitudes larger EMs than the solar flares, they need +significantly higher loop plasma densities or thicknesses. +A significant discrepancy in the EMpeak ratio may sug- +gest that the HYDRAD simulations overestimate the +footpoint EMs and require a mechanism to suppress elec- +tron transport, such as expanded magnetic loops or tur- +bulent magnetic fluctuations. A difference in the cool +component’s EM rise may suggest that both flares are +multiple loop events, as seen in solar flares. +The NICER’s κ1 Ceti observations and the HYDRAD +simulations demonstrate that the time delay of the cool +component and the peak EM ratio of the two temper- +ature plasma components can be used as new, effective +parameters for estimating the flare loop length. +We +should confirm the derived relations with more flare +samples of various luminosities, durations, peak temper- +ature, and stellar types with existing or future NICER +observations. Simultaneous multi-wavelength observa- +tions will also greatly help constrain the flare parame- +ters. In particular, UV and optical observations with +Hubble Space Telescope or the TESS observatory trace +hot chromospheric gas, helping understand the whole +chromospheric and coronal heating process. The HY- +DRAD numerical simulations still have discrepancies +with observations. We should decipher the cause with +further studies and improve the model to explain the +observations. +ACKNOWLEDGMENTS +The material is based upon work supported by NASA +under award number 80GSFC21M0002. JWR was sup- +ported by the Office of Naval Research 6.1 Support +Program. +VSA acknowledges the funds from NICER +GO Cycle 2 project award number 80NSSC21K0101. +This work is supported by JSPS KAKENHI Grant Nos. +JP20KK0072, JP21H01124, and JP21H04492, and by +NINS Grant Nos. 01321802 and 01311904. +This re- +search has made use of data and/or software provided +by the High Energy Astrophysics Science Archive Re- +search Center (HEASARC), which is a service of the As- +trophysics Science Division at NASA/GSFC. We thank +Mr. Craig Gordon for helping resolve a PYXSPEC prob- +lem. +We thank Dr. Andrew Pollock for suggestions +of XMM-Newton RGS data analysis. +We thank Drs. +Stephen Drake, Yuta Notsu, Michael F. Corcoran and +Konstantin V. Getman for discussions about stellar flare +physics. +Facilities: NICER(XTI), XMM(RGS) +Software: +HEASoft (Nasa High Energy Astro- +physics Science Archive Research Center (Heasarc) +2014), xspec (Arnaud 1996), scipy (Virtanen et al. 2020), +astropy (Astropy Collaboration et al. 2013; Scargle et al. +2013), SAS (v19.0; Gabriel et al. 2004), HYDRAD (Brad- +shaw & Mason 2003) + +X-ray Flares from κ1 Ceti +17 +APPENDIX +A. ELEMENTAL ABUNDANCE MEASUREMENT +Telleschi et al. (2005) extensively studied the coronal elemental abundance of κ1 Ceti using XMM/RGS data in +2002. +However, the XMM-Newton instrumental calibration4 and the plasma emission codes (e.g., ATOMDB5) have +significantly improved since then. The elemental abundance of the star might also have changed in 17 years. We, thus, +independently measure the coronal elemental abundance of κ1 Ceti using the XMM/RGS data obtained on 2018 July +30 and 2019 January 29 (ObsID: 0822790901, 0822791001, PI: Wargelin). +We reprocess these datasets with SAS version 19.06. EPIC/MOS2 turns off during these observations, while the +EPIC-pn uses the timing mode with relatively poor spectral resolution. We thus analyze EPIC/MOS1 and RGS data. +For MOS1, we take a 15′′ radius circular source region centered at the X-ray peak position. The MOS1 on-axis CCD +operates with the small window mode so that we take background data from a source-free region from the surrounding +CCDs. The MOS1 light curves of these observations do not show significant time variations. EPIC/MOS1 measures +the 0.6−1.2 keV flux during the second observation at ∼3.5×10−12 ergs cm−2 s−1, which is ∼16% lower than the first +observation. This flux is nearly the lowest among the NICER monitoring observations of κ1 Ceti. +We produce the MOS1 spectra using the same source and background regions. For RGS, we run rgsproc for the +target position measured from the MOS1 image and produce the source and background spectra (Figure A1). We only +use the first-order RGS spectra as the second-order RGS spectra do not have enough photon counts to identify emission +lines. We fit the unbinned MOS1 and RGS spectra simultaneously using Cash statistic (c-stat) built in xspec (Cash +1979). The Cash statistic needs to add background as an additive model component so that we simultaneously fit +background spectra by an empirical model (power-law + 4 Gaussians), convolved with the source response (rmf) +weighted with the background areal scale (backscal). 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Applied Elemental Abundance +Element +κ1 Ceti +Sun +Relative to Sun +Number +Number +H +1.00f +1.00E+00 +1.00E+00 +He +1.00f +8.51E-02 +8.51E-02 +Li +1.00f +1.12E-11 +1.12E-11 +Be +1.00f +2.40E-11 +2.40E-11 +B +1.00f +5.01E-10 +5.01E-10 +C +0.42 +1.12E-04 +2.69E-04 +N +0.44 +2.99E-05 +6.76E-05 +O +0.40 +1.95E-04 +4.90E-04 +F +1.00f +3.63E-08 +3.63E-08 +Ne +0.62 +5.29E-05 +8.51E-05 +Na +1.00f +1.74E-06 +1.74E-06 +Mg +0.68 +2.70E-05 +3.98E-05 +Al +1.00f +2.82E-06 +2.82E-06 +Si +0.55 +1.77E-05 +3.24E-05 +P +1.00f +2.57E-07 +2.57E-07 +S +0.19 +2.51E-06 +1.32E-05 +Cl +1.00f +3.16E-07 +3.16E-07 +Ar +0.19 +4.83E-07 +2.51E-06 +K +1.00f +1.07E-07 +1.07E-07 +Ca +0.58 +1.28E-06 +2.19E-06 +Sc +1.00f +1.41E-09 +1.41E-09 +Ti +1.00f +8.91E-08 +8.91E-08 +V +1.00f +8.51E-09 +8.51E-09 +Cr +1.00f +4.37E-07 +4.37E-07 +Mn +1.00f +2.69E-07 +2.69E-07 +Fe +0.64 +2.03E-05 +3.16E-05 +Co +1.00f +9.77E-08 +9.77E-08 +Ni +1.59 +2.64E-06 +1.66E-06 +Cu +1.00f +1.55E-08 +1.55E-08 +Zn +1.00f +3.63E-08 +3.63E-08 +Note— Abundance numbers relative to H. Solar abun- +dance reference: Asplund et al. (2009). ffixed at the +solar values in the XMM-Newton spectral fits. + +X-ray Flares from κ1 Ceti +21 +5.0 +1 +0.01 +0.1 +1 +10 +counts s−1 keV−1 +Energy (keV) +NVI +NVII +OVII +OVIII +FeXVIII +FeXIX +NeIX +FeXVII +NeX +MgXI +Figure A1. +XMM-Newton RGS1+2 grating spectrum of κ1 Ceti combined from the 2018 and 2019 observations. The spectrum +includes both source and background data (black). The red line shows the best-fit 2T apec model, and the blue line does the +corresponding background model. The black line is the sum of these models. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Sagamihara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Kanagawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 252-5210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Japan 7X-ray Astrophysics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NASA/GSFC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Greenbelt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' MD 20771,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' USA (Received 2022 July 18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Accepted 2022 December 21) ABSTRACT The Neutron star Interior Composition ExploreR (NICER) X-ray observatory observed two powerful X-ray flares equivalent to superflares from the nearby young solar-like star, κ1 Ceti, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER follows each flare from the onset through the early decay, collecting over 30 cnts s−1 near the peak, enabling a detailed spectral variation study of the flare rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The flare in September varies quickly in ∼800 sec, while the flare in December has a few times longer timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In both flares, the hard band (2−4 keV) light curves show typical stellar X-ray flare variations with a rapid rise and slow decay, while the soft X-ray light curves, especially of the September flare, have prolonged flat peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The time-resolved spectra require two temperature plasma components at kT ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−1 keV and ∼2−4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Both components vary similarly, but the cool component lags by ∼200 sec with a 4−6 times smaller emission measure (EM) compared to the hot component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A comparison with hydrodynamic flare loop simulations indicates that the cool component originates from X-ray plasma near the magnetic loop footpoints, which mainly cools via thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The time lag represents the travel time of the evaporated gas through the entire flare loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The cool component has several times smaller EM than its simulated counterpart, suggesting a suppression of conductive cooling possibly by the expansion of the loop cross-sectional area or turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The cool component’s time lag and small EM ratio provide important constraints on the flare loop geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Keywords: Main sequence stars (1000), Solar analogs (1941), Stellar flares (1603), Stellar x-ray flares (1637) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' INTRODUCTION Solar and stellar flares are the most energetic events on low-mass stars (Haisch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' G¨udel & Naz´e 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' They represent the rapid conversion of magnetic energy of active regions into kinetic and thermal ener- gies, radiating from radio to gamma-rays and ejecting high-energy nuclei and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Powerful solar flares have disrupted the Earth’s magnetosphere and human Corresponding author: Kenji Hamaguchi kenji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='hamaguchi@umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='edu activity, yet flares of young Sun-like stars can far sur- pass their solar counterparts in energy and frequency, with their enhanced magnetic dynamos driven by rapid rotations and deep convections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Their intense radiation could impact the exoplanetary environment and habit- ability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Airapetian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' These flares, even with substantial energy variations, share similar behavior and characteristics and arise from the universal magnetic reconnection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Magnetic reconnection efficiently accelerates particles to high energies (≳10 keV), which bombards the footpoints of the loops with high-energy particles and heats the chromosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' the evaporated gas fills the magnetic loop arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='01377v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='HE] 3 Jan 2023 2 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' and gradually cools down via radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The evaporated gas at ≈107 K radiates primarily in soft X-rays between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1−10 keV (≈1−100 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Therefore, soft X-ray obser- vations are crucial in understanding the flare geometry and heating mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' During a typical flare, soft X-ray emission rises quickly as the evaporated gas fills the magnetic loop and de- cays quasi-exponentially as it gradually cools down ra- diatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Earlier studies have focused on the peak and decay phase of flares (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' van den Oord & Mewe 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Reale & Micela 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Tsuboi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Favata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' They suggested that powerful flares tend to decay slowly and originate from larger flare loops, which exceed the stellar radius in extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Direct solar flare imagings, stellar flare occultation observations, or theoretical models support this idea, but the models can significantly overestimate the flare size due to continuous heating, multiple loop structures or subsequent flares during the decay phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Toriumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Schmitt & Favata 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' G¨udel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Reep & Toriumi 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The rising phase holds crucial information on the flare geometry and heating mechanism (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Reale 2007) as it goes through initial heating, evaporation, and loop fill- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, the rising phase is often shorter than the decaying phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Reep & Knizhnik 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Getman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021), and so has been mostly limited to duration or crude hardness ratio studies in the soft X-ray band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' An exception is an XMM-Newton observation of Prox- ima Centauri, which caught a bright flare from the onset to the middle of the decay, recording ≳100 cnts s−1 near the peak (G¨udel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2002, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Reale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The X-ray hardness ratio reached its maximum in the middle of the rise and started to decline near the flux peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The timing of maximum hardness coincides with the U band (3000−3900˚A) flux peak measured with the onboard Optical Monitor, suggesting a connection be- tween the coronal and chromospheric heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The NICER (Neutron star Interior Composition Ex- ploreR) X-ray observatory onboard the International Space Station (ISS) (Gendreau & Arzoumanian 2017) observed two bright X-ray flares from the nearby solar- like star κ(kappa)1 Ceti (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' HD 20630, HIP 15457, d =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='16 pc, mass: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='04 M⊙, radius: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='10 R⊙, ef- fective temperature: 5665 K, Ribas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Rucin- ski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004) during a monitoring program for the Sellers Exoplanet Environments Collaboration (SEEC)1 in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This star shows intense magnetic activity due to its fast stellar rotation (P =9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 days), emitting two 1 https://seec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='gov orders of magnitudes higher coronal X-rays and chro- mospheric UV light than the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In 1986, the star showed a signature of a superflare event in the He I D3 (λ5875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 ˚A) optical line, with an estimated total flare en- ergy of E ≈2×1034 ergs (Schaefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Still, the the radiation from tranition region and coronal plasma satisfies a solar magnetic flux scaling law similar to other Sun-like stars (Toriumi & Airapetian 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' These char- acteristics suggest that κ1 Ceti is a young solar analog at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='8 Gyrs old with an enhanced solar-type coro- nal and chromospheric heating rates (Airapetian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Its global-scale magnetic shear may cause super- flares that eject huge masses of coronal material (Lynch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The NICER X-ray observatory primarily aims at studying rapidly rotating neutron stars with very high timing resolution, but its superb soft X-ray collecting power, wide dynamic range, high throughput and mod- erate background, decent energy resolution, tolerance to optical light, and rapid maneuvering capability make it a powerful tool for observing nearby solar-type stars with sporadic bright X-ray flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This manuscript de- scribes analysis of NICER observations of the two pow- erful X-ray flares from κ1 Ceti and performs hydrody- namic simulations of single loop flares to interpret the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The result provides how X-ray plasmas develop during the flare rising phase in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' OBSERVATION The NICER X-ray Timing Instrument (XTI) is an ar- ray of aligned 56 X-ray modules, each of which consists of an X-ray concentrator (XRC, Okajima et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2016) and a silicon drift detector (SDD, Prigozhin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Each XRC concentrates X-rays within a ∼3′ radius field of view to the paired SDD, which detects each photon with accuracy at ∼84 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The XTI as a whole has one of the largest collecting areas among X-ray instruments between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2−12 keV (∼1900 cm−2 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We use 50 XTI modules as the remaining six (ID: 11, 14, 20, 22, 34, 60) are inactive or noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER can continuously observe a target up to ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 ksec in every ISS orbit (∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 ksec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However tar- get visibility can be limited by Earth or ISS structure occultation, or proximity to high particle regions such as the South Atlantic Anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER can quickly slew the telescope and so observes multiple targets in each ISS orbit to maximize the observing efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This ca- pability enables NICER to visit a target frequently, but it can also cause scheduling conflicts with other timely targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER started monitoring κ1 Ceti on 2019 Septem- ber 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' it has observed the star for ∼180 ksec be- X-ray Flares from κ1 Ceti 3 60000 40000 20000 0 0 5 10 15 20 25 Rate (cnts s 1) Onset: 2019-09-17 09:36:52 4000 2000 0 2000 4000 6000 Time (sec) 0 5 10 15 20 25 Rate (cnts s 1) Onset: 2019-12-10 19:13:50 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Background subtracted light curves of κ1 Ceti between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV on 2019 September 16−17 (top) and 2019 December 10 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER opportunely caught the rising phase of a bright X-ray flare during each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The grey and yellow lines show the background level estimated with the NICER tool nibackgen3C50 and an average count of each Bayesian block derived with the astropy tool bayesian blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The time on the top shows the light curve origin, an estimated flare onset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The color bars present quiescent (magenta), preflare (red), flare rise & peak (blue) and flare decay end (cyan) intervals for the spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Each data bin is 50 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' tween 2019−2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' During these monitoring observa- tions, NICER detected two prominent X-ray flares on 2019 September 17 and December 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Earlier X-ray imaging observations did not detect any X-ray sources at significant X-ray brightness within 3′ from κ1 Ceti (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Telleschi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2005), indicating that the flares originate from κ1 Ceti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We analyze the NICER observations ID: 2300020101, 2300020102, and 2300020114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We reprocess the datasets with the NICER calibration ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' CALDB XTI(20210707), using nicerl2 in HEASoft ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='29c and NICERDAS ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' V008c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We evaluate particle back- ground using nibackgen3C50 ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' v7b with the parame- ters dtmin=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0, dtmax=60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0, hbgcut=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1, s0cut=30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 (Remillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We use python ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='7, numpy ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3, scipy ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 and astropy ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' OBSERVATION RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Light Curves The first flare occurred on 2019 September 17, dur- ing the last snapshot of the one-day observation of κ1 Ceti from September 16 (190917, Figure 1 top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The snapshot only lasts for ∼800 sec, but it covers the rise and beginning of the decay of the flare as the flare varies very quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The Bayesian block analysis tool, bayesian blocks in the astropy package (Scargle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2013) — nonparametric statistical analysis to detect sig- nificant flux change in time-series data — does not sug- gest that the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV count rate changes from the previous snapshot to the first 100 sec of this snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This result suggests that the flare begins around the boundary of the last snapshot’s second and third time bins, 2019 September 17 at 9h 36m 52s UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 4 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1st−5th rows: band-sliced light curves of κ1 Ceti around the first flare on 2019 September 17 (left) and the second flare on December 10 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' These light curves have the same horizontal scales, so the variation timescales are directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The soft band light curves on the upper panels have delayed peaks compared with the 2−4 keV light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The grey line shows the instrumental background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Each light curve has 50 sec bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Bottom row: time-series of the hardness ratio defined by (H − S)/(H + S) where H and S are the net count rates between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2−4 keV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The color bars present preflare (red), flare rise & peak (blue) and flare decay end (cyan) intervals for the spectral analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The first bin of the September flare is below −1 as nibackgen3C50 overestimates the hard band background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In each flare, the hardness ratio peaks before the total (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−4 keV) band count rate maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' X-ray Flares from κ1 Ceti 5 Figure 2 left shows band-sliced light curves of the last snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The 2−4 keV light curve is typical of solar and stellar X-ray flares (Benz & G¨udel 2010, and refer- ences therein), showing a sharp rise in ∼200 sec and a steady decline by a factor of 3 in ∼500 sec after the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, the softer band light curves rise more slowly, peak later, and decay more gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The light curves below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV may not even show a decay during the snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The soft band light curves significantly devi- ate from the hard band light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hardness ratio in the bottom panel rises quickly, peaks before the total (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−4 keV) light curve, and declines gradually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This behavior is similar to the giant flare seen from Proxima Centauri with XMM-Newton (G¨udel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The second flare occurred during the second snapshot on 2019 December 10 (191210, Figure 1 bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This observation (for ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 ksec) is longer than for the first flare, but it similarly covers the rise and beginning of the decay as the second flare develops more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER misses the middle of the decay, but the third snapshot covers the end of the decay as the light curve connects smoothly from the second snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' On the other hand, the first snapshot shows a slight elevation in the middle, but both intervals appear to be in a quiescent state with- out significant variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The light curve before the first flare also shows similar count rate variations (Figure 1 top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A Bayesian block analysis shows that the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV count rate stays at ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 cnts s−1 from the latter half of the first snapshot to the first ∼150 sec of the second snapshot, suggesting that the flare begins during the second snapshot at around 19h 14m 40s UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Mean- while, the last Bayesian block of the third snapshot has almost the same count rate (∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='8 cnts s−1), suggesting that the quiescent emission does not vary significantly during the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Besides the slow variation, the second flare has sim- ilar energy-dependent variations to the first flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The 2−4 keV light curve reaches its peak before other en- ergy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The softer band peaks are delayed: the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 keV light curve does not show a clear peak dur- ing the second snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This delay continues to the third snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The softer band light curves gradually decline, while the 2−4 keV light curve is almost flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hardness ratio reaches a maximum early during the rise, but otherwise the variation is similar to the first flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Time Resolved Spectra To understand the energy-dependent time variations, we analyze the time-resolved spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We first produce a quiescent spectrum from the snapshot directly pre- ceding each flare (the magenta bar in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The snapshot of the September flare shows no significant variation in count rates, while that in December does show a small but significant count rate increase in the middle at −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='7 ksec (Figure 1 bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Therefore, we produce two spectra separated at the time with a sta- tistically significant count rate change (change point) in the Bayesian analysis (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The quiescent spectra show a prominent hump between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='7−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV, consis- tent with emission lines of Fe XVII-XX, Ne IX-X, and OVIII ions seen in the XMM-Newton RGS spectra of κ1 Ceti in 2002 (Telleschi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The spectrum has a steep hard slope, with negligible emission above ∼2 keV, but no absorption cut-off in the soft band down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, thus, apply a thermal plasma emission model (apec) without absorption (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Assum- ing the best-fit elemental abundances of the XMM/RGS spectrum (Appendix A), we find that a 2-temperature (2T) model with kT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='27 & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='62 keV provides an acceptable fit to the quiescent spectrum for the Septem- ber flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The quiescent spectra of the December flare require a 3T model for the former interval and an addi- tional 1T component for the latter interval to account for the flux increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We use this 4T model as the fixed quiescent component for the December flare spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We produce time-resolved spectra during the flares, every 50 sec with a minimum 25 counts per bin for the September flare to track the fast variation and every 100 sec with a minimum of 50 counts per bin for the Decem- ber flare to get good photon statistics (Figures 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' For each flare, we make one spectrum before the flare onset in the same snapshot (the red bars in Figure 2 between −95—−35 sec for the first flare and −80—0 sec for the second flare), which we call the preflare spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the third snapshot of the December flare, due to a de- creased count rate, we increase the time interval of each bin to 400−600 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We also apply longer time intervals for the third snapshot of the December flare near the decay end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The preflare spectrum at the top left panel of each figure matches well the corresponding quiescent spectrum in the solid yellow line, consistent with the Bayesian block analysis of the flare onset timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' During the first flare, the flux in the entire energy band increases for the first 300 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='7−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV hump becomes broader to the high energy side as the flux at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 keV, probably originating from Fe XXII- XXIV or Ne X emission lines, is enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' After that, the hard band emission gradually declines, while the flux at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV, probably produced by Fe XVII-XX lines, strengthens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The second flare evolves more slowly but similarly to the first flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The whole band increases until ∼800 sec, and then the hump at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV begins to strengthen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the third snapshot, the flux declines 6 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Quiescent spectra of the first flare for an exposure of 870 sec from −16921 sec to −16051 sec (left) and the second flare for 789 sec from −4700 sec to −3911 sec (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The solid yellow lines are the best-fit models, and the dotted lines are the individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The upper right corner of each panel shows the best-fit parameters (the units are keV for kT and cm−3 for EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The best-fit models reproduce the observed spectra well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Best-fit Values of the Quiescent Spectra Component 190917 191210 kT EM kT EM (keV) (1051 cm−3) (keV) (1051 cm−3) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='62+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='04+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 ∆χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='f 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='44/64 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='06/175 Note— The errors show 90% confidence ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The 4th com- ponent is required for the latter interval spectrum of the 191210 observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' nearly to the preflare level with some residual in the soft and hard bands in the prior ∼1 ksec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' kT and EM Variations during the Flares The time-resolved spectra do not suggest any signif- icant variation of the quiescent component during the flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, therefore, reproduce each time-resolved spec- trum by a model with variable flare components plus the fixed quiescent component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hard slopes of most spectra require a kT ∼3 keV plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, colli- sional equilibrium plasma at that temperature does not emit Fe XVII-XX emission lines at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV, which are enhanced near the flare peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Non-equilibrium ioniza- tion plasmas emit these lines at τ ∼2×1010 s cm−3, but they do not reproduce the emission lines at ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 keV observed during the rising phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This result suggests that the flare spectra need at least two plasma compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, thus, apply a 2T apec plasma model for the flare emission, with elemental abundances fixed to the best-fit XMM/RGS values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We find that a model with kT ∼2−4 keV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−1 keV components repro- duces each spectrum well and that these temperatures vary monotonically over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, the spectral parameters are poorly constrained near the flare onset and end due to weak flare emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, therefore, fit all spectra simultaneously, assuming that each compo- nent’s plasma temperature varies linearly with time, and find reasonable results with χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' at 685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='96/616 for the first flare and 1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='41/919 for the second flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Fig- ures 4 & 5 plot the best-fit model for individual spectra and Figure 6 shows the best-fit kT and EM values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The third and fourth columns in Table 2 show the best-fit kT slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' X-ray Flares from κ1 Ceti 7 preflare Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Time-resolved spectra of κ1 Ceti during the first flare (190917).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The red/blue line depicts the best-fit cool/hot component of the flare spectrum, and the yellow line does the fixed quiescent component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The solid black line is their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hot component soars in the rising phase, dominating most energy bands, while the cool component is more significant at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV with Fe L emission lines after ∼350 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The top right of each panel shows each spectrum’s time interval in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The cool component EMs are mainly determined by fits of the Fe L emission line complex to the ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We examine whether the line intensity con- straints in the applied model are adequate for this anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' First, the model fixes the elemental abundances during the flares at the best-fit XMM/RGS quiescent spectrum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, some solar or stellar X-ray flares show apparent elemental abundance changes from the pre or post-flare states (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Osten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Au- dard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The best-fit spec- tral models also fit well the ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 keV bump with the Fe L and Ne K lines in the hot component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Since the hard spectral slope determines the hot component’s EM, the hot component’s Fe and Ne abundances are consistent with the assumed abundances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', the coronal abun- dances are not observed to significantly change during the κ1 Ceti flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Second, suppose the cool compo- nent did not reach equilibrium at ionization timescales of ≲1010 s cm−3 as opposed to the model assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In that case, the plasma should emit weaker Fe L lines than the equilibrium case and require a larger EM to account for the observed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 keV bump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, no observed spectra show strong emission below ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='7 keV expected from low ionized oxygen and carbon emission lines from such non-equilibrium plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In addition, preflare loops probably have densities over ∼1011 cm−3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Osten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2006), suggesting that the Fe L line complex develops within ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' These results suggest that the cool component EM measurements are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hot component explains most of the initial flux rise in each flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The component in the second flare is slightly hotter and cools down significantly slower 8 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' preflare Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Time-sliced spectra of κ1 Ceti during the second flare (191210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The flare spectral components behave similarly to those of the first flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The cool component exceeds the hot component at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV after ∼1 ksec but not so much as during the first flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' than the first flare, but the component stays kT ≳2 keV throughout both observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' On the other hand, the cool component develops more slowly than the hot com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The plasma temperature does not vary strongly at ∼1 keV around the flare peak, but it declines to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 keV by the end of the second flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EM time series in Figure 6 bottom panels confirms the similarity of the two flares: i) the hot EM varies with a linear rise and a slow decay, ii) the cool EM varies simi- larly to the hot EM but with a delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' To quantitatively evaluate their variations, we fit the EM time series with X-ray Flares from κ1 Ceti 9 the following conventional formula for stellar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EM(t) = 0 t < tonset = EMpeak t − tonset ∆trise tonset ≤ t < tpeak (1) = EMpeak exp(−t − tpeak τdecay ) tpeak ≤ t where tonset, tpeak, EMpeak and τdecay are free param- eters and ∆trise = tpeak − tonset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' For the fittings, we use curve fit in the scipy package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We fix ∆trise and τdecay of the 190917 flare’s cool component at the best- fit values of the hot component as the cool component does not show a clear EM peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table 2 shows the best- fit result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hot component’s tonset is close to zero, again consistent with the Bayesian blocks measurement of the flare onset in each flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In contrast, the cool component’s tonset is significantly delayed from the hot component’s tonset (hereafter ∆tdelay = tonset(cool) − tonset(hot)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the second flare, the cool component has similar ∆trise to, but a factor of two longer τdecay than, the hot component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The second flare has longer durations in ∆tdelay, ∆trise, and τdecay than the first flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' These behaviors explain the energy-dependent varia- tions of the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The 2−4 keV band light curve is dominated by the hot component’s behavior, showing a conventional stellar flare variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The softer bands add the cool component’s behavior with a conventional flare variation but a time delay compared to the hot component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV light curve deviates most with the strong 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV hump from the cool component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The deviation is stronger in the first flare with a larger relative time delay (∆tdelay/∆trise) and a larger EMpeak ratio (EMpeak(cool)/EMpeak(hot)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' HYDRODYNAMIC SIMULATIONS OF SINGLE LOOP FLARES The hot component constitutes the major part of the flare emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' As discussed in numerous studies, it should originate from radiatively-cooling plasma inside the flare magnetic loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Then, what is the cool compo- nent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Flare spectral fits often require two temperatures or more (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Paudel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021), but the nature of the cool component is poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Our NICER study provides this component’s time variation through the flare rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We run hydrodynamic simula- tions of single magnetic loop flares to understand the cool component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We employ a field-aligned hydrodynamic code, the HYDrodynamics and RADiation code (HYDRAD2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='com/rice-solar-physics/HYDRAD Bradshaw & Mason 2003), used to study heating in the solar corona and solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The code solves the equa- tions for the conservation of mass, momentum, and en- ergy for plasma confined to a magnetic flux tube (Brad- shaw & Cargill 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The loops are assumed to be heated by non-thermal electrons, accelerated by mag- netic reconnection near the loop’s apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' As the electrons propagate, they deposit their energy through Coulomb collisions with the ambient plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The majority of the heat is deposited in the upper chromosphere, caus- ing a rapid increase in temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' It then drives an expansion of material (chromospheric evaporation), carrying hot and dense plasma into the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The assumed form of the heating function that we use was derived by Emslie (1978), with modification for non-uniform ionization in the chromosphere (Hawley & Fisher 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' As the loop evolves, the plasma cools through thermal conduction and radiation, which we calculated using the atomic database CHIANTI (Dere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1997), version 10 (Del Zanna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We use the elemental abundances derived from the XMM/RGS spectra (Table A1), but our preliminary study using solar elemental abundances provides a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Our simulations assume a magnetic loop with a uniform cross-section and injected particles with a power-law en- ergy distribution for 200 sec with an energy flux peak- ing at 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 ergs cm−2 s−1 at 100 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Since the two NICER flares have different flare decay timescales and plausibly different magnetic loop lengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Toriumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Reep & Toriumi 2017), the simulations con- sider loop lengths at 5, 10, 15, 20, and 25×109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The derived EM normalization can be adjusted by changing the cross-section of the magnetic loops (equivalently, the total volume of the loops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The left panel of Figure 7 shows the EM evolution of the 10×109 cm cm flare loop simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The EM is dominated by the hottest plasma emission that peaks near ∼600 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This component represents a radiative- cooling, evaporated plasma that fills the magnetic loop, corresponding to the hot component of the observing flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Since the evaporated plasma cools down grad- ually under thermal equilibrium, a single temperature bucket dominates near its peak and drops to zero quickly once the plasma cools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A secondary component is a group of low-temperature buckets that rises and falls similarly to the main component at one-third the EM of the evaporated plasma component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Each temperature bucket stays in this group until the evaporated plasma cools down to its temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This secondary component represents plasmas at transition regions near the magnetic footpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Because the conductive heat flows from the looptop to the footpoints, the plasma 10 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 190917 191210 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Best-fit kT (top) and EM (bottom) values of the time-resolved flare spectra by the 2T apec models (left: 190917, right: 191210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The red/blue color shows the cool/hot plasma component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the kT plots, the solid lines and filled areas are the best-fit kT linear models and the 90% confidence areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the EM plots, the data points show the best-fit values and their 90% confidence ranges of the combined spectral fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The solid lines show the best-fit linear rise plus exponential decay model to these EM measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In each flare, the cool component rises similarly to the hot component but with a time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Flare Parameters Flare Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' kT(t = 0) kTslope tonset ∆trise τdecay EMpeak LXpeak EX Ebol (keV) (keV/ksec) (sec) (sec) (sec) (1052 cm−3) (1029 ergs s−1) (1032 ergs) (1033 ergs) 190917 cool 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='18+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='24 156 (20) 218 (fix) 572 (fix) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='71 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='061) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 hot 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='46 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='95 14 (7) 218 (15) 572 (86) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 191210 cool 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='02+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='01 227 (35) 923 (99) 2591 (324) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='45 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='032) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 hot 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='32+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='09 33 (37) 1003 (59) 1135 (111) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='8 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='12) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 Note— kT(t = 0), kTslope: Best-fit kT linear time variation model of the combined spectral fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The errors show 90% confidence ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' tonset, ∆trise, τdecay, EMpeak: Best-fit linear rise plus exponential decay model of the EM time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The parentheses show 1σ confidence ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' LXpeak: Peak X-ray luminosity between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EX: Total X-ray flare energy between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−10 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Ebol: Total bolometric flare energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The left values use a relation to the GOES band (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='55-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4 keV) flare-radiated energy for active stars (Table 2 in Osten & Wolk 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The right values use a relation to the GOES band solar flare peak flux (equation (13) in Aschwanden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' has a strong temperature gradient and responds to the evaporated plasma’s variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The other loop length simulations show similar EM variations with different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the first 500 sec, the log T ≥7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 (K) buckets only reflect the evaporated plasma component, while the log T <7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 (K) buckets reflect the footpoint plasma com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, therefore, define two temperature ranges, log T =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 (K) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 (K), and sum up EMs within each range to understand their behaviors near the rising phase (Figure 7 right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' First, the EM[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9] time series of various loop lengths vary similarly for ∼200 sec from the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This EM base origi- nates from the initial heating of the plasma in the up- per chromosphere by the injected particles, and peaks at ∼100 sec in response to the assumed particle injection flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The EM[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0] does not show this component clearly, but the slow rise in the first ∼50 sec originates from the initial plasma heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' All EM[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9] plots except the 5×109 cm simu- lation show the footpoint components’ onsets as clear kinks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', at ∼150 sec for the 10×109 cm simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We measure the timing of each kink from a two linear fit (tonset(foot) in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The onset ranges between ∼80−300 sec, and longer loops have later onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The evaporated component does not have a clear onset sig- nature, so we measure the onset timing from a fit to the first 200 sec of EM[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0] by a linear function starting at tonset(eva).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The onset ranges between 17−64 sec and does not appear to correlate with the loop length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The X-ray Flares from κ1 Ceti 11 log T (K) 5 10 15 20 25 5 20 15 25 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 log T (K) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Left: Whole loop EM variations of the 10×109 cm loop simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The EMs are divided by logarithmic temperature buckets and binned every 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The plot shows two EM components — the evaporated plasma that envelopes individual temperature peaks standing up from the hot side, getting a maximum at ∼600 sec, and the footpoint plasma, a group of low-temperature buckets that rises and falls similarly at one-third of the evaporated plasma component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Right: Whole loop EM variations for the first 500 sec, summed over the temperature ranges log T =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 K (blue) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 K (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The number that labels each line is the loop length in 109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The evaporated component starts to rise at ∼50 sec, while the footpoint component is significantly delayed from the evaporated component, more with longer loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The overlapping triangular base in the red plot up to ∼250 sec originates from the initial heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' HYDRAD Simulation Result Loop Length tonset(eva) tonset(foot) ∆tdelay ∆trise(eva) τdecay(eva) EMpeak ratio kTpeak(cool/hot) (109 cm) (sec) (sec) (sec) (sec) (sec) (keV/keV) 5 63 77 15 330 760 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='32/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='40 10 64 153 89 491 1207 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='31/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='67 15 40 205 164 496 2268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='32/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='66 20 25 248 223 639 3013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='34/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='57 25 17 294 276 752 3601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='35/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='57 190917 142 218 572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='96/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='50 191210 194 1003 1135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='88/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='31 Note—In each simulation, the time origin is the particle injection start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' tonset(eva)/tonset(foot): onset time of the evaporated/footpoint component derived from one/two linear fits to the EM[7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0]/EM[6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9] time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ∆tdelay = tonset(foot) − tonset(eva).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ∆trise(eva)/τdecay(eva): rise/decay time of the evaporated component derived from a fit to the peak EMs of individual temperature buckets by a linear rise plus exponential decay model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EMpeak ratio, kTpeak(cool/hot): EMpeak ratio and plasma temperatures at the cool/hot EM peaks, derived from fits to the synthetic flare spectra with 100 sec bins by a 2T apec model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 12 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' time lags ∆tdelay (=tonset(foot) − tonset(eva)) clearly in- crease with longer loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Figure 8 shows why the footpoint component is de- layed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In the plots, the evaporated component is lo- cated in the middle part where temperature and den- sity quickly increase after the particle injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The footpoint component is located near both ends, whose temperature and density do not increase until the shocks produced by the evaporated gas’s collision at the looptop propagate down to the footpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Figure 9 displays the hydrogen and electron density product and the electron temperature, magnifying the left end on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Both the log T =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 K depth and density product increase between 140 sec and 220 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The time delay corresponds to the travel time of the flare loop by the evaporated flows and the collisional shocks, which is approximately the sound crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' It is, therefore, roughly proportional to the loop length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We also measure the decay timescales of the simu- lated flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' As described above, the evaporated plasma is in a single temperature bucket at each temperature peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, thus, take the peak EM of each tempera- ture bracket and fit them by a linear plus exponential decay model from equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The fits reproduce the EM variations well except around the peak (Figure 10 for the 10×109 cm loop simulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Longer loop flares have longer decay timescales (τdecay(eva) in Table 3), as suggested in earlier studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', van den Oord & Mewe 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Toriumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We make synthetic NICER spectra of the simulated EM distributions to compare the spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' For each simulation, we produce spectral models with 100 sec bins, assuming an apec plasma model for each temperature bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We normalize them to a peak 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−2 keV flux at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2×10−11 ergs cm−2 s−1 to match the observed two NICER flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We then generate a syn- thetic spectrum for each spectral model with the xspec fakeit tool, by convolving the model with the NICER on-axis responses, nixtiref20170601v003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='rmf and nixti- aveonaxis20170601v005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='arf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We increase photon statis- tics by a factor of 10 to reduce statistical uncertainty, equivalent to a 1 ksec exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, then, bin each synthesized spectrum to have ≥50 counts per bin, and fit each spectrum by a 2T apec model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Figure 11 left shows a synthetic spectrum of the 10×109 cm loop be- tween 600−700 sec, adding the quiescent component of the September flare for a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table 3 shows the EMpeak ratios and plasma temperatures at the EM peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The peak plasma temperatures ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='35 keV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='4−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='67 keV are similar among the simulations and significantly lower than the observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The EMpeak ratio is the highest with the 5×109 cm loop sim- ulation at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='21 and smaller with longer loop simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This result is naturally understood since the footpoint plasma volume does not change with the loop length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The numerical simulations demonstrate that the foot- point component is delayed to rise by 100−300 sec from the evaporated component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This result indicates that the cool component in the NICER spectra originates from the footpoint plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The simulations also suggest that longer flare loops have longer delays of the footpoint component rise and smaller EM peak ratios, as well as longer decay timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' All these properties are consis- tent with the properties of the two NICER flares, sug- gesting that the December flare originates from a longer flare loop than the September flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' DISCUSSION The κ1 Ceti flares in 2019 are an order of magnitude more powerful than the most powerful solar flare ever seen, the Carrington Event in 1859 (LX ∼1028 ergs s−1, Cliver & Dietrich 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Sakurai 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Their X-ray lu- minosities are near the upper end of the flare luminosity ranges of solar-type G stars (LX ≲1030 ergs s−1, Schaefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Tsuboi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Their bolometric flare- radiated energies 3−8×1033 ergs, evaluated from two independent empirical relations to the X-ray radiations among solar and active stellar flares (Aschwanden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Osten & Wolk 2015, see Table 2), qualify them as superflares (>1033 ergs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Maehara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2012) and are comparable to the κ1 Ceti superflare recorded in 1986 (∼2×1034 ergs, Schaefer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Nonethe- less, their X-ray luminosities and released X-ray ener- gies are modest among active or young stellar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' (LX ≲1032−33 ergs s−1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Benz & G¨udel 2010, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The other X-ray characteristics — the hot plasma temperature, hard band light curve, and hardness ratio variation — are similar to solar and stel- lar X-ray flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Pye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We conclude that the κ1 Ceti flares in 2019 are conventional magnetic re- connection events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The κ1 Ceti flare spectra require an additional cool (kT ≲1 keV) temperature component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Although such a component has not received much attention, well ex- posed stellar X-ray flare spectra usually require one or more components with kT ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−1 keV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', GT Mus: Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021, EV Lac: Paudel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The high-resolution XMM/RGS spectra of the Prox- ima Centauri flare suggested that the flare EM distri- bution was broad with a peak at ∼30MK and a low- temperature tail during the rise and steadily moved to low-temperatures as the flare developed (G¨udel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Reale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In solar flares, the low- temperature EM (<16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 MK) peaks later than the high- X-ray Flares from κ1 Ceti 13 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Density (top) and temperature (bottom) spatial distribution of the 10 × 109 cm loop simulation at 40, 60, 140 and 220 seconds from the particle injection start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The horizontal axis shows the distance from a footpoint along the loop: the loop top is at 5×109 cm and the two footpoints are at 0, 10×109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' From left, i) at 40 sec, the particle injection heats the footpoint chromosphere, and the evaporated gas soars into the magnetic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The evaporated spectral component starts to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ii) at 60 sec, the upward evaporation flows collide at the loop top, heating the gas further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' iii) at 140 sec, the shock propagate down the other leg, smoothing the corona’s density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' iv) at 220 sec, by the time the shock reaches the footpoints, the loop has enough high density that thermal conduction becomes extremely efficient, and the footpoint spectral component emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The red and blue lines are for electrons and hydrogen, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The black and white double arrows point to the locations of the evaporation or shock fronts from either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1022 1023 1024 1025 1026 Density Product (cm 6) 40 s 60 s 140 s 220 s 10 1 100 Position (109 cm) 105 106 107 108 Electron Temperature (K) Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Electron and hydrogen density product (top) and electron temperature (bottom) distributions of the 10 × 109 cm loop simulation near a footpoint region at t =40 sec (dotted), 60 sec (dash-dotted), 140 sec (dashed), and 220 sec (solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The horizontal axis is the same as Figure 8 but on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The filled grey area in the bottom panel shows the log T =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 K bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The boxes in the top panel show the one-dimensional volumes and density product ranges of this temperature bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Both the density product and the volume significantly increase between t =140 sec and 220 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' temperature EM (McTiernan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1999), perhaps sug- gesting the presence of a similar cool component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The cool component is probably ubiquitous in solar and stellar flares and represents an average of the low- temperature tail in the EM distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The cool component’s EMs of the κ1 Ceti flares in- crease steadily during the rising phase, but the footpoint plasma’s EMs in the HYDRAD simulations rapidly in- crease to half the maximum at the onsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The Septem- ber flare may be statistics limited due to its quick rise, but the December flare clearly shows that the cool com- ponent’s EM steadily increases with a possible step-wise increase in the middle of the rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This may suggest that each flare is an assembly of multiple loops, which is well known from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' spatially-resolved UV and optical imag- ing of solar flares (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Aschwanden & Alexander 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Multiple loop models can reproduce energy-dependent X-ray time variations of solar flares (Reep & Toriumi 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' If the observed flares are multiple loop events, we should ideally convolve the HYDRAD simulations with single loop event rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Very hard X-rays (>20 keV) or microwave emission can trace the flux variation of the injected non-thermal reconnection particles (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Benz 2017), but we do not have simultaneous data in these bands, unfortunately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Earlier flare observations suggest that these emissions drop before the soft X-ray peaks (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Asai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Veronig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2005), which are ∼200 sec in the September flare and ∼1 ksec in the December flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A convolution in each timescale may change the EMpeak ratio and the flare de- 14 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ( ) ( ) ( ) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EM value (blue) at each temperature bucket peak and the corresponding plasma temperature (grey) in the 10×109 cm loop simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The solid green line is the best-fit model of the EM values by a linear plus exponential decay model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The model reproduces the EM variation well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Left: Synthetic NICER spectrum of the 10×109 cm simulation at the flare peak (600−700 sec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The flare spectrum is normalized to have the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−2 keV flux at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2×10−11 ergs cm−2 s−1 and combined with the quiescent spectrum of the September flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Right: The same spectrum but with EMs below the evaporated plasma temperature (log T <7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3) reduced to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The left spectrum has strong emission at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='8 keV, while the right spectrum is close to the observed flare peak spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The upper right corner of each panel shows the best-fit parameters of a 2T apec model (the units are keV for kT and cm−3 for EM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Each plot uses the same color scheme as of Figures 4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' cay timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' It should not change the cool component delay timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We compare the derived κ1 Ceti flare parameters, ∆tdelay, τdecay, and the EMpeak ratio with the simula- tion (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' These values vary monotonically with the loop length in the simulation, so we estimate the loop length for each parameter by linearly interpolating or extrapolating the two neighboring values (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We also list three other estimates from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The first estimate is an empirical relation of the ribbon distance with the decay timescale in solar flares (Tori- umi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017, equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We approximate the decay timescale of the 1−8˚A energy flux with τdecay of the hot component in Table 2 and assume a semi-circular flare loop shape to derive the loop length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The second es- timate is a quasi-statistic cooling model for a constant radiative and conductive timescale ratio (van den Oord & Mewe 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Tsuboi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2000, equation A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A prob- lem with this estimate is that flares never truly cool stat- ically (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Cargill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The third estimate is a magnetic reconnection model, assuming that the gas pressure of a flare loop is comparable to the magnetic pressure (Shibata & Yokoyama 2002, SY02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A caveat is that the model requires the unmeasurable preflare pro- ton density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' All estimates but the EMpeak ratio are con- sistent with ≈1010 cm loop lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' All estimates but X-ray Flares from κ1 Ceti 15 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Flare Loop Length Estimate Flare HYDRAD Sun QS SY02 ∆tdelay EMpeak ratio τdecay 190917 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 191210 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='7 Note— Unit in 109 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' HYDRAD: linear interpolation or extrap- olation of the nearest two values of the HYDRAD simulation in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Sun: Solar flare ribbon distance relation in equation 4 of Toriumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The derived dribbon values are multiplied by π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' QS: Quasi-Static cooling model in Tsuboi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' (2000), equation A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' SY02: Equation 7b in Shibata & Yokoyama (2002) for the preflare proton density at 1011 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' SY02 suggest that the December flare has a longer flare loop than the September flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The derived loop length of ≈1010 cm is near the upper end but still within the range of solar flare loops (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Toriumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Since κ1 Ceti has about the same stellar radius as the Sun, we can safely assume that the observed κ1 Ceti flares have similar magnetic field ge- ometries to moderately large solar flare loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, the peak EMs ∼3×1052 cm−3 are about two orders of magnitudes larger than the EMs of solar flares with sim- ilar loop lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' One solution is that the κ1 Ceti flares have an order of magnitude higher flare plasma density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Such high-density plasma radiatively cools with an order of magnitude shorter timescales, but the κ1 Ceti’s flare decay timescales are consistent with the solar flare’s de- cay time relation (Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The other solution is that the κ1 Ceti flares have two orders of magnitude larger widths and/or have thicker magnetic loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The EMpeak ratio derives inconsistent loop lengths, possibly because the HYDRAD simulation systemati- cally overestimates the footpoint component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The foot- point component is comprised of all temperature buckets below the evaporated plasma temperature (see Figure 7 left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We therefore reduce the footpoint component’s EMs of Figure 11 left simulation — the EMs below the evaporated plasma temperature, log T =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3 (K) — to 10%, as a trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Then, the synthetic spectrum looks more similar to the observed spectra near the flare peaks (Figure 11 right), and the best-fit 2T apec model has a smaller EMpeak ratio at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' As observed, this model also derives a higher hot component temperature at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The footpoint plasma at the height of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6×108 cm is in the transition region (Figures 8, 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The line of sight should have more intervening ma- terial than the evaporated plasma in the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Still, attenuating ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV X-rays by ∼80% requires a hydro- gen column density at NH ∼1022 cm−2, corresponding to a physical depth of ∼1010−11 cm for the density of the transition region (n ∼1011−12 cm−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Flare loops need to be viewed almost edge-on to have this depth, but realizing such geometries for both loops are less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Therefore, the observed flares should have less footpoint plasma to the evaporated plasma than the HYDRAD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The RHESSI observatory found in the hard X-ray band (>10 keV) that solar flares have several times higher electron rates at the looptop than at the footpoints during the impulsive phase3, implying the electrons accumulate in the looptop (Sim˜oes & Kontar 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The κ1 Ceti flares may also have a mechanism to suppress electron transportation to the footpoints and to reduce thermal conduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Such a mechanism may also explain the slower cooling of the evaporated plasma compared to the HYDRAD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' One possible mechanism to suppress electron trans- port is that the flare magnetic loop expands toward the looptop, trapping charged particles in a magnetic mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Solar coronal loops, whether quiescent or flar- ing, do not necessarily show an expansion of the loop width along their lengths (Klimchuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Klim- chuk 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Klimchuk & DeForest 2020, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, the magnetic field strength falls off with height in the corona, implying that there should be an expansion of the cross-sectional area of loops (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Dud´ık et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2014), and models are unable to reproduce both hot and cool emission simultaneously without an area expansion (Warren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Reep et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The loop expansion reduces the thermal conductivity near the footpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A preliminary 10×109 cm loop simulation with the expansion geometry in Reep et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' (2022b) does not produce a small EMpeak ratio, but the cool component EM peaks significantly later than the constant loop simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The other possible mechanism is that the flare loops have turbulent magnetic fluctua- tions, which would increase the frequency of Coulomb collisions, suppressing the energy transport and reduc- ing the thermal conductivity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Allred et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This mechanism increases coronal temperatures compared to those in a model with colli- sionally dominated transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' CONCLUSION NICER observed two moderately strong X-ray flares from κ1 Ceti, a nearby young solar analog, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 3 During the impulsive phase, the magnetic reconnection acceler- ates charged particles, which emit hard non-thermal X-rays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Benz 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This phase occurs mostly before the cool component rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 16 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' NICER’s excellent soft X-ray sensitivity, good energy resolution, and large collecting area bring rare details of bright X-ray flares from the onsets through the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Both flares show conventional stellar flare variations above 2 keV with a rapid rise and decay, having sim- ilar X-ray fluxes at ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2×10−11 ergs cm−2 s−1 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='3−2 keV and high plasma temperatures at ∼3 keV near the peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Their bolometric energies estimated from the X-ray radiated energies, ∼3−9×1032 ergs, are comparable to superflares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The flare on September 17 varies in several hundred seconds in X-rays, with an in- teresting flat soft X-ray flux peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The flare on Decem- ber 10 varies 2−4 times more slowly, showing a similar but less extreme variation in the soft band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The time-resolved spectra show that, in the rising phase, the hard band slope increases first, and a hump at ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV, originating from the Fe L line complex, fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Most spectra require two temperature optically- thin thermal plasma components at kT ∼1 keV and ∼3 keV on top of the quiescent component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The hot component mainly reproduces the hard band slope, and the cool component does the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='9 keV hump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Both com- ponents’ EMs rise linearly on similar timescales, but the cool component is delayed by 100−200 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The Septem- ber flare has a longer delay time relative to the flare rise duration and a more substantial cool component than the December flare, producing a heavily rounded flare peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The HYDRAD field-aligned numerical simulations demonstrate that the cooler footpoint plasmas start to increase a few hundred seconds after the hot evaporated plasmas increase — longer flare loops have longer time delays and weaker cool components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This result indi- cates that the cool components in the κ1 Ceti flares orig- inate primarily from the footpoint plasma and that the September flare stems from a shorter flare loop than the December flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The estimated loop lengths of ≈1010 cm are large but still within the range of solar flare loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Since the κ1 Ceti flares have more than two orders of magnitudes larger EMs than the solar flares, they need significantly higher loop plasma densities or thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A significant discrepancy in the EMpeak ratio may sug- gest that the HYDRAD simulations overestimate the footpoint EMs and require a mechanism to suppress elec- tron transport, such as expanded magnetic loops or tur- bulent magnetic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A difference in the cool component’s EM rise may suggest that both flares are multiple loop events, as seen in solar flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The NICER’s κ1 Ceti observations and the HYDRAD simulations demonstrate that the time delay of the cool component and the peak EM ratio of the two temper- ature plasma components can be used as new, effective parameters for estimating the flare loop length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We should confirm the derived relations with more flare samples of various luminosities, durations, peak temper- ature, and stellar types with existing or future NICER observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Simultaneous multi-wavelength observa- tions will also greatly help constrain the flare parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' In particular, UV and optical observations with Hubble Space Telescope or the TESS observatory trace hot chromospheric gas, helping understand the whole chromospheric and coronal heating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The HY- DRAD numerical simulations still have discrepancies with observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We should decipher the cause with further studies and improve the model to explain the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ACKNOWLEDGMENTS The material is based upon work supported by NASA under award number 80GSFC21M0002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' JWR was sup- ported by the Office of Naval Research 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 Support Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' VSA acknowledges the funds from NICER GO Cycle 2 project award number 80NSSC21K0101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This work is supported by JSPS KAKENHI Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' JP20KK0072, JP21H01124, and JP21H04492, and by NINS Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 01321802 and 01311904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This re- search has made use of data and/or software provided by the High Energy Astrophysics Science Archive Re- search Center (HEASARC), which is a service of the As- trophysics Science Division at NASA/GSFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We thank Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Craig Gordon for helping resolve a PYXSPEC prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Andrew Pollock for suggestions of XMM-Newton RGS data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We thank Drs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Stephen Drake, Yuta Notsu, Michael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Corcoran and Konstantin V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Getman for discussions about stellar flare physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Facilities: NICER(XTI), XMM(RGS) Software: HEASoft (Nasa High Energy Astro- physics Science Archive Research Center (Heasarc) 2014), xspec (Arnaud 1996), scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2020), astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Scargle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2013), SAS (v19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Gabriel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2004), HYDRAD (Brad- shaw & Mason 2003) X-ray Flares from κ1 Ceti 17 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ELEMENTAL ABUNDANCE MEASUREMENT Telleschi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' (2005) extensively studied the coronal elemental abundance of κ1 Ceti using XMM/RGS data in 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' However, the XMM-Newton instrumental calibration4 and the plasma emission codes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', ATOMDB5) have significantly improved since then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The elemental abundance of the star might also have changed in 17 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We, thus, independently measure the coronal elemental abundance of κ1 Ceti using the XMM/RGS data obtained on 2018 July 30 and 2019 January 29 (ObsID: 0822790901, 0822791001, PI: Wargelin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We reprocess these datasets with SAS version 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EPIC/MOS2 turns off during these observations, while the EPIC-pn uses the timing mode with relatively poor spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We thus analyze EPIC/MOS1 and RGS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' For MOS1, we take a 15′′ radius circular source region centered at the X-ray peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The MOS1 on-axis CCD operates with the small window mode so that we take background data from a source-free region from the surrounding CCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The MOS1 light curves of these observations do not show significant time variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' EPIC/MOS1 measures the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='6−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='2 keV flux during the second observation at ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='5×10−12 ergs cm−2 s−1, which is ∼16% lower than the first observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' This flux is nearly the lowest among the NICER monitoring observations of κ1 Ceti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We produce the MOS1 spectra using the same source and background regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' For RGS, we run rgsproc for the target position measured from the MOS1 image and produce the source and background spectra (Figure A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We only use the first-order RGS spectra as the second-order RGS spectra do not have enough photon counts to identify emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We fit the unbinned MOS1 and RGS spectra simultaneously using Cash statistic (c-stat) built in xspec (Cash 1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The Cash statistic needs to add background as an additive model component so that we simultaneously fit background spectra by an empirical model (power-law + 4 Gaussians), convolved with the source response (rmf) weighted with the background areal scale (backscal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' For the source spectra, we assume a 2T thermal plasma model with various abundance values (vapec) and fit all MOS1/RGS source/background spectra of the two observations simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We allow for varying spectral normalization between MOS1 and RGS to account for calibration uncertainty and kT and normalization between the two observations for time variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table A1 lists the derived elemental abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' We use these values for the NICER data analysis and numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' REFERENCES Airapetian, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Barnes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Cohen, O.' 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482 Virtanen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Gommers, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Oliphant, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Haberland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Reddy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Cournapeau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Burovski, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Peterson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Weckesser, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Bright, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', van der Walt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Brett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Wilson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Millman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Mayorov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Nelson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Jones, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Kern, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Larson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Carey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Polat, ˙I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Moore, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', VanderPlas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Laxalde, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Perktold, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Cimrman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Henriksen, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Quintero, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Harris, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Archibald, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Ribeiro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Pedregosa, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', van Mulbregt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', & SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 Contributors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2020, Nature Methods, 17, 261 Warren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Winebarger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', & Brooks, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 2010, ApJ, 711, 228 White, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Culhane, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Parmar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Kellett, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', Kahn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', van den Oord, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=', & Kuijpers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' 1986, ApJ, 301, 262 20 Hamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Applied Elemental Abundance Element κ1 Ceti Sun Relative to Sun Number Number H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00E+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00E+00 He 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='51E-02 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='51E-02 Li 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='12E-11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='12E-11 Be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='40E-11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='40E-11 B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='01E-10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='01E-10 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='12E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='69E-04 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='99E-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='76E-05 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='95E-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='90E-04 F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='63E-08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='63E-08 Ne 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='62 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='29E-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='51E-05 Na 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='74E-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='74E-06 Mg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='70E-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='98E-05 Al 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='00f 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='82E-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='82E-06 Si 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='55 1.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='63E-08 Note— Abundance numbers relative to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' Solar abun- dance reference: Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' ffixed at the solar values in the XMM-Newton spectral fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' X-ray Flares from κ1 Ceti 21 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content='1 1 10 counts s−1 keV−1 Energy (keV) NVI NVII OVII OVIII FeXVIII FeXIX NeIX FeXVII NeX MgXI Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' XMM-Newton RGS1+2 grating spectrum of κ1 Ceti combined from the 2018 and 2019 observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The spectrum includes both source and background data (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The red line shows the best-fit 2T apec model, and the blue line does the corresponding background model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The black line is the sum of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} +page_content=' The prominent emission lines are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQfa_xP/content/2301.01377v1.pdf'} diff --git a/iNAyT4oBgHgl3EQfkPi4/content/tmp_files/2301.00431v1.pdf.txt b/iNAyT4oBgHgl3EQfkPi4/content/tmp_files/2301.00431v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..28d73d8dee10acfba529aef979e6a9683bf82a1c --- /dev/null +++ b/iNAyT4oBgHgl3EQfkPi4/content/tmp_files/2301.00431v1.pdf.txt @@ -0,0 +1,1937 @@ +arXiv:2301.00431v1 [math.GR] 1 Jan 2023 +ON WONDERFUL COMPACTIFICATIONS OF SL(2, F) +FOR NON-ARCHIMEDEAN LOCAL FIELDS F +CORINA CIOBOTARU +Abstract. We compute the wonderful compactification of symmetric varieties of SL(2, F), where +F is a finite field-extension of Qp with p ̸= 2, that comes from either an abstract or F-involutions +of SL(2, F). For each of those wonderful compactifications we find the SL(2, F)-stabilizers of the +accumulation points of the corresponding symmetric varieties and compare them to the Chabauty +limits found in [CL22]. +1. Introduction +The wonderful compactifications emerged from specific problems in enumerative geometry. In +laymen’s terms a wonderful compactification is a way of compactifying a symmetric variety G/H +via an irreducible linear representation of the group G, where G is an algebraic group and H ≤ G +is the fixed point set of an involutorial automorphism of G. This procedure will provide remarkable +properties of the G-closed orbits of that compactification. In principal, the wonderful compactifi- +cation of G/H does not depend on the chosen irreducible linear representation of G, and a Borel +subgroup of G will only have one open and dense orbit in G/H. The study of wonderful compactifi- +cations was initiated by DeConcini and Procesi [DCP83] in the case of a complex semi-simple group +G of adjoint type (i.e. a semi-simple group with trivial center). Since then wonderful compacti- +fications have been extensively studied in algebraic geometry and have important applications in +many fields of Mathematics (see the Introduction of [EJ08]). If one replaces the complex numbers +by any (not necessarily algebraically closed) field k of characteristic ̸= 2, the results of [DCP83] +were further expanded by DeConcini and Springer in [DCS99] to any adjoint semi-simple group G +defined over k and H the fixed point group of an involutorial automorphism of G defined over k. +In this article we focus on the wonderful compactifications of symmetric varieties of SL(2, F), +where F a finite field-extension of Qp with p ̸= 2, that come from various abstract and F-involutions of +SL(2, F). Notice that SL(2, F) is not of adjoint type. In particular, we are interested in computing the +SL(2, F)-stabilizers of the accumulation points in the wonderful compactifications of those symmetric +varieties and we compare them to the Chabauty limits found in [CL22]. Although some of the +techniques used in this article to build the wonderful compactifications are similar to the well +established literature (e.g. [EJ08]), we will make extensive use of some results from [BH09], the +Bruhat–Tits tree and its visual boundary associated with SL(2, F), which contrast with the methods +employed in [DCS99]. +Let us first fix the notation. Throughout this article we restrict to a prime p ̸= 2. Let F be a finite +field-extension of Qp and E be any quadratic extension of F. Let kF, kE be the residue fields of F, E, +respectively, and ωF, ωE be uniformizers of F, E, respectively. Recall k∗ +F/(k∗ +F)2 = {1, S}, for some +non-square S ∈ k∗ +F. Then F∗/(F∗)2 = {1, ωF, S, SωF} ([Ser12, Corollaries to Theorems 3 and 4] or +[SJ98, page 41, Section 12]). If E is ramified then E = F(√ωF), or E = F(√SωF) (where ωF ̸= ωE), +or if E is unramified then E = F( +√ +S) (where ωF = ωE). We choose the unique valuation | · |E on E +Date: January 2, 2023. +corina.ciobotaru@gmail.com. +1 + +that extends the given valuation | · |F on F. Choose α ∈ {√ωF, +√ +S, √SωF} and so E = F(α). Notice +each element x ∈ E can be uniquely written as x = a+bα, with a, b ∈ F. For the ramified extensions +we can consider ω2 +E = ωF. Let OF := {x ∈ F | |x|F ≤ 1} denote the ring of integers of F, then OF is +compact and open in F. For F = Qp we have OQp = Zp, ωQp = p, kQp = Fp, F∗ +p/(F∗ +p)2 = {1, Sp}. +We denote by TF the Bruhat–Tits tree for SL(2, F) whose vertices are equivalence classes of OF- +lattices in F2 (for its construction see [Ser12]). The tree TF is a regular, infinite tree with valence +|kF|+1 at every vertex. The boundary at infinity ∂TF of TF is the projective space P 1(F) ∼= F∪{∞}. +Moreover, the end ∞ ∈ ∂TF corresponds to the vector +� 0 +1 +� +∈ P 1(F). The rest of the ends ξ ∈ ∂TF +correspond to the vectors +� 1 +x +� +∈ P 1(F), where x ∈ F. To give a concrete example, the Bruhat–Tits +tree of SL(2, Qp) is the p + 1 = |Fp| + 1-regular tree. The boundary at infinity ∂TQp of TQp is the +projective space P 1(Qp) = Qp ∪ {∞}. +Let ρ : SL(2, F) → GL(F2) be the representation of SL(2, F) over F given by g ∈ SL(2, F) �→ +ρ(g) := g. One can easily prove ρ is an irreducible and continuous representation of SL(2, F) with +respect to the compact-open topologies of the locally compact groups SL(2, F) and GL(F2). By the +known theory ρ has a dominant weight (see [Tit71, Theorem 2.5]). +For an abstract involution θ : SL(2, F) → SL(2, F), i.e. θ is an abstract automorphism of SL(2, F) +with θ2 = Id, we consider: +(1) the action of SL(2, F) on End(F2) given by g ·θ D := ρ(g)Dρ(θ(g−1)), for every g ∈ SL(2, F) +and D ∈ End(F2), +(2) the continuous map ψθ : SL(2, F) → P(End(F2)) given by g �→ ψθ(g) := [ρ(g)ρ(θ(g−1))] = +[ρ(gθ(g−1))], +(3) and the fixed point group of θ denoted by Hθ := {g ∈ SL(2, F) | θ(g) = g}. +Definition 1.1. We take Xθ := ψθ(SL(2, F)) = [SL(2, F) ·θ IdF2] ⊂ P(End(F2)) and, by abuse of +terminology, call it the wonderful compactification of SL(2, F)/Hθ with respect to the involution θ. +We will see in the proofs below that a Borel subgroup of SL(2, F) might have more open orbits +on SL(2, F)/Hθ, that are pairwise disjoint and not dense in SL(2, F)/Hθ. This is because SL(2, F) +is not of adjoint type. By abuse of notation, for any g ∈ SL(2, F) and any [D] ∈ Xθ ⊂ P(End(F2)) +we use g ·θ [D] to mean [ρ(g)Dρ(θ(g−1))] ∈ Xθ for some, thus any, representative D ∈ End(F2) of +[D] ∈ P(End(F2)). +By the known theory, those wonderful compactifications do not depend on the irreducible repre- +sentation ρ of SL(2, F) (for a proof see [AB18, Section 7] or [DCS99, Proposition 3.10]). +The three main results of the paper are as follows. They are proved in Sections 2, 3, 4, respectively, +where the reader can refer for the notation. +Theorem 1.2. Consider the involution θ(x, y) := (y, x) of SL(2, F)×SL(2, F) where Hθ := Diag(G) = +SL(2, F) is the fixed point group, and SL(2, F)×SL(2, F) acts on End(F2) by (g1, g2)·θA = ρ(g1)Aρ(g−1 +2 ). +Then there are 2 orbits of SL(2, F) × SL(2, F) in Xθ with respect to the action ·θ: +(1) the open SL(2, F)×SL(2, F)-orbit of IdF2, whose SL(2, F)×SL(2, F)-stabilizer is Diag(SL(2, F)× +SL(2, F)) = SL(2, F) +(2) the closed SL(2, F) × SL(2, F)-orbit of [1, 0; 0, 0] ∈ P(End(F2)), whose SL(2, F) × SL(2, F)- +stabilizer is B+ × B−. +Theorem 1.3. Let F be a finite field-extension of Qp and E = F(α) be any quadratic extension +of F. Take θ to be the involution of SL(2, E) induced by the ‘conjugation’ in E with respect to F, +θ(a + αb) := a − αb, for any a, b ∈ F. There are 2 orbits of SL(2, E) in Xθ with respect to the action +·θ: +(1) the open SL(2, E)-orbit of IdE2, whose SL(2, E)-stabilizer is SL(2, F) +(2) the closed SL(2, E)-orbit of [0, 1; 0, 0] ∈ P(End(E2)), whose SL(2, E)-stabilizer is the subgroup +{ +� a−αb +z +0 +a+αb +� +| a, b ∈ F with a2 − α2b2 = 1, z ∈ E} ≤ B+ +E . +2 + +Theorem 1.4. Let m ∈ F∗/(F∗)2 = {1, ωF, S, SωF}. Consider Am := ( 0 1 +m 0 ) and the corresponding +inner F-involution of SL(2, F) given by σm := ιAm (see Section 4). There are 2 orbits of SL(2, F) in +Xσm with respect to the action ·σm: +(1) the open SL(2, F)-orbit of IdF2, whose SL(2, F)-stabilizer is Hσm +(2) the closed SL(2, F)-orbit of [1, 0; 0, 0] ∈ P(End(F2)), whose SL(2, F)-stabilizer is the subgroup +{µ2 · ( 1 b +0 1 ) | b ∈ F} of the Borel B+ ≤ SL(2, F), where µ2 is the group of 2nd roots of unity +in F. +Comparing the stabilizers of the accumulation points [0, 1; 0, 0] and [1, 0; 0, 0] computed in Theo- +rems 1.3 and 1.4, respectively, with the Chabauty limits computed in [CL22] Theorems 1.4 and 1.6, +respectively, the reader can notice we found the same subgroups. The general case of SL(n, F) will +be treated in a future research project. +Acknowledgements. Ciobotaru is supported by the European Union’s Horizon 2020 research +and innovation programme under the Marie Sklodowska-Curie grant agreement No 754513 and +The Aarhus University Research Foundation. She was partially supported by The Mathematis- +ches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics). She +would like to thank those two institutions for the perfect working conditions they provide, and to +Linus Kramer and Maneesh Thakur for very helpful discussions regarding reference [Tit71]. As well, +Ciobotaru thanks Arielle Leitner for the wonderful collaboration on our joint paper [CL22] that gave +rise to this article, for reading it and providing useful comments. +2. The wonderful compactification of SL(2, F) +First recall the usual wonderful compactification associated with the involution θ(x, y) := (y, x) +of G := SL(2, F) × SL(2, F), having Diag(G) = SL(2, F) as a fixed point group of the involution +θ. We compactify SL(2, F) = G/Diag(G) in the following way. Consider ρ(SL(2, F)) ≤ GL(F2) ⊆ +End(F2) and SL(2, F)× SL(2, F) acting on End(F2) by (g1, g2)·θ A = ρ(g1)Aρ(g−1 +2 ), where (g1, g2) ∈ +SL(2, F) × SL(2, F) and A ∈ End(F2). Then the stabilizer in SL(2, F) × SL(2, F) of IdF2 is exactly +Diag(G) = SL(2, F), and ρ(SL(2, F)) = (SL(2, F) × SL(2, F)) ·θ IdF2. +Next we take the map +ψθ : SL(2, F) × SL(2, F) → P(End(F2)), given by (g1, g2) �→ ψθ(g1, g2) := [ρ(g1g−1 +2 )] +and we want to understand the closure +Xθ := ψθ(SL(2, F) × SL(2, F)) = [(SL(2, F) × SL(2, F)) ·θ IdF2] = [ρ(SL(2, F))] in P(End(F2)) +called the wonderful compactification of SL(2, F) = G/Diag(G) with respect to the involution θ, +where P(End(F2)) is endowed with the usual topology. +Denote +T := +��x +0 +0 +x−1 +� +| x ∈ F∗ +� +U + := +��1 +y +0 +1 +� +| y ∈ F +� +U − := +��1 +0 +y +1 +� +| y ∈ F +� +B+ := +��x +y +0 +x−1 +� +| y ∈ F, x ∈ F∗ +� +B− = +��x +0 +y +x−1 +� +| y ∈ F, x ∈ F∗ +� +, +(1) +where T is the maximal tori of SL(2, F), U + is the unipotent radical of the Borel subgroup B+, and +U − is the unipotent radical of the opposite Borel B−. By the definition of Xθ: +Lemma 2.1. The SL(2, F) × SL(2, F) orbit of IdF2 in P(End(F2)) with respect to the action ·θ is +dense in Xθ. +In order to prove Theorem 1.2 we need the following notation. Let +P0 := {x = [1, x2; x3, x4] ∈ P(End(F2))} +3 + +and notice this is open in P(End(F2)). Denote Xθ,0 := P0 ∩ Xθ and notice this is again open in Xθ. +The set Xθ,0 is called the big cell in Xθ. +Recall that B+ = T U + as well as the following decomposition in SL(2, F) +(2) +� +a +b +c +d +� += +� +1 +0 +c/a +1 +� � +a +0 +0 +a−1 +� � +1 +b/a +0 +1 +� +, for a ̸= 0. +From here it is easy to verify the following facts. +Lemma 2.2. The set U −T U + is dense in SL(2, F), and in particular [(U −T × U +) ·θ IdF2] is dense +in ψθ(SL(2, F) × SL(2, F)) = [ρ(SL(2, F))], and so in [ρ(SL(2, F))]. +Proof. The first part of the lemma follows from an easy approximation of matrices having a = 0 +using the decomposition given in (2). The rest of the lemma follows directly from the density of +U −T U + in SL(2, F). +□ +Lemma 2.3. We have that +Xθ,0 ∩ [ρ(SL(2, F))] = [(U −T × U +) ·θ IdF2] +Xθ = +� +(g1,g2)∈SL(2,F)×SL(2,F) +(g1, g2) ·θ Xθ,0 +Xθ,0 = [(U −T × U +) ·θ IdF2] ∩ P0. +Proof. By the decomposition 2 it is clear that +[(U −T × U +) ·θ IdF2] = Xθ,0 ∩ [ρ(SL(2, F))] = P0 ∩ [ρ(SL(2, F))]. +Since [(U −T × U +) ·θ IdF2] = Xθ we immediately have Xθ,0 = [(U −T × U +) ·θ IdF2] ∩ P0, i.e. the +closure of the set [(U −T × U +) ·θ IdF2] in P0. +Let C ∈ Xθ. As C ∈ P(End(F2)) we have C = [x1, x2; x3, x4]. If x1 ̸= 0 then C ∈ Xθ,0 and we are +done. If x1 = 0, then by some easy calculations one can arrange for some (g1, g2) ∈ SL(2, F)×SL(2, F) +such that ρ(g1)Cρ(g−1 +2 ) ∈ P0. Then we are done. +□ +Proof of Theorem 1.2. By Lemma 2.3 it is enough to compute the set (U −T × U +)·θ IdF2, and then +to precisely compute its closure [(U −T × U +) ·θ IdF2] ∩ P0. We have +(U −T × U +) ·θ IdF2 = +��1 +0 +a +1 +� �x +0 +0 +x−1 +� �1 +−b +0 +1 +� +| x ∈ F∗, a, b ∈ F +� +�� x +−xb +ax +−axb + x−1 +� +| x ∈ F∗, a, b ∈ F +� +⇒ +⇒ [(U −T × U +) ·θ IdF2] = {[1, −b; a, −ab + x−2] | x ∈ F∗, a, b ∈ F}. +(3) +Then the only accumulation points of the set [(U −T × U +) ·θ IdF2] in P0 are the points of the +form [1, −b; a, −ab], for any a, b ∈ F. Moreover, it is very easy to see that the only accumulation +point of the set [T ·θ IdF2] in P0 is just [1, 0; 0, 0]. Notice, an element [1, −b; a, −ab] respresents a +matrix in P(End(F2)) of determinant zero. As well, {[1, −b; a, −ab] | a, b ∈ F} is the (U −T × U +)- +orbit of the element [1, 0; 0, 0] ∈ P(End(F2)). From the above calculation and since the action ·θ is +continuous, it is immediate that the SL(2, F) × SL(2, F)-orbit of IdF2 is open. It is also clear that +the SL(2, F) × SL(2, F)-orbit of [1, 0; 0, 0] in P(End(F2)) is closed and, by in easy computation, the +stabilizer of [1, 0; 0, 0] in SL(2, F) × SL(2, F) is exactly B+ × B−. The theorem follows. +□ +4 + +3. The wonderful compactification of SL(2, E)/ SL(2, F) for quadratic E/F +Let F be a finite field-extension of Qp and E = F(α) be any quadratic extension of F. Denote by +θ : E → E the ‘conjugation’ in E with respect to F, θ(a + αb) := a − αb, for any a, b ∈ F. This map +θ induces an abstract involution of SL(2, E) given by +( x y +z t ) ∈ SL(2, E) �→ θ(( x y +z t )) := +� +θ(x) θ(y) +θ(z) θ(t) +� +whose fixed point group Hθ := {g ∈ SL(2, E) | θ(g) = g} is SL(2, F). +The stabiliser in SL(2, F), resp. SL(2, E), of the endpoint +� 1 +0 +� +is the Borel subgroup +B+ +F := { +� a +b +0 a−1 +� +| b ∈ F, a ∈ F×}, +resp. B+ +E := { +� x +y +0 x−1 +� +| y ∈ E, x ∈ E×}. +The stabiliser in SL(2, F), resp. SL(2, E), of the endpoint +� 0 +1 +� +is the opposite Borel subgroup +B− +F := { +� a +0 +b a−1 +� +| b ∈ F, a ∈ F ×}, +resp. B− +E := { +� +x +0 +y x−1 +� +| y ∈ E, x ∈ E×}. +Let +TE := +�� +x +0 +0 +x−1 +� +| x ∈ E∗ +� +UE := +�� +1 +y +0 +1 +� +| y ∈ E +� +U − +E := +�� +1 +0 +y +1 +� +| y ∈ E +� +. +Given the abstract involution θ : SL(2, E) → SL(2, E), we consider the action of SL(2, E) on +End(E2) given by g ·θ D := ρ(g)Dρ(θ(g−1)), for every g ∈ SL(2, E) and D ∈ End(E2). The stabilizer +in SL(2, E), with respect to the action ·θ, of the element IdE2 is the subgroup SL(2, F). +Then +SL(2, E) ·θ IdE2 ∼= SL(2, E)/ SL(2, F) and we consider the continuous map +ψθ : SL(2, E) → P(End(E2)), given by g �→ ψθ(g) := [ρ(g)ρ(θ(g−1))] = [ρ(gθ(g−1))]. +We want to understand the following closure with respect to the involution θ and the map ψθ: +Xθ := ψθ(SL(2, E)) = [SL(2, E) ·θ IdE2] = [SL(2, E)/ SL(2, F)] in P(End(E2)). +Definition 3.1. We regard the above closure Xθ as a compactification of SL(2, E)/ SL(2, F) and +call it the the wonderful compactification of SL(2, E)/ SL(2, F) with respect to the involution θ. +Just from the definition, we have the following trivial lemma: +Lemma 3.2. The SL(2, E)-orbit of IdE2 in P(End(E2)), with respect to the action ·θ, is dense in +Xθ. +In order to compute the wonderful compactification of SL(2, E)/ SL(2, F) we first need to find the +number of orbits of SL(2, F) on the boundary ∂TE. +Lemma 3.3. There are at most 5 SL(2, F)-orbits on the boundary ∂TE ∼= P 1E = E ∪ {∞}: +(1) the SL(2, F)-orbit of +� 1 +0 +� +which is closed +(2) for each m ∈ F∗/(F∗)2, the SL(2, F)-orbit of +� +1 +mα +� +is open and might coincide with the orbit +of a different m. +Proof. It is clear that the SL(2, F)-orbit of +� 1 +0 +� +is closed in ∂TE with respect to the cone topology +on ∂TE (see [FTN91] page 4). +Consider the subgroup B := { +� +d−1 0 +c +d +� +| c ∈ F, d ∈ F∗} in SL(2, F). Then the image of +� +1 +mα +� +under B is +� +1 +cd+d2mα +� +, which covers the entire ∂TE − ∂TF when m takes all values from F∗/(F∗)2. +Moreover, for each m ∈ F∗/(F∗)2, it is easy to notice that the set { +� +1 +cd+d2mα +� +| c ∈ OF, d ∈ O∗ +F} is +open in ∂TE with respect to the cone topology on ∂TE. Thus the SL(2, F)-orbit of +� +1 +mα +� +is open, for +each m ∈ F∗/(F∗)2. +□ +5 + +Consider the matrices in SL(2, E) = SL(2, F(α)) +g1 := IdE2, +gm := +� +1 − +1 +mα +0 +1 +� +for m ∈ F∗/(F∗)2 +such that +gm +�� +1 +mα +�� += +� 0 +1 +� +. +Let I := {IdF2, gm | m ∈ F∗/(F∗)2}, and Im := {gm | m ∈ F∗/(F∗)2}. +Lemma 3.4. With the above notation we have that: +(1) SL(2, E) = � +gi∈I +B− +E gi SL(2, F) +(2) for any gm ∈ Im, the set B− +E gm SL(2, F) = U − +E TEgm SL(2, F) is open in SL(2, E), and +moreover the set +� +gm∈Im +U − +E TEgm SL(2, F) is dense in SL(2, E) +(3) ψθ( +� +gm∈Im +U − +E TEgm) is open and dense in ψθ(SL(2, E)) = [SL(2, E) ·θ IdE2], and so +ψθ( +� +gm∈Im +U − +E TEgm) = [SL(2, E) ·θ IdE2]. +Proof. Let g ∈ SL(2, E) and let ξ0 := +� 0 +1 +� +. The SL(2, E)-stabilizer of ξ0 is B− +E . +If g−1(ξ0) is in the SL(2, F)-orbit of ξ0 then g ∈ B− +E SL(2, F). +If not then there are 4 cases +to consider, each corresponding to the open SL(2, F)-orbits in ∂TE from Lemma 3.3. By applying +accordingly an element h ∈ SL(2, F) we get that hg−1(ξ0) ∈ { +� +1 +mα +� +| m ∈ F∗/(F∗)2}. Further, by +multiplying again accordingly with the element gm from Im we obtain gmhg−1(ξ0) = ξ0, and thus +g−1 ∈ SL(2, F)g−1 +m B− +F , giving g ∈ B− +F gm SL(2, F) as required. This proves part (1) of the lemma. +Let us prove part (2). Take gm ∈ Im. By Lemma 3.3 the SL(2, F)-orbit of +� +1 +mα +� += g−1 +m +�� 0 +1 +�� +is open in ∂TE. Because the cone topology on SL(2, E)/B− +E ∼= ∂TE is equivalent to the quotient +topology on SL(2, E)/B− +E induced from the continuous and open canonical projection of SL(2, E) to +SL(2, E)/B− +E , we have SL(2, F)g−1 +m B− +E is open in SL(2, E), and by taking the inverse map which is +continuous, the set B− +E gm SL(2, F) is open in SL(2, E) as well. +It remains to prove that +� +gm∈Im +U − +E TEgm SL(2, F) is dense in SL(2, F), and by part (1) it is enough +to show that for any fixed gm ∈ Im, the accumulation points of the open set B− +E gm SL(2, F) are +exactly the elements of B− +E SL(2, F). Using again the cone topology on SL(2, E)/B− +E ∼= ∂TE and the +group B from the proof of Lemma 3.3 one easily gets part (2) of the lemma. +Part (3) follows easily from part (2), the fact that B− +E = U − +E TE, and the continuity of the map +ψθ. +□ +In this section we will consider the set +P0 := {x = [x1, 1; x3, x4] ∈ P(End(E2))} +which is open in P(End(E2)). Then we denote +Xθ,0 := P0 ∩ Xθ +and this is open in Xθ. As before, the set Xθ,0 is called the big cell in Xθ. +By Lemma 3.4 it is again advised to study the sets (U − +E TEgi) ·θ IdE2, for any gi ∈ I, as well as +their closure. +6 + +We have +(U − +E TE) ·θ IdE2 = +��x +0 +a +x−1 +� +θ +��x−1 +0 +−a +x +�� +| x ∈ E∗, a ∈ E +� += +�� +x +0 +a +x−1 +� � +θ(x−1) +0 +θ(−a) +θ(x) +� +| x ∈ E∗, a ∈ E +� += +�� +xθ(x−1) +0 +aθ(x−1) + x−1θ(−a) +x−1θ(x) +� +| x ∈ E∗, a ∈ E +� +⇒ +⇒ [(U − +E TE) ·θ IdE2] = {[xθ(x−1), 0; aθ(x−1) + x−1θ(−a), x−1θ(x)] | x ∈ E∗, a ∈ E}. +(4) +For any gm ∈ Im we also have: +(U − +E TEgm) ·θ IdE2 = +�� +x +0 +a +x−1 +� � +1 +− 1 +mα +0 +1 +� +θ +�� +1 +1 +mα +0 +1 +� � +x−1 +0 +−a +x +�� +| x ∈ E∗, a ∈ E +� += +�� +x +− x +mα +a +− a +mα + x−1 +� � +θ(x−1) + θ(a) +mα +− θ(x) +mα +−θ(a) +θ(x) +� +| x ∈ E∗, a ∈ E +� +multiply +=======⇒ +x−1θ(x−1) +��� +1 +− 1 +mα +a +x +− +a +mxα + x−2 +� � +θ(x−2) + +θ(a) +mθ(x)α +− 1 +mα +− θ(a) +θ(x) +1 +�� +| x ∈ E∗, a ∈ E +� += += [(U − +E TE) ·θ IdE2] ⊂ P0. +(5) +Then the only accumulation points of the set [(U − +E TEgm) ·θ IdE2] are the points [−θ(b), 1; −bθ(b), b] +with b ∈ E. Those elements appear when taking the sequences {xnk := ω−nk +E +}k≥0 and {ank := +bxnk}k≥0, with 0 < nk → ∞ and b ∈ E. For sequences of the form {xnk := ωnk +E }k≥0 we obtain no +limits. As well, notice that the set {[−θ(b), 1; −bθ(b), b] | b ∈ E} is the U − +E TE-orbit of the element +[0, 1; 0, 0] ∈ P(End(E2)) with respect to the action ·θ. +As a result of Lemma 3.4 and the above calculations in (4) and (5) we get: +Lemma 3.5. Recall the notation Im := {gm | m ∈ E∗/(E∗)2}. Then: +Xθ,0 ∩ ψθ(SL(2, E)) = [( +� +gm∈Im +U − +E TEgm) ·θ IdE2] +Xθ,0 = [( +� +gm∈Im +U − +E TEgm) ·θ IdE2] ∩ P0 +Xθ = +� +g∈SL(2,E) +g ·θ Xθ,0. +Proof. It is clear from the above calculation that +[( +� +gm∈Im +U − +E TEgm) ·θ IdE2] = Xθ,0 ∩ ψθ(SL(2, E)) = P0 ∩ ψθ(SL(2, E)). +Since [( +� +gm∈Im +U − +E TEgm) ·θ IdE2] = Xθ by Lemma 3.4(3), we immediately have +Xθ,0 = [( +� +gm∈Im +U − +E TEgm) ·θ IdE2] ∩ P0. +Let C ∈ Xθ. As C ∈ P(End(E2)) we have C = [x1, x2; x3, x4]. If x2 ̸= 0 then C ∈ Xθ,0 and we +are done. If x1 = 0, then by some easy calculations one can arrange for some g ∈ SL(2, E), such +that ρ(g)Cρ(θ(g−1)) ∈ P0. Then we are done. +□ +7 + +We are now ready to prove the main result of this section. +Proof of Theorem 1.3. By Lemma 3.5, the SL(2, E)-orbits in Xθ with respect to the ·θ action are +determined by the SL(2, E)-orbits given by the points of the set Xθ,0. +Moreover, the calcula- +tions in (5) show the only accumulation points of the sets [(U − +E TEgm) ·θ IdE2] in P0 are the points +[−θ(b), 1; −bθ(b), b] with b ∈ E, which is the U − +E TE-orbit of the element [0, 1; 0, 0] ∈ P(End(E2)) with +respect to the action ·θ. In addition, it is very easy to see that the only accumulation point of the +set [TE ·θ IdE2] in P0 is just [0, 1; 0, 0]. +Further, by an easy computation +� +a +b +c +d +� � +0 +1 +0 +0 +� � +θ(d) +−θ(b) +−θ(c) +θ(a) +� += +� +−aθ(c) +aθ(a) +−cθ(c) +cθ(a) +� +showing that the SL(2, E)-orbit of the point [0, 1; 0, 0] is indeed closed in Xθ. +As [( +� +gm∈Im +U − +E TEgm) ·θ IdE2] is already part of the SL(2, E)-orbit of the point [1, 0; 0, 1] = [IdE2] +with respect to the ·θ action, the theorem follows. +□ +Corollary 3.6. With respect to the action ·θ, the stabilizer in SL(2, E) of the point [1, 0; 0, 1] = [IdE2] +is SL(2, F), and the stabilizer in SL(2, E) of the point [0, 1; 0, 0] is the subgroup { +� a−αb +z +0 +a+αb +� +| a, b ∈ +F with a2 − α2b2 = 1, z ∈ E} ≤ B+ +E . +Proof. The second part just follows from the computation +� +a +b +c +d +� � +0 +1 +0 +0 +� � +θ(d) +−θ(b) +−θ(c) +θ(a) +� += +� +−aθ(c) +aθ(a) +−cθ(c) +cθ(a) +� += +� +0 +1 +0 +0 +� +. +□ +4. The wonderful compactification of SL(2, F)/Hσ +The third wonderful compactification is given by the family of inner involutions σ of SL(2, F) +with fixed point groups Hσ = {g ∈ SL(2, F)| σ(g) = g}. In this section we will use the notation +from (1). +In the next few paragraphs we summarize the corresponding results from [HW02] for k-involutions +of SL(2, k) when k is a field of characteristic not 2. Let k be the algebraic closure of k. +Recall, a mapping φ : SL(2, k) → SL(2, k) is a k-automorphism (or equivalently, an automorphism +defined over k) if φ is a bijective rational k-homomorphism whose inverse is also a rational k- +homomorphism, [Hel00, Sec. 2.2]. An abstract automorphism θ of SL(2, k) of order two is called +an abstract involution of SL(2, k). A k-involution θ of SL(2, k) is an involution defined over k of +SL(2, k), and the restriction of θ to SL(2, k) is a k-involution of SL(2, k). Given g ∈ SL(2, k) denote +by ιg the inner automorphism of SL(2, k) defined by x �→ ιg(x) := gxg−1. +The classification of the isomorphism classes of k-involutions of a connected reductive algebraic +group defined over k is given in [Hel00]. +A simple characterization of the isomorphism classes +of k-involutions of SL(n, k) is given in [HWD06]. We record the classification of k-involutions of +SL(2, k): +Theorem 4.1. [[HW02] Theorem 1, Corollary 1, Corollary 2]. Every k-isomorphism class of k- +involution of SL(2, k) is of the form ιA with A = ( 0 1 +m 0 ) ∈ GL(2, k). Two such k-involutions ιA with +A ∈ {( 0 1 +m 0 ) , +� 0 +1 +m′ 0 +� +} ⊂ GL(2, k) of SL(2, k) are conjugate if and only if m and m′ are in the same +square class of k∗. In particular, there are order(k∗/(k∗)2) k-isomorphism classes of k-involutions +of SL(2, k). +Definition 4.2. Given an involution σ of a group G the fixed point group of σ is Hσ := {x ∈ +G | σ(x) = x}. +8 + +For σ a k-involution of SL(2, k) the quotient SL(2, k)/Hσ is called a k-symmetric variety, and +much of the structure of SL(2, k)/Hσ is determined by Hσ. +Proposition 4.3 ([HW02] Section 3). Let σ = ιA, with A = ( 0 1 +m 0 ) ∈ GL(2, k), be a k-involution of +SL(2, k). Then Hσ = {( x +y +my x ) ∈ SL(2, k) | x2 − my2 = 1}. +Theorem 4.4 ([HW02] Section 3.2). Let k = Qp, σ = ιA with A = ( 0 1 +m 0 ) and m ∈ Q∗ +p/(Q∗ +p)2. +Then Hσ is anisotropic if and only if ¯m ̸= ¯1. If ¯m = ¯1, then Hσ is isotropic and conjugate to the +maximal Qp-split torus of SL(2, Qp), i.e. the diagonal subgroup of SL(2, Qp). +Theorem 4.5 ([BH09] Theorem 4.18). Let k = Qp, σ = ιA with A = ( 0 1 +m 0 ) and m ∈ Q∗ +p/(Q∗ +p)2. +Then +(1) |B+\ SL(2, Qp)/Hσ| = 2 if m ̸= 1 +(2) |B+\ SL(2, Qp)/Hσ| = 6 if m = 1. +The same results as in Theorem 4.5 should hold true for any finite field-extension F of Qp. Below +we will give a purely geometric proof of such results. +We also record the following geometric interpretation of the fixed point group Hσ, when k = F +is a finite field-extension of Qp. To fix notation, let F be a finite field-extension of Qp. We denote +by TF the Bruhat–Tits tree for SL(2, F) whose vertices are equivalence classes of OF-lattices in F2 +(for its construction see [Ser12]). The tree TF is a regular, infinite tree with valence |kF| + 1 at every +vertex, where kF is the residue field of F. The boundary at infinity ∂TF of TF is the projective space +P 1(F) ∼= F ∪ {∞}. Moreover, the endpoint ∞ ∈ ∂TF corresponds to the vector +� 0 +1 +� +∈ P 1(F). The +rest of the endpoints ξ ∈ ∂TF correspond to the vectors +� 1 +x +� +∈ P 1(F), where x ∈ F. +Theorem 4.6 ([CL22] Corollary 4.8). Let F be a finite field-extension of Qp, A = ( 0 1 +m 0 ), with +m ∈ F∗/(F∗)2, and σ := ιA the corresponding F-involution of SL(2, F). Take Km := F(√m) a field +extension. Then the only solutions of the equation A(ξ) = ξ with ξ ∈ P 1Ka are ξ± := +� +1 +±√m +� +and +Hσ = FixSL(2,Km)({ξ−, ξ+}) ∩ SL(2, F) = {( x +y +my x ) ∈ SL(2, F) | x2 − my2 = 1}. Moreover, +(1) if m = 1 then ξ± := +� 1 +±1 +� +and Hσ contains all the hyperbolic elements of SL(2, F) with ξ± +as their repelling and attracting endpoints. In particular, Hσ is GL(2, F)-conjugate to the +entire diagonal subgroup of SL(2, F), thus Hσ is noncompact and abelian. +(2) if m ̸= 1 then ξ± := +� +1 +±√m +� +∈ P 1Km − P 1F, and Hσ is compact and abelian. +Now, given an involution σ := ιA as in Theorem 4.1, we consider the following action of SL(2, F) +on End(F2) given by g ·σ D := ρ(g)Dρ(σ(g−1)), for every g ∈ SL(2, F) and D ∈ End(F2). The +stabilizer in SL(2, F), with respect to the action ·σ, of the element IdF2 is the subgroup Hσ. Then +SL(2, F) ·σ IdF2 ∼= SL(2, F)/Hσ and we consider the continuous map +ψσ : SL(2, F) → P(End(F2)), given by g �→ ψσ(g) := [ρ(g)ρ(σ(g−1))] = [ρ(gσ(g−1))]. +We want to understand the following closure with respect to the involution σ and the map ψσ: +Xσ := ψσ(SL(2, F)) = [SL(2, F) ·σ IdF2] = [SL(2, F)/Hσ] in P(End(F2)). +Definition 4.7. We regard the above closure Xσ as a compactification of SL(2, F)/Hσ and call it +the the wonderful compactification of SL(2, F)/Hσ with respect to the involution σ. +Just from the definition, we have the following trivial lemma: +Lemma 4.8. The SL(2, F)-orbit of IdF2 in P(End(F2)), with respect to the action ·σ, is dense in +Xσ. +In order to prove Theorem 1.4 we will again need the set +P0 := {x = [1, x2; x3, x4] ∈ P(End(F2))} +9 + +which is open in P(End(F2)). Then denote +Xσ,0 := P0 ∩ Xσ +and this is open in Xσ. The set Xσ,0 is also called the big cell in Xσ. +Instead of using the density of U −T ×U in SL(2, F) as in Section 2 it will be much more convenient +to employ the fact that the double coset B+\ SL(2, F)/Hσ has a finite number of elements. Below +we will explicitly compute those double coset for each m ∈ F∗/(F∗)2, and find the double cosets that +are open in SL(2, F). Along the way we will apply the geometric picture provided by Theorem 4.6. +Since the constuction below works by replacing the Borel B+ with any of its SL(2, F)-conjugates, +we will use the opposite B−. +Recall that for any locally compact group G and any closed subgroup H ≤ G, the quotient +topology on the homogeneous space G/H is defined by the canonical projection p : G → G/H being +continuous and open. Moreover, one can prove that for any compact subset Q of G/H, there exists +a compact subset K of G with p(K) = Q (see [BdlHV08, Appendices B, Lemma B.1.1]). +The cases m = 1 and m ̸= 1 behave differently, since for the later we will use a quadratic extension +of F. +Case m = 1. +Recall from [CL22] the following trivial lemma. +Lemma 4.9. Let K be a finite field-extension of Qp. There are 6 orbits of the diagonal subgroup +Diag(K) := { +� d−1 0 +0 +d +� +| d ∈ K∗} ≤ SL(2, K) on the boundary ∂TK: +(1) the Diag(K)-orbit of +� 1 +0 +� +and the Diag(K)-orbit of +� 0 +1 +� +(2) the Diag(K)-orbits of +� 1 +m +� +, for each m ∈ K∗/(K∗)2. +Proof. The subgroup Diag(K) fixes pointwise the ends +� 1 +0 +� +and +� 0 +1 +� +. The Diag(K)-orbit of +� 1 +m +� +, +for each m ∈ K∗/(K∗)2, consists of vectors of the form +� +1 +d2m +� +, these cover the entire boundary +∂TE − { +� 0 +1 +� +, +� 1 +0 +� +}. +□ +As a trivial consequence of Lemma 4.9 one obtains: +Lemma 4.10. Let F be a finite field-extension of Qp, A = ( 0 1 +1 0 ), and σ1 := ιA the corresponding +F-involution of SL(2, F) with associated fixed point group Hσ1. Then +Hσ1 = +� 1 −1 +1 +1 +� +Diag(F) +� 1 −1 +1 +1 +�−1 , +Hσ1 has exaclty 6 orbits on the boundary ∂TF, and the corresponding Hσ1-orbits on the boundary +∂TF are given by the following 6 representatives: { +� 1−m +1+m +� +| m ∈ F∗/(F∗)2} ∪ { +� 1 +−1 +� +, +� 1 +1 +� +} ⊂ ∂TF. +Let be the matrices in SL(2, F) +g1 := +� −1 −1 +2 +1 +� +, +g2 := +� −1 −1 +1 +0 +� +, +gm := +� +1+m m−1 +m +m−1 +1 +� +for m ∈ F∗/(F∗)2 with m ̸= 1 +such that +g1 +�� 1 +−1 +�� += +� 0 +1 +� +, +g2 +�� 1 +1 +�� += +� 0 +1 +� +, +gm +�� 1−m +1+m +�� += +� 0 +1 +� +. +Let I := {IdF2, g1, g2, gm | m ∈ F∗/(F∗)2, m ̸= 1}, and Im := {gm | m ∈ F∗/(F∗)2, m ̸= +1} ∪ {IdF2}, with the convention that for m = 1 ∈ F∗/(F∗)2 we take gm := IdF2. +Lemma 4.11. With the above notation we have that: +(1) SL(2, F) = � +gi∈I +B−giHσ1 +(2) for any gm ∈ Im, the set B−gmHσ1 = U −T gmHσ1 is open in SL(2, F), and moreover the +set +� +gm∈Im +U −T gmHσ1 is dense in SL(2, F) +10 + +(3) ψσ1( +� +gm∈Im +U −T gm) is open and dense in ψσ1(SL(2, F)) = [SL(2, F) ·σ1 IdF2], and so +ψσ1( +� +gm∈Im +U −T gm) = [SL(2, F) ·σ1 IdF2]. +Proof. Let g ∈ SL(2, F) and let ξ0 := +� 0 +1 +� +. Notice the SL(2, F)-stabilizer of ξ0 is exactly B− = +�� +x +0 +y +x−1 +� +| y ∈ F, x ∈ F∗ +� +. +If g−1(ξ0) = ξ0 then g ∈ B− and thus g ∈ B−Hσ1. If g−1(ξ0) ̸= ξ0 then there are 6 cases to +consider each corresponding to the Hσ1-orbits in ∂TF. By applying accordingly an element h ∈ Hσ1 +we get that hg−1(ξ0) ∈ { +� 1−m +1+m +� +| m ∈ F∗/(F∗)2} ∪ { +� 1 +−1 +� +, +� 1 +1 +� +}. Further, by multiplying again +accordingly with some element gi from I we obtain gihg−1(ξ0) = ξ0, and thus g−1 ∈ Hσ1g−1 +i +B−, +giving g ∈ B−giHσ1 as required. This proves part (1) of the lemma. +Let us prove part (2). Take gm ∈ Im. Since by Lemma 4.10 the group Hσ1 is conjugate to +Diag(F), and since the Diag(F)-orbit of the vector +� 1 +m +� +∈ ∂TF, where m ∈ F∗/(F∗)2, is clearly +open with respect to the cone topology on SL(2, F)/B− ∼= ∂TF, then the Hσ1-orbit of +� 1−m +1+m +� +∈ ∂TF +is open in ∂TF. Because the cone topology on SL(2, F)/B− ∼= ∂TF is equivalent to the quotient +topology on SL(2, F)/B− induced from the continuous and open canonical projection of SL(2, F) +to SL(2, F)/B−, we have Hσ1g−1 +m B− is open in SL(2, F), and by taking the inverse map which is +continuous, the set B−gmHσ1 is open in SL(2, F) as well. +It remains to prove that +� +gm∈Im +U −T gmHσ1 is dense in SL(2, F), and by part (1) it is enough to +show that for any fixed gm ∈ Im, the accumulation points of the open set B−gmHσ1 are exactly +the elements of B−g1Hσ1 ∪ B−g2Hσ1. Indeed, since Hσ1 is conjugate to the diagonal subgroup +Diag(F), its attacting and repealing endpoints are { +� 1 +−1 +� +, +� 1 +1 +� +}. Notice that for any fixed gm ∈ Im, +the open set Hσ1 +� 1−m +1+m +� +has { +� 1 +−1 +� +, +� 1 +1 +� +} as its unique accumulation points in ∂TF. Using again the +cone topology on SL(2, F)/B− ∼= ∂TF, there exist sequences {bn}n≥1 ⊂ B− and {hn}n≥1 ⊂ Hσ1 +such that {bngmhn}n≥1 converges to g1, or g2, with respect to the topology on SL(2, F). Then part +(2) follows. +Part (3) follows easily from part (2), the fact that B− = U −T , and the continuity of the map +ψσ1. +□ +By Lemma 4.11 it is advised to study the sets (U −T gi) ·σ1 IdF2, for any gi ∈ I, as well as their +closure. +We have +(U −T g1) ·σ1 IdF2 = +��x +0 +a +x−1 +� �−1 +−1 +2 +1 +� �0 +1 +1 +0 +� � 1 +1 +−2 +−1 +� �x−1 +0 +−a +x +� �0 +1 +1 +0 +� +| x ∈ F∗, a ∈ F +� += +�� +x +0 +a +x−1 +� � +1 +0 +−3 +−1 +� � +0 +x−1 +x +−a +� +| x ∈ F∗, a ∈ F +� += +�� 0 +1 +−1 +2ax−1 − 3x−2 +� +| x ∈ F∗, a ∈ F +� +⇒ +⇒ [(U −T g1) ·σ1 IdF2] = {[0, 1; −1, 2ax−1 − 3x−2] | x ∈ F∗, a ∈ F} ⊂ [SL(2, F) ·σ1 IdF2]. +(6) +11 + +(U −T g2) ·σ1 IdF2 = +�� +x +0 +a +x−1 +� � +−1 +−1 +1 +0 +� � +0 +1 +1 +0 +� � +0 +1 +−1 +−1 +� � +x−1 +0 +−a +x +� � +0 +1 +1 +0 +� +| x ∈ F∗, a ∈ F +� += +��x +0 +a +x−1 +� � 1 +0 +−1 +−1 +� �0 +x−1 +x +−a +� +| x ∈ F∗, a ∈ F +� += +�� +0 +1 +−1 +2ax−1 − x−2 +� +| x ∈ F∗, a ∈ F +� +⇒ +⇒ [(U −T g2) ·σ1 IdF2] = {[0, 1; −1, ax−1 − x−2] | x ∈ F∗, a ∈ F} ⊂ [SL(2, F) ·σ1 IdF2]. +(7) +Now for every m ∈ F∗/(F∗)2, with m ̸= 1, we have: +(U −T gm) ·σ1 IdF2 = +��x +0 +a +x−1 +� �1 + m +m − 1 +m +m−1 +1 +� �0 +1 +1 +0 +� � 1 +1 − m +m +1−m +1 + m +� �x−1 +0 +−a +x +� �0 +1 +1 +0 +� +| x ∈ F∗, a ∈ F +� += +�� +x +0 +a +x−1 +� � 3m−1 +1−m +4m +1−2m +(1−m)2 +3m−1 +m−1 +� � +0 +x−1 +x +−a +� +| x ∈ F∗, a ∈ F +� += +�� +4mx2 +3m−1 +1−m − 4max +4max + 3m−1 +m−1 +2ax−1 3m−1 +1−m + x−2 1−2m +(1−m)2 − 4ma2 +� +| x ∈ F∗, a ∈ F +� +⇒ +⇒ [(U −T gm) ·σ1 IdF2] = += {[1, +3m − 1 +(1 − m)4mx2 − ax−1; ax−1 + +3m − 1 +(m − 1)4mx2 , +2a +4mx2 +3m − 1 +1 − m + +1 − 2m +(1 − m)24mx4 − a2x−2] | x ∈ F∗, a ∈ F}. +(8) +Then the only accumulation points of the set [(U −T gm) ·σ1 IdF2] are the points [1, −b; b, −b2] with +b ∈ F. Those elements appear when taking sequences {xnk = ω−nk +F +}k≥0 and {ank := bxnk}k≥0, with +0 < nk → ∞ and b ∈ F. For sequences of the form {xnk = ωnk +F }k≥0 we obtain no limits. +For the case m ∈ F∗/(F∗)2 with m = 1, by our convention above, we take gm := IdF2 and we +have: +(U −T ) ·σ1 IdF2 = +�� +x +0 +a +x−1 +� � +0 +1 +1 +0 +� � +x−1 +0 +−a +x +� � +0 +1 +1 +0 +� +| x ∈ F∗, a ∈ F +� += +�� 0 +x +x−1 +a +� �0 +x−1 +x +−a +� +| x ∈ F∗, a ∈ F +� += +�� +x2 +−xa +ax +−a2 + x−2 +� +| x ∈ F∗, a ∈ F +� +⇒ +⇒ [(U −T ) ·σ1 IdF2] = {[1, −x−1a; ax−1, −a2x−2 + x−4] | x ∈ F∗, a ∈ F}. +(9) +Then the only accumulation points of the set [(U −T gm) ·σ1 IdF2] are the points [1, −b; b, −b2] with +b ∈ F. Those elements appear when taking sequences {xnk = ω−nk +F +}k≥0 and {ank := bxnk}k≥0, with +0 < nk → ∞ and b ∈ F. For sequences of the form {xnk = ωnk +F }k≥0 we obtain no limits. +As well, notice that the set {[1, −b; b, −b2] | b ∈ F} is the U −T -orbit of the element [1, 0; 0, 0] ∈ +P(End(F2)) with respect to the action ·σ1. +As a result of Lemma 4.11 and the above calculations in (6) to (9) we have: +Lemma 4.12. Recall the notation Im := {gm | m ∈ F∗/(F∗)2, m ̸= 1} ∪ {IdF2}. Then: +Xσ1,0 ∩ ψσ1(SL(2, F)) = [( +� +gm∈Im +U −T gm) ·σ1 IdF2] +12 + +Xσ1,0 = [( +� +gm∈Im +U −T gm) ·σ1 IdF2] ∩ P0 +Xσ1 = +� +g∈SL(2,F) +g ·σ1 Xσ1,0. +Proof. It is clear from the above calculation that +[( +� +gm∈Im +U −T gm) ·σ1 IdF2] = Xσ1,0 ∩ ψσ1(SL(2, F)) = P0 ∩ ψσ1(SL(2, F)). +Since [( +� +gm∈Im +U −T gm) ·σ1 IdF2] = Xσ1 by Lemma 4.11(3), we immediately have +Xσ1,0 = [( +� +gm∈Im +U −T gm) ·σ1 IdF2] ∩ P0. +Let C ∈ Xσ1. As C ∈ P(End(F2)) we have C = [x1, x2; x3, x4]. If x1 ̸= 0 then C ∈ Xσ1,0 and +we are done. If x1 = 0, then by some easy calculations one can arrange for some g ∈ SL(2, F), such +that ρ(g)Cρ(σ1(g−1)) ∈ P0. Then we are done. +□ +Proof of Theorem 1.4. By Lemma 4.12, the SL(2, F)-orbits in Xσ1 with respect to the ·σ1 action are +determined by the SL(2, F)-orbits given by the points of the set Xσ1,0. Moreover, the calculations +in (8) and (9) show the only accumulation points of the set [(U −T gm) ·σ1 IdF2] in P0 are the points +[1, −b; b, −b2] with b ∈ F, and the only accumulation point of the set [T ·σ1 IdF2] in P0 is [1, 0; 0, 0]. +As well, the set {[1, −b; b, −b2] | b ∈ F} is the U −T -orbit of the element [1, 0; 0, 0] ∈ P(End(F2)) with +respect to the action ·σ1. Further, an easy computation +�a +b +c +d +� �1 +0 +0 +0 +� �0 +1 +1 +0 +� � d +−b +−c +a +� �0 +1 +1 +0 +� += +�a +b +c +d +� �1 +0 +0 +0 +� � a +−c +−b +d +� += +�a2 +−ac +ca +−c2 +� +shows that the SL(2, F)-orbit of the point [1, 0; 0, 0] is indeed closed in Xσ1. +As [( +� +gm∈Im +U −T gm) ·σ1 IdF2] is already part of the SL(2, F)-orbit of the point [1, 0; 0, 1] = [IdF2] +with respect to the ·σ1 action, the theorem follows. +□ +Corollary 4.13. With respect to the action ·σ1, the stabilizer in SL(2, F) of the point [1, 0; 0, 1] = +[IdF2] is Hσ1, and the stabilizer in SL(2, F) of the point [1, 0; 0, 0] is the subgroup {µ2 · ( 1 b +0 1 ) | b ∈ F} +of the Borel B+ ≤ SL(2, F), where µ2 is the group of 2nd roots of unity in F. +Proof. This just follows from the computation +�a +b +c +d +� �1 +0 +0 +0 +� � a +−c +−b +d +� += +�a2 +−ac +ca +−c2 +� +. +□ +Case m ̸= 1. +Recall for m ∈ F∗/(F∗)2 with m ̸= 1 we consider A := ( 0 1 +m 0 ) which has associated the F-involution +of SL(2, F) given by σm := ιA. +From [CL22] also recall Lemma 5.3: +Lemma 4.14. Let F be a finite field extension of Qp, A = ( 0 1 +m 0 ), with m ∈ F∗/(F∗)2, m ̸= 1 and +σm := ιA the corresponding F-involution of SL(2, F). Then Hσm := {g ∈ SL(2, F) | σm(g) = g} has +at most 8 orbits on the boundary ∂TF. +Choose now a set I ⊂ SL(2, F) of representatives such that the Hσm-orbits of g−1 +i +�� 0 +1 +�� +∈ ∂TF, +for gi ∈ I, are all disjoint and cover the boundary ∂TF. By Lemma 4.14 we know that I is a finite +set. +13 + +Lemma 4.15. With the above notation we have that: +(1) SL(2, F) = � +gi∈I +B−giHσm +(2) for any gi ∈ I, the set B−giHσm = U −T giHσm is open in SL(2, F) +(3) ψσm( � +gi∈I +U −T gi) is open and dense in ψσm(SL(2, F)) = [SL(2, F) ·σm IdF2], and so +ψσm( +� +gi∈I +U −T gi) = [SL(2, F) ·σm IdF2]. +Proof. Let g ∈ SL(2, F) and let ξ0 := +� 0 +1 +� +. Notice the SL(2, F)-stabilizer of ξ0 is exactly B− = +�� +x +0 +y +x−1 +� +| y ∈ F, x ∈ F∗ +� +. +If g−1(ξ0) = ξ0 then g ∈ B− and thus g ∈ B−Hσm. If g−1(ξ0) ̸= ξ0 then there are exactly +|I| < ∞ cases to consider, each corresponding to the Hσm-orbits in ∂TF. By applying accordingly +an element h ∈ Hσm we get that hg−1(ξ0) ∈ {g−1 +i +(ξ0) | gi ∈ I}. Further, by multiplying again +accordingly with some element gi from I we obtain gihg−1(ξ0) = ξ0, and thus g−1 ∈ Hσmg−1 +i +B−, +giving g ∈ B−giHσm as required. This proves part (1) of the lemma. +Let us prove part (2). Consider the quadratic field extention E := F(√m). By Theorem 4.6(2) we +know that the two ends ξ± := +� +1 +±√m +� +are in P 1E−P 1F = ∂TE −∂TF. Moreover, the group SL(2, F) +acts on TE, it preserves the subset of ends ∂TF, and in fact the subset ∂TF is closed in ∂TE with +respect to the cone topology on ∂TE. Now, again by Theorem 4.6 we know that FixSL(2,E)({ξ−, ξ+}) +is a conjugate of the diagonal Diag(E) ≤ SL(2, E), and by the same proof as in Lemma 4.11(2), +FixSL(2,E)({ξ−, ξ+}) has 4 open orbits on ∂TE − {ξ±}. In particular, this means the open orbits of +FixSL(2,E)({ξ−, ξ+}) cover ∂TF, and the intersection of an open orbit of FixSL(2,E)({ξ−, ξ+}) in ∂TE +with the closed set ∂TF remains open with respect to the cone topology on ∂TF and TF. In addition, +by the proof of Lemma 4.14 from [CL22][ Lemma 5.3] we know that Hσm = FixSL(2,E)({ξ−, ξ+}) ∩ +SL(2, F), has at most two orbits on any of the open orbits of FixSL(2,E)({ξ−, ξ+}) interesecting ∂TF. +Since Hσm is compact and the action of SL(2, F) on ∂TF is continuous, any Hσm-orbit on ∂TF is open +with respect to the cone topololgy on ∂TF. Because the cone topology on SL(2, F)/B− ∼= ∂TF is +equivalent to the quotient topology on SL(2, F)/B− induced from the continuous and open canonical +projection of SL(2, F) to SL(2, F)/B−, we have Hσmg−1 +i +B− is open in SL(2, F), and by taking the +inverse map which is continuous, the set B−giHσ1 is open in SL(2, F) as well. +Part (3) easily follows from parts (1) and (2). +□ +Take g = +�a +b +c +d +� +∈ SL(2, F) then we have: +[(U −T g) ·σm IdF2] = +��� +x +0 +y +x−1 +� � +a +b +c +d +� � +0 +1 +m +0 +� � +d +−b +−c +a +� � +x−1 +0 +−y +x +� � +0 +1 +m +1 +0 +�� +| x ∈ F∗, y ∈ F +� += +��� +xa +xb +ya + x−1c +yb + x−1d +� �−c +a +md +−mb +� �0 +1 +mx +x +−y +m +�� +| x ∈ F∗, y ∈ F +� += +��� +xa +xb +ya + c +x +yb + d +x +� � xa +− c +mx − ya +m +−mxb +d +x + by +�� +| x ∈ F∗, y ∈ F +� +multiply +======= +x−2 +��� +a +b +ya +x + +c +x2 +yb +x + d +x2 +� � a +− +c +mx2 − ya +mx +−mb +d +x2 + by +x +�� +| x ∈ F∗, y ∈ F +� +(10) +Then the only accumulation points of the set [(U −T g) ·σm IdF2] ⊂ P0 are the points +��� a +b +za +zb +� � a +− za +m +−mb +bz +�� += [1, − z +m; z, −z2 +m ] | z ∈ F +� +14 + +since we always have a2−mb2 ̸= 0. Those elements appear when taking sequences {xnk = ω−nk +F +}k≥0 +and {ynk := zxnk}k≥0, with 0 < nk → ∞ and z ∈ F. For sequences of the form {xnk = ωnk +F }k≥0 we +obtain no limits. +As well, notice that the set {[1, − z +m; z, − z2 +m ] | z ∈ F} is the U −T -orbit of the element [1, 0; 0, 0] ∈ +P(End(F2)) with respect to the action ·σm. +Remark 4.16. We combine Lemma 4.15(1) and (2) with the calculation (10). Therefore, since +ψσm( � +gi∈I +U −T gi) ⊂ P0 is open and equals ψσm(SL(2, F)) = [SL(2, F) ·σm IdF2] we notice that the big +cell Xσ,0 := P0 ∩ Xσm in Xσm is exactly the set Xσm. +Proof of Theorem 1.4. By Remark 4.16, the SL(2, F)-orbits in Xσm with respect to the ·σm action are +determined by the SL(2, F)-orbits given by the points of the set Xσm,0. Moreover, the calculations in +(10) show the only acumulation points of [( � +g∈I +U −T g) ·σm IdF2] in P0 are the points [1, − z +m; z, − z2 +m ], +with z ∈ F, which are exactly the U −T -orbit of the element [1, 0; 0, 0] ∈ P(End(F2)) with respect to +the action ·σm. In addition, the only accumulation point of the set [T ·σm IdF2] in P0 is [1, 0; 0, 0]. +Further, by an easy computation +� +a +b +c +d +� � +1 +0 +0 +0 +� � +0 +1 +m +0 +� � +d +−b +−c +a +� � +0 +1 +m +1 +0 +� += +� +a +b +c +d +� � +1 +0 +0 +0 +� � +a +−c +m +−bm +d +� += +�a2 +−ac +m +ca +−c2 +m +� +. +showing that the SL(2, F)-orbit of the point [1, 0; 0, 0] is indeed closed in Xσm. +As [( � +gi∈I +U −T gi)·σm IdF2] is already the SL(2, F)-orbit of the point [1, 0; 0, 1] = [IdF2] with respect +to the ·σm action, the theorem follows. +□ +Corollary 4.17. With respect to the action ·σm, the stabilizer in SL(2, F) of the point [1, 0; 0, 1] = +[IdF2] is Hσm, and the stabilizer in SL(2, F) of the point [1, 0; 0, 0] is the subgroup {µ2 ·( 1 b +0 1 ) | b ∈ F} +of the Borel B+ ≤ SL(2, F), where µ2 is the group of 2nd roots of unity in F. +Proof. This just follows from the computation +�a +b +c +d +� �1 +0 +0 +0 +� � 0 +1 +m +0 +� � d +−b +−c +a +� � +0 +1 +m +1 +0 +� += +�a +b +c +d +� �1 +0 +0 +0 +� � a +−c +m +−bm +d +� += +�a2 +−ac +m +ca +−c2 +m +� +by taking +�a2 +−ac +m +ca +−c2 +m +� += +� +1 +0 +0 +0 +� +. +□ +References +[AB18] Ana B˘alibanu, Part II: The wonderful compactification, 2018. Lectures notes +https://people.math.harvard.edu/~ana/part2.pdf. ↑2 +[BdlHV08] Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Mono- +graphs, vol. 11, Cambridge University Press, Cambridge, 2008. MR2415834 ↑10 +[BH09] Stacy L Beun and Aloysius G Helminck, On the classification of orbits of symmetric subgroups acting on +flag varieties of SL(2, k), Communications in Algebra 37 (2009), no. 4, 1334–1352. ↑1, 9 +[CL22] Corina Ciobotaru and Arielle Leitner, Chabauty Limits of Groups of Involutions In SL(2, F ) for local +fields, 2022. arXiv:2208.12247v1. ↑1, 3, 9, 10, 13, 14 +[DCP83] C. De Concini and C. Procesi, Complete symmetric varieties, In: Invariant Theory, Lect. Notes in Math., +vol. 996, Springer (1983), 1–44. ↑1 +[DCS99] C. +De Concini and T.A. Springer, Compactification of symmetric varieties, Transformation Groups 4 +(1999), 273–300, DOI https://doi.org/10.1007/BF01237359. ↑1, 2 +[EJ08] Sam Evens and Benjamin F. Jones, On the wonderful compactification, 2008. arXiv:0801.0456. ↑1 +[FTN91] Alessandro Fig´a-Talamanca and Claudio Nebbia, Harmonic Analysis and Representation Theory for +Groups Acting on Homogenous Trees, London Mathematical Society Lecture Note Series, Cambridge +University Press, Cambridge, 1991. ↑5 +[Hel00] A. G. Helminck, On the classification of k-involutions, Adv. Math. 153 (2000), no. 1, 1–117, DOI +10.1006/aima.1998.1884. MR1771689 ↑8 +15 + +[HWD06] Aloysius G. Helminck, Ling Wu, and Christopher E. Dometrius, Involutions of SL(n, k), (n > 2), Acta +Appl. Math. 90 (2006), no. 1-2, 91–119, DOI 10.1007/s10440-006-9032-7. MR2242950 ↑8 +[HW02] Aloysius G Helminck and Ling Wu, Classification of involutions of SL(2, k), Communications in Algebra +30 (2002), no. 1, 193–203. ↑8, 9 +[SJ98] Paul J. Sally Jr., An Introduction to p-adic Fields, Harmonic Analysis and the Representation Theory of +SL2, Letters in Mathematical Physics 46 (1998), 1–47, DOI doi.org/10.1023/A:1007583108067. ↑1 +[Ser12] Jean-Pierre Serre, A course in arithmetic, Vol. 7, Springer Science & Business Media, 2012. ↑1, 2, 9 +[Ste74] Robert Steinberg, Abstract homomorphisms of simple algebraic groups (after A. Borel and J. Tits), +S´eminaire Bourbaki, 25`eme ann´ee (1972/1973), Exp. No. 435, Springer, Berlin, 1974, pp. 307–326. Lecture +Notes in Math., Vol. 383. MR0414732 ↑ +[Tit71] J. Tits, Repr´esentations lin´eaires irr´eductibles d’un groupe r´eductif sur un corps quelconque, J. Reine +Angew. Math. 247 (1971), 196–220, DOI 10.1515/crll.1971.247.196 (French). MR277536 ↑2, 3 +16 + diff --git a/iNAyT4oBgHgl3EQfkPi4/content/tmp_files/load_file.txt b/iNAyT4oBgHgl3EQfkPi4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b35af4c88670532243885897c06794583da5baf6 --- /dev/null +++ b/iNAyT4oBgHgl3EQfkPi4/content/tmp_files/load_file.txt @@ -0,0 +1,592 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf,len=591 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='00431v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='GR] 1 Jan 2023 ON WONDERFUL COMPACTIFICATIONS OF SL(2, F) FOR NON-ARCHIMEDEAN LOCAL FIELDS F CORINA CIOBOTARU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We compute the wonderful compactification of symmetric varieties of SL(2, F), where F is a finite field-extension of Qp with p ̸= 2, that comes from either an abstract or F-involutions of SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For each of those wonderful compactifications we find the SL(2, F)-stabilizers of the accumulation points of the corresponding symmetric varieties and compare them to the Chabauty limits found in [CL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Introduction The wonderful compactifications emerged from specific problems in enumerative geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In laymen’s terms a wonderful compactification is a way of compactifying a symmetric variety G/H via an irreducible linear representation of the group G, where G is an algebraic group and H ≤ G is the fixed point set of an involutorial automorphism of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This procedure will provide remarkable properties of the G-closed orbits of that compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In principal, the wonderful compactifi- cation of G/H does not depend on the chosen irreducible linear representation of G, and a Borel subgroup of G will only have one open and dense orbit in G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The study of wonderful compactifi- cations was initiated by DeConcini and Procesi [DCP83] in the case of a complex semi-simple group G of adjoint type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a semi-simple group with trivial center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since then wonderful compacti- fications have been extensively studied in algebraic geometry and have important applications in many fields of Mathematics (see the Introduction of [EJ08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If one replaces the complex numbers by any (not necessarily algebraically closed) field k of characteristic ̸= 2, the results of [DCP83] were further expanded by DeConcini and Springer in [DCS99] to any adjoint semi-simple group G defined over k and H the fixed point group of an involutorial automorphism of G defined over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In this article we focus on the wonderful compactifications of symmetric varieties of SL(2, F), where F a finite field-extension of Qp with p ̸= 2, that come from various abstract and F-involutions of SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notice that SL(2, F) is not of adjoint type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In particular, we are interested in computing the SL(2, F)-stabilizers of the accumulation points in the wonderful compactifications of those symmetric varieties and we compare them to the Chabauty limits found in [CL22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Although some of the techniques used in this article to build the wonderful compactifications are similar to the well established literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' [EJ08]), we will make extensive use of some results from [BH09], the Bruhat–Tits tree and its visual boundary associated with SL(2, F), which contrast with the methods employed in [DCS99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let us first fix the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Throughout this article we restrict to a prime p ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let F be a finite field-extension of Qp and E be any quadratic extension of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let kF, kE be the residue fields of F, E, respectively, and ωF, ωE be uniformizers of F, E, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall k∗ F/(k∗ F)2 = {1, S}, for some non-square S ∈ k∗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then F∗/(F∗)2 = {1, ωF, S, SωF} ([Ser12, Corollaries to Theorems 3 and 4] or [SJ98, page 41, Section 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If E is ramified then E = F(√ωF), or E = F(√SωF) (where ωF ̸= ωE), or if E is unramified then E = F( √ S) (where ωF = ωE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We choose the unique valuation | · |E on E Date: January 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' corina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='ciobotaru@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 1 that extends the given valuation | · |F on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Choose α ∈ {√ωF, √ S, √SωF} and so E = F(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notice each element x ∈ E can be uniquely written as x = a+bα, with a, b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For the ramified extensions we can consider ω2 E = ωF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let OF := {x ∈ F | |x|F ≤ 1} denote the ring of integers of F, then OF is compact and open in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For F = Qp we have OQp = Zp, ωQp = p, kQp = Fp, F∗ p/(F∗ p)2 = {1, Sp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We denote by TF the Bruhat–Tits tree for SL(2, F) whose vertices are equivalence classes of OF- lattices in F2 (for its construction see [Ser12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The tree TF is a regular, infinite tree with valence |kF|+1 at every vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The boundary at infinity ∂TF of TF is the projective space P 1(F) ∼= F∪{∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, the end ∞ ∈ ∂TF corresponds to the vector � 0 1 � ∈ P 1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The rest of the ends ξ ∈ ∂TF correspond to the vectors � 1 x � ∈ P 1(F), where x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' To give a concrete example, the Bruhat–Tits tree of SL(2, Qp) is the p + 1 = |Fp| + 1-regular tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The boundary at infinity ∂TQp of TQp is the projective space P 1(Qp) = Qp ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let ρ : SL(2, F) → GL(F2) be the representation of SL(2, F) over F given by g ∈ SL(2, F) �→ ρ(g) := g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' One can easily prove ρ is an irreducible and continuous representation of SL(2, F) with respect to the compact-open topologies of the locally compact groups SL(2, F) and GL(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By the known theory ρ has a dominant weight (see [Tit71, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For an abstract involution θ : SL(2, F) → SL(2, F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' θ is an abstract automorphism of SL(2, F) with θ2 = Id, we consider: (1) the action of SL(2, F) on End(F2) given by g ·θ D := ρ(g)Dρ(θ(g−1)), for every g ∈ SL(2, F) and D ∈ End(F2), (2) the continuous map ψθ : SL(2, F) → P(End(F2)) given by g �→ ψθ(g) := [ρ(g)ρ(θ(g−1))] = [ρ(gθ(g−1))], (3) and the fixed point group of θ denoted by Hθ := {g ∈ SL(2, F) | θ(g) = g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We take Xθ := ψθ(SL(2, F)) = [SL(2, F) ·θ IdF2] ⊂ P(End(F2)) and, by abuse of terminology, call it the wonderful compactification of SL(2, F)/Hθ with respect to the involution θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We will see in the proofs below that a Borel subgroup of SL(2, F) might have more open orbits on SL(2, F)/Hθ, that are pairwise disjoint and not dense in SL(2, F)/Hθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This is because SL(2, F) is not of adjoint type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By abuse of notation, for any g ∈ SL(2, F) and any [D] ∈ Xθ ⊂ P(End(F2)) we use g ·θ [D] to mean [ρ(g)Dρ(θ(g−1))] ∈ Xθ for some, thus any, representative D ∈ End(F2) of [D] ∈ P(End(F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By the known theory, those wonderful compactifications do not depend on the irreducible repre- sentation ρ of SL(2, F) (for a proof see [AB18, Section 7] or [DCS99, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The three main results of the paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' They are proved in Sections 2, 3, 4, respectively, where the reader can refer for the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Consider the involution θ(x, y) := (y, x) of SL(2, F)×SL(2, F) where Hθ := Diag(G) = SL(2, F) is the fixed point group, and SL(2, F)×SL(2, F) acts on End(F2) by (g1, g2)·θA = ρ(g1)Aρ(g−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then there are 2 orbits of SL(2, F) × SL(2, F) in Xθ with respect to the action ·θ: (1) the open SL(2, F)×SL(2, F)-orbit of IdF2, whose SL(2, F)×SL(2, F)-stabilizer is Diag(SL(2, F)× SL(2, F)) = SL(2, F) (2) the closed SL(2, F) × SL(2, F)-orbit of [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)), whose SL(2, F) × SL(2, F)- stabilizer is B+ × B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let F be a finite field-extension of Qp and E = F(α) be any quadratic extension of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Take θ to be the involution of SL(2, E) induced by the ‘conjugation’ in E with respect to F, θ(a + αb) := a − αb, for any a, b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' There are 2 orbits of SL(2, E) in Xθ with respect to the action θ: (1) the open SL(2, E)-orbit of IdE2, whose SL(2, E)-stabilizer is SL(2, F) (2) the closed SL(2, E)-orbit of [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(E2)), whose SL(2, E)-stabilizer is the subgroup { � a−αb z 0 a+αb � | a, b ∈ F with a2 − α2b2 = 1, z ∈ E} ≤ B+ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let m ∈ F∗/(F∗)2 = {1, ωF, S, SωF}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Consider Am := ( 0 1 m 0 ) and the corresponding inner F-involution of SL(2, F) given by σm := ιAm (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' There are 2 orbits of SL(2, F) in Xσm with respect to the action ·σm: (1) the open SL(2, F)-orbit of IdF2, whose SL(2, F)-stabilizer is Hσm (2) the closed SL(2, F)-orbit of [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)), whose SL(2, F)-stabilizer is the subgroup {µ2 · ( 1 b 0 1 ) | b ∈ F} of the Borel B+ ≤ SL(2, F), where µ2 is the group of 2nd roots of unity in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Comparing the stabilizers of the accumulation points [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] and [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] computed in Theo- rems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4, respectively, with the Chabauty limits computed in [CL22] Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='6, respectively, the reader can notice we found the same subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The general case of SL(n, F) will be treated in a future research project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Ciobotaru is supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 754513 and The Aarhus University Research Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' She was partially supported by The Mathematis- ches Forschungsinstitut Oberwolfach (MFO, Oberwolfach Research Institute for Mathematics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' She would like to thank those two institutions for the perfect working conditions they provide, and to Linus Kramer and Maneesh Thakur for very helpful discussions regarding reference [Tit71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As well, Ciobotaru thanks Arielle Leitner for the wonderful collaboration on our joint paper [CL22] that gave rise to this article, for reading it and providing useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The wonderful compactification of SL(2, F) First recall the usual wonderful compactification associated with the involution θ(x, y) := (y, x) of G := SL(2, F) × SL(2, F), having Diag(G) = SL(2, F) as a fixed point group of the involution θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We compactify SL(2, F) = G/Diag(G) in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Consider ρ(SL(2, F)) ≤ GL(F2) ⊆ End(F2) and SL(2, F)× SL(2, F) acting on End(F2) by (g1, g2)·θ A = ρ(g1)Aρ(g−1 2 ), where (g1, g2) ∈ SL(2, F) × SL(2, F) and A ∈ End(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then the stabilizer in SL(2, F) × SL(2, F) of IdF2 is exactly Diag(G) = SL(2, F), and ρ(SL(2, F)) = (SL(2, F) × SL(2, F)) ·θ IdF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Next we take the map ψθ : SL(2, F) × SL(2, F) → P(End(F2)), given by (g1, g2) �→ ψθ(g1, g2) := [ρ(g1g−1 2 )] and we want to understand the closure Xθ := ψθ(SL(2, F) × SL(2, F)) = [(SL(2, F) × SL(2, F)) ·θ IdF2] = [ρ(SL(2, F))] in P(End(F2)) called the wonderful compactification of SL(2, F) = G/Diag(G) with respect to the involution θ, where P(End(F2)) is endowed with the usual topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Denote T := ��x 0 0 x−1 � | x ∈ F∗ � U + := ��1 y 0 1 � | y ∈ F � U − := ��1 0 y 1 � | y ∈ F � B+ := ��x y 0 x−1 � | y ∈ F, x ∈ F∗ � B− = ��x 0 y x−1 � | y ∈ F, x ∈ F∗ � , (1) where T is the maximal tori of SL(2, F), U + is the unipotent radical of the Borel subgroup B+, and U − is the unipotent radical of the opposite Borel B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By the definition of Xθ: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The SL(2, F) × SL(2, F) orbit of IdF2 in P(End(F2)) with respect to the action ·θ is dense in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2 we need the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let P0 := {x = [1, x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' x3, x4] ∈ P(End(F2))} 3 and notice this is open in P(End(F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Denote Xθ,0 := P0 ∩ Xθ and notice this is again open in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The set Xθ,0 is called the big cell in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall that B+ = T U + as well as the following decomposition in SL(2, F) (2) � a b c d � = � 1 0 c/a 1 � � a 0 0 a−1 � � 1 b/a 0 1 � , for a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' From here it is easy to verify the following facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The set U −T U + is dense in SL(2, F), and in particular [(U −T × U +) ·θ IdF2] is dense in ψθ(SL(2, F) × SL(2, F)) = [ρ(SL(2, F))], and so in [ρ(SL(2, F))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The first part of the lemma follows from an easy approximation of matrices having a = 0 using the decomposition given in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The rest of the lemma follows directly from the density of U −T U + in SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We have that Xθ,0 ∩ [ρ(SL(2, F))] = [(U −T × U +) ·θ IdF2] Xθ = � (g1,g2)∈SL(2,F)×SL(2,F) (g1, g2) ·θ Xθ,0 Xθ,0 = [(U −T × U +) ·θ IdF2] ∩ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By the decomposition 2 it is clear that [(U −T × U +) ·θ IdF2] = Xθ,0 ∩ [ρ(SL(2, F))] = P0 ∩ [ρ(SL(2, F))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since [(U −T × U +) ·θ IdF2] = Xθ we immediately have Xθ,0 = [(U −T × U +) ·θ IdF2] ∩ P0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' the closure of the set [(U −T × U +) ·θ IdF2] in P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let C ∈ Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As C ∈ P(End(F2)) we have C = [x1, x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' x3, x4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If x1 ̸= 0 then C ∈ Xθ,0 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If x1 = 0, then by some easy calculations one can arrange for some (g1, g2) ∈ SL(2, F)×SL(2, F) such that ρ(g1)Cρ(g−1 2 ) ∈ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3 it is enough to compute the set (U −T × U +)·θ IdF2, and then to precisely compute its closure [(U −T × U +) ·θ IdF2] ∩ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We have (U −T × U +) ·θ IdF2 = ��1 0 a 1 � �x 0 0 x−1 � �1 −b 0 1 � | x ∈ F∗, a, b ∈ F � �� x −xb ax −axb + x−1 � | x ∈ F∗, a, b ∈ F � ⇒ ⇒ [(U −T × U +) ·θ IdF2] = {[1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a, −ab + x−2] | x ∈ F∗, a, b ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (3) Then the only accumulation points of the set [(U −T × U +) ·θ IdF2] in P0 are the points of the form [1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a, −ab], for any a, b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, it is very easy to see that the only accumulation point of the set [T ·θ IdF2] in P0 is just [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notice, an element [1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a, −ab] respresents a matrix in P(End(F2)) of determinant zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As well, {[1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a, −ab] | a, b ∈ F} is the (U −T × U +)- orbit of the element [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' From the above calculation and since the action ·θ is continuous, it is immediate that the SL(2, F) × SL(2, F)-orbit of IdF2 is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' It is also clear that the SL(2, F) × SL(2, F)-orbit of [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] in P(End(F2)) is closed and, by in easy computation, the stabilizer of [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] in SL(2, F) × SL(2, F) is exactly B+ × B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The wonderful compactification of SL(2, E)/ SL(2, F) for quadratic E/F Let F be a finite field-extension of Qp and E = F(α) be any quadratic extension of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Denote by θ : E → E the ‘conjugation’ in E with respect to F, θ(a + αb) := a − αb, for any a, b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This map θ induces an abstract involution of SL(2, E) given by ( x y z t ) ∈ SL(2, E) �→ θ(( x y z t )) := � θ(x) θ(y) θ(z) θ(t) � whose fixed point group Hθ := {g ∈ SL(2, E) | θ(g) = g} is SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The stabiliser in SL(2, F), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' SL(2, E), of the endpoint � 1 0 � is the Borel subgroup B+ F := { � a b 0 a−1 � | b ∈ F, a ∈ F×}, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' B+ E := { � x y 0 x−1 � | y ∈ E, x ∈ E×}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The stabiliser in SL(2, F), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' SL(2, E), of the endpoint � 0 1 � is the opposite Borel subgroup B− F := { � a 0 b a−1 � | b ∈ F, a ∈ F ×}, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' B− E := { � x 0 y x−1 � | y ∈ E, x ∈ E×}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let TE := �� x 0 0 x−1 � | x ∈ E∗ � UE := �� 1 y 0 1 � | y ∈ E � U − E := �� 1 0 y 1 � | y ∈ E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Given the abstract involution θ : SL(2, E) → SL(2, E), we consider the action of SL(2, E) on End(E2) given by g ·θ D := ρ(g)Dρ(θ(g−1)), for every g ∈ SL(2, E) and D ∈ End(E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The stabilizer in SL(2, E), with respect to the action ·θ, of the element IdE2 is the subgroup SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then SL(2, E) ·θ IdE2 ∼= SL(2, E)/ SL(2, F) and we consider the continuous map ψθ : SL(2, E) → P(End(E2)), given by g �→ ψθ(g) := [ρ(g)ρ(θ(g−1))] = [ρ(gθ(g−1))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We want to understand the following closure with respect to the involution θ and the map ψθ: Xθ := ψθ(SL(2, E)) = [SL(2, E) ·θ IdE2] = [SL(2, E)/ SL(2, F)] in P(End(E2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We regard the above closure Xθ as a compactification of SL(2, E)/ SL(2, F) and call it the the wonderful compactification of SL(2, E)/ SL(2, F) with respect to the involution θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Just from the definition, we have the following trivial lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The SL(2, E)-orbit of IdE2 in P(End(E2)), with respect to the action ·θ, is dense in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In order to compute the wonderful compactification of SL(2, E)/ SL(2, F) we first need to find the number of orbits of SL(2, F) on the boundary ∂TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' There are at most 5 SL(2, F)-orbits on the boundary ∂TE ∼= P 1E = E ∪ {∞}: (1) the SL(2, F)-orbit of � 1 0 � which is closed (2) for each m ∈ F∗/(F∗)2, the SL(2, F)-orbit of � 1 mα � is open and might coincide with the orbit of a different m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' It is clear that the SL(2, F)-orbit of � 1 0 � is closed in ∂TE with respect to the cone topology on ∂TE (see [FTN91] page 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Consider the subgroup B := { � d−1 0 c d � | c ∈ F, d ∈ F∗} in SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then the image of � 1 mα � under B is � 1 cd+d2mα � , which covers the entire ∂TE − ∂TF when m takes all values from F∗/(F∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, for each m ∈ F∗/(F∗)2, it is easy to notice that the set { � 1 cd+d2mα � | c ∈ OF, d ∈ O∗ F} is open in ∂TE with respect to the cone topology on ∂TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Thus the SL(2, F)-orbit of � 1 mα � is open, for each m ∈ F∗/(F∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ 5 Consider the matrices in SL(2, E) = SL(2, F(α)) g1 := IdE2, gm := � 1 − 1 mα 0 1 � for m ∈ F∗/(F∗)2 such that gm �� 1 mα �� = � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let I := {IdF2, gm | m ∈ F∗/(F∗)2}, and Im := {gm | m ∈ F∗/(F∗)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' With the above notation we have that: (1) SL(2, E) = � gi∈I B− E gi SL(2, F) (2) for any gm ∈ Im, the set B− E gm SL(2, F) = U − E TEgm SL(2, F) is open in SL(2, E), and moreover the set � gm∈Im U − E TEgm SL(2, F) is dense in SL(2, E) (3) ψθ( � gm∈Im U − E TEgm) is open and dense in ψθ(SL(2, E)) = [SL(2, E) ·θ IdE2], and so ψθ( � gm∈Im U − E TEgm) = [SL(2, E) ·θ IdE2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let g ∈ SL(2, E) and let ξ0 := � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The SL(2, E)-stabilizer of ξ0 is B− E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If g−1(ξ0) is in the SL(2, F)-orbit of ξ0 then g ∈ B− E SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If not then there are 4 cases to consider, each corresponding to the open SL(2, F)-orbits in ∂TE from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By applying accordingly an element h ∈ SL(2, F) we get that hg−1(ξ0) ∈ { � 1 mα � | m ∈ F∗/(F∗)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Further, by multiplying again accordingly with the element gm from Im we obtain gmhg−1(ξ0) = ξ0, and thus g−1 ∈ SL(2, F)g−1 m B− F , giving g ∈ B− F gm SL(2, F) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This proves part (1) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let us prove part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Take gm ∈ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3 the SL(2, F)-orbit of � 1 mα � = g−1 m �� 0 1 �� is open in ∂TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Because the cone topology on SL(2, E)/B− E ∼= ∂TE is equivalent to the quotient topology on SL(2, E)/B− E induced from the continuous and open canonical projection of SL(2, E) to SL(2, E)/B− E , we have SL(2, F)g−1 m B− E is open in SL(2, E), and by taking the inverse map which is continuous, the set B− E gm SL(2, F) is open in SL(2, E) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' It remains to prove that � gm∈Im U − E TEgm SL(2, F) is dense in SL(2, F), and by part (1) it is enough to show that for any fixed gm ∈ Im, the accumulation points of the open set B− E gm SL(2, F) are exactly the elements of B− E SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Using again the cone topology on SL(2, E)/B− E ∼= ∂TE and the group B from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3 one easily gets part (2) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Part (3) follows easily from part (2), the fact that B− E = U − E TE, and the continuity of the map ψθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ In this section we will consider the set P0 := {x = [x1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' x3, x4] ∈ P(End(E2))} which is open in P(End(E2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then we denote Xθ,0 := P0 ∩ Xθ and this is open in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As before, the set Xθ,0 is called the big cell in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4 it is again advised to study the sets (U − E TEgi) ·θ IdE2, for any gi ∈ I, as well as their closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 6 We have (U − E TE) ·θ IdE2 = ��x 0 a x−1 � θ ��x−1 0 −a x �� | x ∈ E∗, a ∈ E � = �� x 0 a x−1 � � θ(x−1) 0 θ(−a) θ(x) � | x ∈ E∗, a ∈ E � = �� xθ(x−1) 0 aθ(x−1) + x−1θ(−a) x−1θ(x) � | x ∈ E∗, a ∈ E � ⇒ ⇒ [(U − E TE) ·θ IdE2] = {[xθ(x−1), 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' aθ(x−1) + x−1θ(−a), x−1θ(x)] | x ∈ E∗, a ∈ E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (4) For any gm ∈ Im we also have: (U − E TEgm) ·θ IdE2 = �� x 0 a x−1 � � 1 − 1 mα 0 1 � θ �� 1 1 mα 0 1 � � x−1 0 −a x �� | x ∈ E∗, a ∈ E � = �� x − x mα a − a mα + x−1 � � θ(x−1) + θ(a) mα − θ(x) mα −θ(a) θ(x) � | x ∈ E∗, a ∈ E � multiply =======⇒ x−1θ(x−1) ��� 1 − 1 mα a x − a mxα + x−2 � � θ(x−2) + θ(a) mθ(x)α − 1 mα − θ(a) θ(x) 1 �� | x ∈ E∗, a ∈ E � = = [(U − E TE) ·θ IdE2] ⊂ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (5) Then the only accumulation points of the set [(U − E TEgm) ·θ IdE2] are the points [−θ(b), 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' −bθ(b), b] with b ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Those elements appear when taking the sequences {xnk := ω−nk E }k≥0 and {ank := bxnk}k≥0, with 0 < nk → ∞ and b ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For sequences of the form {xnk := ωnk E }k≥0 we obtain no limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As well, notice that the set {[−θ(b), 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' −bθ(b), b] | b ∈ E} is the U − E TE-orbit of the element [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(E2)) with respect to the action ·θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As a result of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4 and the above calculations in (4) and (5) we get: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall the notation Im := {gm | m ∈ E∗/(E∗)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then: Xθ,0 ∩ ψθ(SL(2, E)) = [( � gm∈Im U − E TEgm) ·θ IdE2] Xθ,0 = [( � gm∈Im U − E TEgm) ·θ IdE2] ∩ P0 Xθ = � g∈SL(2,E) g ·θ Xθ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' It is clear from the above calculation that [( � gm∈Im U − E TEgm) ·θ IdE2] = Xθ,0 ∩ ψθ(SL(2, E)) = P0 ∩ ψθ(SL(2, E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since [( � gm∈Im U − E TEgm) ·θ IdE2] = Xθ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4(3), we immediately have Xθ,0 = [( � gm∈Im U − E TEgm) ·θ IdE2] ∩ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let C ∈ Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As C ∈ P(End(E2)) we have C = [x1, x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' x3, x4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If x2 ̸= 0 then C ∈ Xθ,0 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If x1 = 0, then by some easy calculations one can arrange for some g ∈ SL(2, E), such that ρ(g)Cρ(θ(g−1)) ∈ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ 7 We are now ready to prove the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='5, the SL(2, E)-orbits in Xθ with respect to the ·θ action are determined by the SL(2, E)-orbits given by the points of the set Xθ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, the calcula- tions in (5) show the only accumulation points of the sets [(U − E TEgm) ·θ IdE2] in P0 are the points [−θ(b), 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' −bθ(b), b] with b ∈ E, which is the U − E TE-orbit of the element [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(E2)) with respect to the action ·θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In addition, it is very easy to see that the only accumulation point of the set [TE ·θ IdE2] in P0 is just [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Further, by an easy computation � a b c d � � 0 1 0 0 � � θ(d) −θ(b) −θ(c) θ(a) � = � −aθ(c) aθ(a) −cθ(c) cθ(a) � showing that the SL(2, E)-orbit of the point [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] is indeed closed in Xθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As [( � gm∈Im U − E TEgm) ·θ IdE2] is already part of the SL(2, E)-orbit of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 1] = [IdE2] with respect to the ·θ action, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' With respect to the action ·θ, the stabilizer in SL(2, E) of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 1] = [IdE2] is SL(2, F), and the stabilizer in SL(2, E) of the point [0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] is the subgroup { � a−αb z 0 a+αb � | a, b ∈ F with a2 − α2b2 = 1, z ∈ E} ≤ B+ E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The second part just follows from the computation � a b c d � � 0 1 0 0 � � θ(d) −θ(b) −θ(c) θ(a) � = � −aθ(c) aθ(a) −cθ(c) cθ(a) � = � 0 1 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The wonderful compactification of SL(2, F)/Hσ The third wonderful compactification is given by the family of inner involutions σ of SL(2, F) with fixed point groups Hσ = {g ∈ SL(2, F)| σ(g) = g}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In this section we will use the notation from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In the next few paragraphs we summarize the corresponding results from [HW02] for k-involutions of SL(2, k) when k is a field of characteristic not 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let k be the algebraic closure of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall, a mapping φ : SL(2, k) → SL(2, k) is a k-automorphism (or equivalently, an automorphism defined over k) if φ is a bijective rational k-homomorphism whose inverse is also a rational k- homomorphism, [Hel00, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' An abstract automorphism θ of SL(2, k) of order two is called an abstract involution of SL(2, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' A k-involution θ of SL(2, k) is an involution defined over k of SL(2, k), and the restriction of θ to SL(2, k) is a k-involution of SL(2, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Given g ∈ SL(2, k) denote by ιg the inner automorphism of SL(2, k) defined by x �→ ιg(x) := gxg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The classification of the isomorphism classes of k-involutions of a connected reductive algebraic group defined over k is given in [Hel00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' A simple characterization of the isomorphism classes of k-involutions of SL(n, k) is given in [HWD06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We record the classification of k-involutions of SL(2, k): Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' [[HW02] Theorem 1, Corollary 1, Corollary 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Every k-isomorphism class of k- involution of SL(2, k) is of the form ιA with A = ( 0 1 m 0 ) ∈ GL(2, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Two such k-involutions ιA with A ∈ {( 0 1 m 0 ) , � 0 1 m′ 0 � } ⊂ GL(2, k) of SL(2, k) are conjugate if and only if m and m′ are in the same square class of k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In particular, there are order(k∗/(k∗)2) k-isomorphism classes of k-involutions of SL(2, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Given an involution σ of a group G the fixed point group of σ is Hσ := {x ∈ G | σ(x) = x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 8 For σ a k-involution of SL(2, k) the quotient SL(2, k)/Hσ is called a k-symmetric variety, and much of the structure of SL(2, k)/Hσ is determined by Hσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3 ([HW02] Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let σ = ιA, with A = ( 0 1 m 0 ) ∈ GL(2, k), be a k-involution of SL(2, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then Hσ = {( x y my x ) ∈ SL(2, k) | x2 − my2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4 ([HW02] Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let k = Qp, σ = ιA with A = ( 0 1 m 0 ) and m ∈ Q∗ p/(Q∗ p)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then Hσ is anisotropic if and only if ¯m ̸= ¯1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If ¯m = ¯1, then Hσ is isotropic and conjugate to the maximal Qp-split torus of SL(2, Qp), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' the diagonal subgroup of SL(2, Qp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='5 ([BH09] Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let k = Qp, σ = ιA with A = ( 0 1 m 0 ) and m ∈ Q∗ p/(Q∗ p)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then (1) |B+\\ SL(2, Qp)/Hσ| = 2 if m ̸= 1 (2) |B+\\ SL(2, Qp)/Hσ| = 6 if m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The same results as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='5 should hold true for any finite field-extension F of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Below we will give a purely geometric proof of such results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We also record the following geometric interpretation of the fixed point group Hσ, when k = F is a finite field-extension of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' To fix notation, let F be a finite field-extension of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We denote by TF the Bruhat–Tits tree for SL(2, F) whose vertices are equivalence classes of OF-lattices in F2 (for its construction see [Ser12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The tree TF is a regular, infinite tree with valence |kF| + 1 at every vertex, where kF is the residue field of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The boundary at infinity ∂TF of TF is the projective space P 1(F) ∼= F ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, the endpoint ∞ ∈ ∂TF corresponds to the vector � 0 1 � ∈ P 1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The rest of the endpoints ξ ∈ ∂TF correspond to the vectors � 1 x � ∈ P 1(F), where x ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='6 ([CL22] Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let F be a finite field-extension of Qp, A = ( 0 1 m 0 ), with m ∈ F∗/(F∗)2, and σ := ιA the corresponding F-involution of SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Take Km := F(√m) a field extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then the only solutions of the equation A(ξ) = ξ with ξ ∈ P 1Ka are ξ± := � 1 ±√m � and Hσ = FixSL(2,Km)({ξ−, ξ+}) ∩ SL(2, F) = {( x y my x ) ∈ SL(2, F) | x2 − my2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, (1) if m = 1 then ξ± := � 1 ±1 � and Hσ contains all the hyperbolic elements of SL(2, F) with ξ± as their repelling and attracting endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In particular, Hσ is GL(2, F)-conjugate to the entire diagonal subgroup of SL(2, F), thus Hσ is noncompact and abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (2) if m ̸= 1 then ξ± := � 1 ±√m � ∈ P 1Km − P 1F, and Hσ is compact and abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Now, given an involution σ := ιA as in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1, we consider the following action of SL(2, F) on End(F2) given by g ·σ D := ρ(g)Dρ(σ(g−1)), for every g ∈ SL(2, F) and D ∈ End(F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The stabilizer in SL(2, F), with respect to the action ·σ, of the element IdF2 is the subgroup Hσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then SL(2, F) ·σ IdF2 ∼= SL(2, F)/Hσ and we consider the continuous map ψσ : SL(2, F) → P(End(F2)), given by g �→ ψσ(g) := [ρ(g)ρ(σ(g−1))] = [ρ(gσ(g−1))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We want to understand the following closure with respect to the involution σ and the map ψσ: Xσ := ψσ(SL(2, F)) = [SL(2, F) ·σ IdF2] = [SL(2, F)/Hσ] in P(End(F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We regard the above closure Xσ as a compactification of SL(2, F)/Hσ and call it the the wonderful compactification of SL(2, F)/Hσ with respect to the involution σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Just from the definition, we have the following trivial lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The SL(2, F)-orbit of IdF2 in P(End(F2)), with respect to the action ·σ, is dense in Xσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4 we will again need the set P0 := {x = [1, x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' x3, x4] ∈ P(End(F2))} 9 which is open in P(End(F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then denote Xσ,0 := P0 ∩ Xσ and this is open in Xσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The set Xσ,0 is also called the big cell in Xσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Instead of using the density of U −T ×U in SL(2, F) as in Section 2 it will be much more convenient to employ the fact that the double coset B+\\ SL(2, F)/Hσ has a finite number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Below we will explicitly compute those double coset for each m ∈ F∗/(F∗)2, and find the double cosets that are open in SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Along the way we will apply the geometric picture provided by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since the constuction below works by replacing the Borel B+ with any of its SL(2, F)-conjugates, we will use the opposite B−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall that for any locally compact group G and any closed subgroup H ≤ G, the quotient topology on the homogeneous space G/H is defined by the canonical projection p : G → G/H being continuous and open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, one can prove that for any compact subset Q of G/H, there exists a compact subset K of G with p(K) = Q (see [BdlHV08, Appendices B, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The cases m = 1 and m ̸= 1 behave differently, since for the later we will use a quadratic extension of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Case m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall from [CL22] the following trivial lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let K be a finite field-extension of Qp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' There are 6 orbits of the diagonal subgroup Diag(K) := { � d−1 0 0 d � | d ∈ K∗} ≤ SL(2, K) on the boundary ∂TK: (1) the Diag(K)-orbit of � 1 0 � and the Diag(K)-orbit of � 0 1 � (2) the Diag(K)-orbits of � 1 m � , for each m ∈ K∗/(K∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The subgroup Diag(K) fixes pointwise the ends � 1 0 � and � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' The Diag(K)-orbit of � 1 m � , for each m ∈ K∗/(K∗)2, consists of vectors of the form � 1 d2m � , these cover the entire boundary ∂TE − { � 0 1 � , � 1 0 � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ As a trivial consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='9 one obtains: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let F be a finite field-extension of Qp, A = ( 0 1 1 0 ), and σ1 := ιA the corresponding F-involution of SL(2, F) with associated fixed point group Hσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then Hσ1 = � 1 −1 1 1 � Diag(F) � 1 −1 1 1 �−1 , Hσ1 has exaclty 6 orbits on the boundary ∂TF, and the corresponding Hσ1-orbits on the boundary ∂TF are given by the following 6 representatives: { � 1−m 1+m � | m ∈ F∗/(F∗)2} ∪ { � 1 −1 � , � 1 1 � } ⊂ ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let be the matrices in SL(2, F) g1 := � −1 −1 2 1 � , g2 := � −1 −1 1 0 � , gm := � 1+m m−1 m m−1 1 � for m ∈ F∗/(F∗)2 with m ̸= 1 such that g1 �� 1 −1 �� = � 0 1 � , g2 �� 1 1 �� = � 0 1 � , gm �� 1−m 1+m �� = � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let I := {IdF2, g1, g2, gm | m ∈ F∗/(F∗)2, m ̸= 1}, and Im := {gm | m ∈ F∗/(F∗)2, m ̸= 1} ∪ {IdF2}, with the convention that for m = 1 ∈ F∗/(F∗)2 we take gm := IdF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' With the above notation we have that: (1) SL(2, F) = � gi∈I B−giHσ1 (2) for any gm ∈ Im, the set B−gmHσ1 = U −T gmHσ1 is open in SL(2, F), and moreover the set � gm∈Im U −T gmHσ1 is dense in SL(2, F) 10 (3) ψσ1( � gm∈Im U −T gm) is open and dense in ψσ1(SL(2, F)) = [SL(2, F) ·σ1 IdF2], and so ψσ1( � gm∈Im U −T gm) = [SL(2, F) ·σ1 IdF2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let g ∈ SL(2, F) and let ξ0 := � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notice the SL(2, F)-stabilizer of ξ0 is exactly B− = �� x 0 y x−1 � | y ∈ F, x ∈ F∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If g−1(ξ0) = ξ0 then g ∈ B− and thus g ∈ B−Hσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If g−1(ξ0) ̸= ξ0 then there are 6 cases to consider each corresponding to the Hσ1-orbits in ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By applying accordingly an element h ∈ Hσ1 we get that hg−1(ξ0) ∈ { � 1−m 1+m � | m ∈ F∗/(F∗)2} ∪ { � 1 −1 � , � 1 1 � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Further, by multiplying again accordingly with some element gi from I we obtain gihg−1(ξ0) = ξ0, and thus g−1 ∈ Hσ1g−1 i B−, giving g ∈ B−giHσ1 as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This proves part (1) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let us prove part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Take gm ∈ Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='10 the group Hσ1 is conjugate to Diag(F), and since the Diag(F)-orbit of the vector � 1 m � ∈ ∂TF, where m ∈ F∗/(F∗)2, is clearly open with respect to the cone topology on SL(2, F)/B− ∼= ∂TF, then the Hσ1-orbit of � 1−m 1+m � ∈ ∂TF is open in ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Because the cone topology on SL(2, F)/B− ∼= ∂TF is equivalent to the quotient topology on SL(2, F)/B− induced from the continuous and open canonical projection of SL(2, F) to SL(2, F)/B−, we have Hσ1g−1 m B− is open in SL(2, F), and by taking the inverse map which is continuous, the set B−gmHσ1 is open in SL(2, F) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' It remains to prove that � gm∈Im U −T gmHσ1 is dense in SL(2, F), and by part (1) it is enough to show that for any fixed gm ∈ Im, the accumulation points of the open set B−gmHσ1 are exactly the elements of B−g1Hσ1 ∪ B−g2Hσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Indeed, since Hσ1 is conjugate to the diagonal subgroup Diag(F), its attacting and repealing endpoints are { � 1 −1 � , � 1 1 � }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notice that for any fixed gm ∈ Im, the open set Hσ1 � 1−m 1+m � has { � 1 −1 � , � 1 1 � } as its unique accumulation points in ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Using again the cone topology on SL(2, F)/B− ∼= ∂TF, there exist sequences {bn}n≥1 ⊂ B− and {hn}n≥1 ⊂ Hσ1 such that {bngmhn}n≥1 converges to g1, or g2, with respect to the topology on SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then part (2) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Part (3) follows easily from part (2), the fact that B− = U −T , and the continuity of the map ψσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='11 it is advised to study the sets (U −T gi) ·σ1 IdF2, for any gi ∈ I, as well as their closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We have (U −T g1) ·σ1 IdF2 = ��x 0 a x−1 � �−1 −1 2 1 � �0 1 1 0 � � 1 1 −2 −1 � �x−1 0 −a x � �0 1 1 0 � | x ∈ F∗, a ∈ F � = �� x 0 a x−1 � � 1 0 −3 −1 � � 0 x−1 x −a � | x ∈ F∗, a ∈ F � = �� 0 1 −1 2ax−1 − 3x−2 � | x ∈ F∗, a ∈ F � ⇒ ⇒ [(U −T g1) ·σ1 IdF2] = {[0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' −1, 2ax−1 − 3x−2] | x ∈ F∗, a ∈ F} ⊂ [SL(2, F) ·σ1 IdF2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (6) 11 (U −T g2) ·σ1 IdF2 = �� x 0 a x−1 � � −1 −1 1 0 � � 0 1 1 0 � � 0 1 −1 −1 � � x−1 0 −a x � � 0 1 1 0 � | x ∈ F∗, a ∈ F � = ��x 0 a x−1 � � 1 0 −1 −1 � �0 x−1 x −a � | x ∈ F∗, a ∈ F � = �� 0 1 −1 2ax−1 − x−2 � | x ∈ F∗, a ∈ F � ⇒ ⇒ [(U −T g2) ·σ1 IdF2] = {[0, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' −1, ax−1 − x−2] | x ∈ F∗, a ∈ F} ⊂ [SL(2, F) ·σ1 IdF2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (7) Now for every m ∈ F∗/(F∗)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' with m ̸= 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' we have: (U −T gm) ·σ1 IdF2 = ��x 0 a x−1 � �1 + m m − 1 m m−1 1 � �0 1 1 0 � � 1 1 − m m 1−m 1 + m � �x−1 0 −a x � �0 1 1 0 � | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a ∈ F � = �� x 0 a x−1 � � 3m−1 1−m 4m 1−2m (1−m)2 3m−1 m−1 � � 0 x−1 x −a � | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a ∈ F � = �� 4mx2 3m−1 1−m − 4max 4max + 3m−1 m−1 2ax−1 3m−1 1−m + x−2 1−2m (1−m)2 − 4ma2 � | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' a ∈ F � ⇒ ⇒ [(U −T gm) ·σ1 IdF2] = = {[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 3m − 1 (1 − m)4mx2 − ax−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ax−1 + 3m − 1 (m − 1)4mx2 , 2a 4mx2 3m − 1 1 − m + 1 − 2m (1 − m)24mx4 − a2x−2] | x ∈ F∗, a ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (8) Then the only accumulation points of the set [(U −T gm) ·σ1 IdF2] are the points [1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' b, −b2] with b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Those elements appear when taking sequences {xnk = ω−nk F }k≥0 and {ank := bxnk}k≥0, with 0 < nk → ∞ and b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For sequences of the form {xnk = ωnk F }k≥0 we obtain no limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For the case m ∈ F∗/(F∗)2 with m = 1, by our convention above, we take gm := IdF2 and we have: (U −T ) ·σ1 IdF2 = �� x 0 a x−1 � � 0 1 1 0 � � x−1 0 −a x � � 0 1 1 0 � | x ∈ F∗, a ∈ F � = �� 0 x x−1 a � �0 x−1 x −a � | x ∈ F∗, a ∈ F � = �� x2 −xa ax −a2 + x−2 � | x ∈ F∗, a ∈ F � ⇒ ⇒ [(U −T ) ·σ1 IdF2] = {[1, −x−1a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ax−1, −a2x−2 + x−4] | x ∈ F∗, a ∈ F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' (9) Then the only accumulation points of the set [(U −T gm) ·σ1 IdF2] are the points [1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' b, −b2] with b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Those elements appear when taking sequences {xnk = ω−nk F }k≥0 and {ank := bxnk}k≥0, with 0 < nk → ∞ and b ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For sequences of the form {xnk = ωnk F }k≥0 we obtain no limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As well, notice that the set {[1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' b, −b2] | b ∈ F} is the U −T -orbit of the element [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)) with respect to the action ·σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As a result of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='11 and the above calculations in (6) to (9) we have: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall the notation Im := {gm | m ∈ F∗/(F∗)2, m ̸= 1} ∪ {IdF2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then: Xσ1,0 ∩ ψσ1(SL(2, F)) = [( � gm∈Im U −T gm) ·σ1 IdF2] 12 Xσ1,0 = [( � gm∈Im U −T gm) ·σ1 IdF2] ∩ P0 Xσ1 = � g∈SL(2,F) g ·σ1 Xσ1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' It is clear from the above calculation that [( � gm∈Im U −T gm) ·σ1 IdF2] = Xσ1,0 ∩ ψσ1(SL(2, F)) = P0 ∩ ψσ1(SL(2, F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since [( � gm∈Im U −T gm) ·σ1 IdF2] = Xσ1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='11(3), we immediately have Xσ1,0 = [( � gm∈Im U −T gm) ·σ1 IdF2] ∩ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let C ∈ Xσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As C ∈ P(End(F2)) we have C = [x1, x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' x3, x4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If x1 ̸= 0 then C ∈ Xσ1,0 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If x1 = 0, then by some easy calculations one can arrange for some g ∈ SL(2, F), such that ρ(g)Cρ(σ1(g−1)) ∈ P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='12, the SL(2, F)-orbits in Xσ1 with respect to the ·σ1 action are determined by the SL(2, F)-orbits given by the points of the set Xσ1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, the calculations in (8) and (9) show the only accumulation points of the set [(U −T gm) ·σ1 IdF2] in P0 are the points [1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' b, −b2] with b ∈ F, and the only accumulation point of the set [T ·σ1 IdF2] in P0 is [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As well, the set {[1, −b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' b, −b2] | b ∈ F} is the U −T -orbit of the element [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)) with respect to the action ·σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Further, an easy computation �a b c d � �1 0 0 0 � �0 1 1 0 � � d −b −c a � �0 1 1 0 � = �a b c d � �1 0 0 0 � � a −c −b d � = �a2 −ac ca −c2 � shows that the SL(2, F)-orbit of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] is indeed closed in Xσ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As [( � gm∈Im U −T gm) ·σ1 IdF2] is already part of the SL(2, F)-orbit of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 1] = [IdF2] with respect to the ·σ1 action, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' With respect to the action ·σ1, the stabilizer in SL(2, F) of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 1] = [IdF2] is Hσ1, and the stabilizer in SL(2, F) of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] is the subgroup {µ2 · ( 1 b 0 1 ) | b ∈ F} of the Borel B+ ≤ SL(2, F), where µ2 is the group of 2nd roots of unity in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This just follows from the computation �a b c d � �1 0 0 0 � � a −c −b d � = �a2 −ac ca −c2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Case m ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Recall for m ∈ F∗/(F∗)2 with m ̸= 1 we consider A := ( 0 1 m 0 ) which has associated the F-involution of SL(2, F) given by σm := ιA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' From [CL22] also recall Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let F be a finite field extension of Qp, A = ( 0 1 m 0 ), with m ∈ F∗/(F∗)2, m ̸= 1 and σm := ιA the corresponding F-involution of SL(2, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Then Hσm := {g ∈ SL(2, F) | σm(g) = g} has at most 8 orbits on the boundary ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Choose now a set I ⊂ SL(2, F) of representatives such that the Hσm-orbits of g−1 i �� 0 1 �� ∈ ∂TF, for gi ∈ I, are all disjoint and cover the boundary ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='14 we know that I is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' With the above notation we have that: (1) SL(2, F) = � gi∈I B−giHσm (2) for any gi ∈ I, the set B−giHσm = U −T giHσm is open in SL(2, F) (3) ψσm( � gi∈I U −T gi) is open and dense in ψσm(SL(2, F)) = [SL(2, F) ·σm IdF2], and so ψσm( � gi∈I U −T gi) = [SL(2, F) ·σm IdF2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let g ∈ SL(2, F) and let ξ0 := � 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notice the SL(2, F)-stabilizer of ξ0 is exactly B− = �� x 0 y x−1 � | y ∈ F, x ∈ F∗ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If g−1(ξ0) = ξ0 then g ∈ B− and thus g ∈ B−Hσm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' If g−1(ξ0) ̸= ξ0 then there are exactly |I| < ∞ cases to consider, each corresponding to the Hσm-orbits in ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By applying accordingly an element h ∈ Hσm we get that hg−1(ξ0) ∈ {g−1 i (ξ0) | gi ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Further, by multiplying again accordingly with some element gi from I we obtain gihg−1(ξ0) = ξ0, and thus g−1 ∈ Hσmg−1 i B−, giving g ∈ B−giHσm as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This proves part (1) of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Let us prove part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Consider the quadratic field extention E := F(√m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='6(2) we know that the two ends ξ± := � 1 ±√m � are in P 1E−P 1F = ∂TE −∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, the group SL(2, F) acts on TE, it preserves the subset of ends ∂TF, and in fact the subset ∂TF is closed in ∂TE with respect to the cone topology on ∂TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Now, again by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='6 we know that FixSL(2,E)({ξ−, ξ+}) is a conjugate of the diagonal Diag(E) ≤ SL(2, E), and by the same proof as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='11(2), FixSL(2,E)({ξ−, ξ+}) has 4 open orbits on ∂TE − {ξ±}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In particular, this means the open orbits of FixSL(2,E)({ξ−, ξ+}) cover ∂TF, and the intersection of an open orbit of FixSL(2,E)({ξ−, ξ+}) in ∂TE with the closed set ∂TF remains open with respect to the cone topology on ∂TF and TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In addition, by the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='14 from [CL22][ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='3] we know that Hσm = FixSL(2,E)({ξ−, ξ+}) ∩ SL(2, F), has at most two orbits on any of the open orbits of FixSL(2,E)({ξ−, ξ+}) interesecting ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Since Hσm is compact and the action of SL(2, F) on ∂TF is continuous, any Hσm-orbit on ∂TF is open with respect to the cone topololgy on ∂TF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Because the cone topology on SL(2, F)/B− ∼= ∂TF is equivalent to the quotient topology on SL(2, F)/B− induced from the continuous and open canonical projection of SL(2, F) to SL(2, F)/B−, we have Hσmg−1 i B− is open in SL(2, F), and by taking the inverse map which is continuous, the set B−giHσ1 is open in SL(2, F) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Part (3) easily follows from parts (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Take g = �a b c d � ∈ SL(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' F) then we have: [(U −T g) ·σm IdF2] = ��� x 0 y x−1 � � a b c d � � 0 1 m 0 � � d −b −c a � � x−1 0 −y x � � 0 1 m 1 0 �� | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' y ∈ F � = ��� xa xb ya + x−1c yb + x−1d � �−c a md −mb � �0 1 mx x −y m �� | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' y ∈ F � = ��� xa xb ya + c x yb + d x � � xa − c mx − ya m −mxb d x + by �� | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' y ∈ F � multiply ======= x−2 ��� a b ya x + c x2 yb x + d x2 � � a − c mx2 − ya mx −mb d x2 + by x �� | x ∈ F∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' y ∈ F � (10) Then the only accumulation points of the set [(U −T g) ·σm IdF2] ⊂ P0 are the points ��� a b za zb � � a − za m −mb bz �� = [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' − z m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' z, −z2 m ] | z ∈ F � 14 since we always have a2−mb2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Those elements appear when taking sequences {xnk = ω−nk F }k≥0 and {ynk := zxnk}k≥0, with 0 < nk → ∞ and z ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' For sequences of the form {xnk = ωnk F }k≥0 we obtain no limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As well, notice that the set {[1, − z m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' z, − z2 m ] | z ∈ F} is the U −T -orbit of the element [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)) with respect to the action ·σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' We combine Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='15(1) and (2) with the calculation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Therefore, since ψσm( � gi∈I U −T gi) ⊂ P0 is open and equals ψσm(SL(2, F)) = [SL(2, F) ·σm IdF2] we notice that the big cell Xσ,0 := P0 ∩ Xσm in Xσm is exactly the set Xσm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='16, the SL(2, F)-orbits in Xσm with respect to the ·σm action are determined by the SL(2, F)-orbits given by the points of the set Xσm,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Moreover, the calculations in (10) show the only acumulation points of [( � g∈I U −T g) ·σm IdF2] in P0 are the points [1, − z m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' z, − z2 m ], with z ∈ F, which are exactly the U −T -orbit of the element [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] ∈ P(End(F2)) with respect to the action ·σm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' In addition, the only accumulation point of the set [T ·σm IdF2] in P0 is [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Further, by an easy computation � a b c d � � 1 0 0 0 � � 0 1 m 0 � � d −b −c a � � 0 1 m 1 0 � = � a b c d � � 1 0 0 0 � � a −c m −bm d � = �a2 −ac m ca −c2 m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' showing that the SL(2, F)-orbit of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] is indeed closed in Xσm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' As [( � gi∈I U −T gi)·σm IdF2] is already the SL(2, F)-orbit of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 1] = [IdF2] with respect to the ·σm action, the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' With respect to the action ·σm, the stabilizer in SL(2, F) of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 1] = [IdF2] is Hσm, and the stabilizer in SL(2, F) of the point [1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 0, 0] is the subgroup {µ2 ·( 1 b 0 1 ) | b ∈ F} of the Borel B+ ≤ SL(2, F), where µ2 is the group of 2nd roots of unity in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' This just follows from the computation �a b c d � �1 0 0 0 � � 0 1 m 0 � � d −b −c a � � 0 1 m 1 0 � = �a b c d � �1 0 0 0 � � a −c m −bm d � = �a2 −ac m ca −c2 m � by taking �a2 −ac m ca −c2 m � = � 1 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' □ References [AB18] Ana B˘alibanu, Part II: The wonderful compactification, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lectures notes https://people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='edu/~ana/part2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ↑2 [BdlHV08] Bachir Bekka, Pierre de la Harpe, and Alain Valette, Kazhdan’s property (T), New Mathematical Mono- graphs, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 11, Cambridge University Press, Cambridge, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' MR2415834 ↑10 [BH09] Stacy L Beun and Aloysius G Helminck, On the classification of orbits of symmetric subgroups acting on flag varieties of SL(2, k), Communications in Algebra 37 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 4, 1334–1352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ↑1, 9 [CL22] Corina Ciobotaru and Arielle Leitner, Chabauty Limits of Groups of Involutions In SL(2, F ) for local fields, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='12247v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ↑1, 3, 9, 10, 13, 14 [DCP83] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' De Concini and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Procesi, Complete symmetric varieties, In: Invariant Theory, Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Notes in Math.' metadata={'source': 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+page_content=' Helminck, Ling Wu, and Christopher E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Dometrius, Involutions of SL(n, k), (n > 2), Acta Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 90 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 1-2, 91–119, DOI 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1007/s10440-006-9032-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' MR2242950 ↑8 [HW02] Aloysius G Helminck and Ling Wu, Classification of involutions of SL(2, k), Communications in Algebra 30 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 1, 193–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ↑8, 9 [SJ98] Paul J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Sally Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=', An Introduction to p-adic Fields, Harmonic Analysis and the Representation Theory of SL2, Letters in Mathematical Physics 46 (1998), 1–47, DOI doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1023/A:1007583108067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ↑1 [Ser12] Jean-Pierre Serre, A course in arithmetic, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 7, Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' ↑1, 2, 9 [Ste74] Robert Steinberg, Abstract homomorphisms of simple algebraic groups (after A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Borel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Tits), S´eminaire Bourbaki, 25`eme ann´ee (1972/1973), Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 435, Springer, Berlin, 1974, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' 307–326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=', Vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1515/crll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content='196 (French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} +page_content=' MR277536 ↑2, 3 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAyT4oBgHgl3EQfkPi4/content/2301.00431v1.pdf'} diff --git a/itE1T4oBgHgl3EQffwSj/content/tmp_files/2301.03222v1.pdf.txt b/itE1T4oBgHgl3EQffwSj/content/tmp_files/2301.03222v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..156285ee50bd2e456fecf7a87fa4ae46b76d0b10 --- /dev/null +++ b/itE1T4oBgHgl3EQffwSj/content/tmp_files/2301.03222v1.pdf.txt @@ -0,0 +1,745 @@ +1 | P a g e + + +Machine Learning Algorithms for Depression Detection and Their Comparison. + +Danish Muzafar1, Furqan Yaqub Khan2, Mubashir Qayoom3 +1Department of Information Technology, Central University of Kashmir, J&K, India +2Department of CSE, IIT Patna +3Department of Physics, University of Kashmir, J&K, India +Email: furkaan309@gmail.com + +Abstract –Textual emotional intelligence is playing a ubiquitously important role in +leveraging human emotions on social media platforms. Social media platforms are privileged +with emotional contents and are leveraged for various purposes like opinion mining, emotion +mining, and sentiment analysis. This data analysis is also levered for prevention of online +bullying, suicide prevention and depression detection among social media users. In this +article we have designed an automatic depression detection of online social media users by +analyzing their social media behavior. The designed depression detection classification can +be effectively used to mine user’s social media interactions and one can determine whether +a social media user is suffering from depression or not. The underlying classifier is made +using state-of-art technology in emotional artificial intelligence which includes LSTM (Long +Short Term Memory) and other machine learning classifiers. The highest accuracy of the +classifier is around 70% of LSTM and for SVM the highest accuracy is 81.79%. We trained +the classifier on the datasets that are widely used in literature for the emotion mining tasks. +Confusion matrix of results is also given. + +Keyword---Emotion Artificial Intelligence, Deep Learning, Machine Learning Algorithms, +Word Embedding’s, Twitter. + +I. INTRODUCTION +Depression triggers sorrow, frustration, lack of interest, annoyance. It will contribute to +variety of emotional and physical problems and might cut back the power of a person to +operate at full potential [1]. The matter of mental state in humans is incredibly serious. At +its worst, depression is the main cause of death for 80000 human beings leading them to +suicide. Teenagers are most prone to depression as a result of they often need Psychotherapy, + +2 | P a g e + +Teenagers limitlessly use mobile phones and social media and spend less time in real world +and more in virtuality which causes a gap between their internal state of mind and outward +world, causing a strong sense of loneliness which in-turn leads to depression[2]. Bullying +via social media based on factors like social status, standard of language, race, ethnicity, +immigration standards, health, behavioral ethics, etc. are the main drawbacks of social media +which cause depression to user.[4] The definition of problem dealt in this research article is +to find whether the person under consideration does have symptoms of depression or not. +With the use of neural networks we have used users tweets to diagnose whether the person +under consideration is behaving positively or negatively. If the person is shows particular +standard of positivity in his tweets then we can predict him or her to be depression free, +otherwise he or she is suffering from depression. The standard used in this article for a person +to be considered under depression is that if 80% of his tweets are classified negative. We do +the analysis of user feeds posted on a twitter account via state-of-art machine learning and +deep learning techniques based on scientifically accepted 80% criterion.[6][7]. +1.1. Some Initial Symptoms of depression are listed below: +o Feeling crazy, sad or in a depressing mood +o Weight loss +o Increase fatigue and sleep difficulties. +o Loss of energy and increased fatigue with urge to suicide. +o Thinking very difficultly, concentrating, or making decisions based on the +assumption based on the idea of not liking anything. +o Frequent and continuous of death or suicide +Symptoms must last at least two weeks or longer than that. Depression may have a different +effect on the sexes and shows that men with depression can have signs such as irritability, +escape or dangerous behavior, or that everyone is really angry. Symptoms must last at least +two weeks or even more for a diagnosis of depression [1] +1.2. Risk Factors for depression- Depression can affect anyone, even a person who seems to +be living in reasonably ideal conditions. +Depression can be caused by a variety of circumstances: +o Biochemistry: Harmonal differences may contribute to symptoms of depression in +certain cases due to chemicals imbalance in the brain. + +3 | P a g e + +o Genetics: Genetics also plays an essential role in co-depression for e.g. if one +identical twin has depression, for instance, the other has a 70 percent risk of +developing the disorder sometime in life. +o Personality: People with low self-confidence are more likely to experience +depression, causing them to be speechless by stress, or who are generally pessimistic. +o Environmental factors: Continuous exposure to violence, negligence, abuse or +poverty can make some individuals more susceptible to depression. +1.3. Role of Social Media and Depression +Social networks are becoming an important part of the lives of individuals, they mirror the +personal life of the user on social media, people like to share happiness, joy and sorrow [4]. +Researchers are using these platforms to classify and detect the causes of depression, reading +an old article on the News website about how Twitter can tell if you're depressed and the +possibility of developing an artificial intelligence model that can scan your Twitter feed and +tell you if an individual is at risk of depression, or accepting notifications from third parties, +such as those advising you to seek help based on an automatic scan of your tweets [1]. The +day has come to an end. Early detection of depression can be a big step toward addressing +the mental illness and providing help to those who are suffering from it. Various approaches +for sentiment analysis in Machine Learning, including decision-based systems, Bayesian +classifiers, support vector machines, neural networks, and sample-based methods can be +used to detect symptoms of depression before it is too late. +II. Depression and Emotion Intelligence +If a person is attached with someone closely then it is clear the person is emotionally attached +to that person and if this emotional dependence is too high that the decisions of attached +persons are dependent on each other this is called emotion bonding. This is when these +persons take emotions seriously sometimes their emotions are not positive in nature hence +leading to depression in one or all of them. +2.1. Emotional intelligence is commonly defined by four attributes: +a. Self-management – You can control impulsive emotions and actions, handle your +emotions in healthy ways, take action, meet commitments, and respond to changing +conditions. + +4 | P a g e + +b. Self-awareness – You recognize your own feelings and how your thoughts and +conduct are affected by surrounding conditions. You are aware of your strengths and +weaknesses, and have faith and confidence in ones self. +c. Social awareness – You've got empathy. You should understand other people's +thoughts, desires, and concerns, pick up on emotional signals, socially feel relaxed, +and know the dynamics of power in a group or organization. +d. Relationship management – You know how to build and maintain good +relationships, clearly communicate, inspire and influence others, work well as a +team, and resolve conflicts. +III. DISCUSSION +3.1. Dataset +We have developed a depression detection dataset by manually annotating the twitter data. +We used two classes for this research one is depressive and other non-depressive. The +Tweets were downloads from the twitter Platform and some from the kaggle data and initial +dataset had 25,000 tweets from which foreign language tweets and re-tweets were removed. +This dataset contains three columns: First Column contains tweet ID, second column +contains text and third column contain labels these columns are features. The dataset was +then annotated by three annotators and majority voting was kept. Thus if a tweet was +annotated depressive by more than one annotator only then it was given a depressive label +likewise was done for non-depressive tweets. In the final dataset we had 7,500 depressive +tweets and 7,500 non-depressive tweets. +Different algorithms are used we discuss one by one: +3.2. Support Vector Machine (SVM) +Support Vector Machine(SVM) is a supervised algorithm for machine learning that can be +used for problems with classification and regression. In classification issues, however, it is +mostly used [1]. In the SVM algorithm, we depict each data item as a point in n-dimensional +space (where n is the number of characteristics you have), with the value of each + +5 | P a g e + +characteristic being the coordinate value. Then, we carry out classification by determining +the hyper-plane that clearly distinguishes the two classes [12]. The Support Vector Machine +is a discriminative classifier that is formally described by a distinct hyperplane (SVM). In +other words, given labelled training data, the algorithm generates an ideal hyperplane that +categorizes fresh examples (supervised learning). This hyperplane is a line in two- +dimensional space that divides a plane into two sections, with each class on either side. The +learning of the hyperplane in linear SVM is finished by changing the issue using some linear +algebra [10]. This is when the kernel comes into play. In our experiment we gave the linear +SVM using train test split of 80-20%. We employed scikit-learn kernels for the classification +purpose. +3.3. Random Forest Classifier +The Random Forest Classifier an ensemble algorithm, produces a set of decision trees from +the training set's randomly selected subset. To evaluate the final class of the test object, it +then aggregates the votes from various decision trees. Different decision trees can create +subsets that overlapping in nature. To deal with this weighting principle is used to consider +the impact of any decision tree's result. Trees with a high mistake rate are given a low weight +value, whereas trees with a low mistake rate are given a high weight value. Low error-rate +trees would have a greater impact on decision-making as a result of this. The total number +of trees to be constructed, are as well the important characteristics for the decision tree. The +minimum split, split criteria, etc., are all basic Random Forest Classifier parameters. We +used Random Forest classifier with 40 estimators. The training and testing sets as inputs to +classifier were divided into 80-20 train test splits of total data entries. We employed scikit- +learn kernels for the classification purpose as well. +3.4. Multinomial Naïve Bayes +Naive Bayes is a family of algorithms based on applying the Bayes theorem with the strong +(naive) assumption that each feature is independent of the others in order to predict the +category of a given sample. Because they are probabilistic classifiers, the Bayes theorem +will be used to compute the likelihood of each category, and the category with the highest +probability will be created. Naive Bayes classifiers have been used successfully in a variety +of fields, including Natural Language Processing (NLP).Other options, such as Support +Vector Machines (SVM) and neural networks, are available when dealing with NLP issues. + +6 | P a g e + +Their simplicity of Naive Bayes classifiers, on the other hand, makes them particularly +desirable for such classifiers. Furthermore, they have been proved to be fast, reliable, and +exact in a variety of NLP applications. We use a non-naive Bayes approach to look at +sentences in their whole, thus if a sentence does not appear in the training set, we will get a +zero probability, making further computations impossible. However, in the case of Naive +Bayes, each word is assumed to be independent of the others [10]. We now examine +individual words as a phrase rather than the complete sentence. +3.5 Long Short Term Memory (LSTM) +LSTMs have proven to be effective in tasks involving sequential knowledge. The input +vectors used for the other classifiers, such as the TF-IDF weights vector, do not, however, +maintain any information on the sequential relationship between each document's terms and +phrases. Thus, the more appropriate input for LSTMs are sequences of word embedding +vectors; in fact, word embedding are the preferred option for textual representation in +modern deep learning. +We tuned the number of memory units, number of epochs, batch size, and input and recurring +dropout rates of LSTM by using the training and tuning set. The number of epochs is the +number of times in the neural network the entire training set is moved forward and backward +[12]. The network can be fit and the resulting neuron weights may be far from optimal if the +number of epochs is too low. Before each change to the weights, the batch size is simply the +number of samples fed into the network and helps to make training more effective by +reducing the memory requirements and the number of iterations needed to achieve the +optimum weights. Finally, in an attempt to minimize overfitting, the dropout rate is simply +the likelihood that an input or recurrent node will be omitted from consideration during a +weight update. Input to the classifier in the 80-20 train test format was given to the training +and testing set respectively. + + +7 | P a g e + + +Fig. 1: Mechanism of LSTM Classifier +A cell, an input gate, an output gate, and a forget gate make up a standard LSTM unit. The +cell recalls values across arbitrary time intervals, and the three gates govern the flow of +information into and out of the cell. Because there might be lags of undetermined duration +between critical occurrences in a time series, LSTM networks are well suited to categorizing, +processing, and making predictions based on time series data. LSTMs were created to deal +with the vanishing gradient problem that might occur when training regular RNNs. The +relative insensitivity to gap length of LSTM over RNNs, hidden Markov models, and other +sequence learning approaches provides a benefit a variety of applications. +A. LSTM architecture: Multiple architectures of LSTM units exist. The standard +architecture consists of a cell (the LSTM unit's memory portion) and three information flow +regulators inside the LSTM unit, typically referred to as gates: an input gate, an output gate, +and a forgotten gate. Any of the versions of the LSTM device may not have one or both of +these gates, or even other gates. Gated Recurrent Units (GRUs) do not, for example, have an +output gate. The input gate determines how much a new value enters the cell, the forgotten +gate determines how much the cell value remains, and the output gate determines how much +the cell value is used to generate the LSTM unit's output activation. The activation function +of the LSTM gates is also the logistic sigmoid function. + +Fig. 2: Process of LSTM +B. Word Embedding +A word embedding is a group of ways for representing words and texts that use a dense +vector representation. Instead, in an embedding, words are represented by dense vectors, + +Input +Feed1-hotvectorsonaString +SequenceLength:15 +Embed +0 +Reduce Order +S66 +sa +LSTM +Featur +0 +Do Magic +Output +0 +Predict +H +0 +877 +sa +pappin +reversed +0 +S +unordered +0 +0 +966 +H +08 | P a g e + +each of which reflects the word's projection into a continuous vector space [2]. The +placement of a word is based on the words that surround it when it is used and is learned +from text inside the vector space. The embedding of a word is its position in the learned +vector space. Two common examples of learning methods for embedding words from text +include: +• +Word2Vec. +• +GloVe. +In this Research article Word2Vec and Doc2Vec are used. + +C. Keras Embedding layer +Keras provides an embedding layer which can be used on text data for neural networks. This +includes integer encoding of the input data, so that each word is represented by a unique +integer. The Tokenizer API that is also supplied with Keras can be used to perform this data +preparation stage. All of the words in the training dataset will learn an embedding since the +Embedding layer is initialized with random weights. It's a versatile layer that can be applied +in a variety of ways, like it can be used on its own to teach a word integration that can then +be stored and reused in a different model. It can be utilized as part of a deep learning model +where the embedding is taught alongside the model. It is a type of transfer learning, and can +be used to load a pre-trained word embedding model. Weights are learned in the Embedding +layer. If you save your model as a file, the weights for the Embedding layer will be included. +An embedding layer's output is a 2D vector in the input word sequence (input document), +with one embedding for each word. You must first flatten the 2D output matrix to a 1D +vector if you want to link a dense layer directly to an embedded layer using the Flatten layer. +Finally we train the LSTM Classifier for training and testing data classification. + +IV. METHOLODOLOGY +In this section we discuss the overall methodology that is used for depression detection using +online social profiles of the user. We employ emotional artificial intelligence for the +depression detection task. We create a depression classifier using Deep-Learning framework +and machine learning and use that classifier for classifying tweets from an online social + +9 | P a g e + +profile for automatic depression detection. Figure 5 shows the overall methodology used for +the task. + +Fig. 3: Overall methodology of the depression detection system. +The system receives as input an emotion labeled dataset which is used for the training of +supervised deep-learning classifier or machine learning classifier. Once the classifier is +trained on emotion labeled dataset we use the tweets from online social profiles for the +depression detection. If the tweets from the profile contain more than 75% tweets classified +as depressive, our system classifies gives the red flag for the profile that it may be having +depression symptoms. Figure 6 shows the components of the system in detail. + +Fig. 4: Architecture of the system +The system has 5 modules and uses emotion labeled dataset. +4.1. Data Normalization + +TESTDATASET +DATASETWISENTIMENT +X +Downioad +DataFile +Feeddatato +TwitterAP! +Saveastxt +Tweet +Training& +Downloader +Testing +PREDICTANDSHOWACCURACT +NaiveBayesian +DTree +SVM +ForTesting +Ra +fomForestData +Data +Dataset +Normalization +Preprocessing +Supervised +YES +Classification +Deep-Learning +>75% +Classifier +Depressive +NO +OnlineSocial +Profile +Non-Depressive10 | P a g e + +As part of data preparation for machine learning, standardization is a technique sometimes +used. The aim of adjustment is to adjust the numeric column values in the dataset to use a +common scale, without distorting disparities in the value ranges or losing information. You +also deleted all retweets, all non-alphanumeric characters, URLs, and @mentions. Besides, +all stop words were abolished, with the exception of first, second and third person pronouns. +About 40% work done by this through NLTK automatically and all libraries. +4.2. Data Preprocessing +Preprocessing each tweet before passing it to the automated classifier is how the +preprocessing module prepares it for the classifier. Slangs and abbreviations are eliminated +by the use of English directories from the tweets. If a term is identified that does not have a +meaningful definition, all of the words are passed into the dictionary module to be looked +up, and then into the word replacer module to be replaced with the right word. In the word +replacer module, we employed the SMS dictionary, Netlingo, and the urban dictionary. +Preprocessing stages are made up of the following steps: +4.3. Tokenization +Is performed using Stanford Core NLP package to break tweets into sentences and words, +such as units of words, sentences, or themes? The first column of the csv file containing the +tweet is extracted and converted to individual tokens in this case. +4.4. Stemming +Stemming is the process of reducing words to their simplest form. This would allow us to +group terms that are similar together. Poster Stemmer is utilized for implementation. +4.5. Stop Words Removal +Because they are of little use in training, the widely used words known as stop words need +to be omitted and may also lead to incorrect results if not ignored. There is a collection of +stop words in the NLTK library which can be used as a guide for removing stop words from +the dataset. Stop words that are deleted from tweets using the Tf-Idf, Stanford, and wiki +features. +4.6. POS Tagger + +11 | P a g e + +The tokenized text is allocated to the respective sections of the speech by using a POS tagger +to improve the quality of the remaining data. Since other parts of speech are not of much +importance, this can be used to remove only adjectives, nouns and adverbs. Example: 'I +LOVE CODING' is extracted-'I 'is a pronoun, rest is omitted. +4.7. Lemmatization +It's used to apply stem words to their stems. Lemmatization refers to performing things +correctly utilizing a vocabulary and morphological word analysis, mainly aimed at merely +removing inflectional endings and restoring the root or dictionary form of a word known as +the lemma, utilizing the Stanford core NLP kit. +A bag of words is generated after all of these pre-processing stages. The number of +occurrences of each word is calculated in a bag of words, which is subsequently utilized as +a feature to train a classifier. +4.8. Bag of words or One Hot Encoding +Each part of the vector corresponds to one word or n-gram (token) within the corpus +vocabulary during this methodology. Then if the token at a selected index exists within the +document, that part is marked as one, else it’s marked as zero. + +Fig.5: BOW Representation + +In the above example, our corpus consists of every unique word across the three documents. +A bag of Words model, or BOW for short, is a method of extracting text properties for use in +modeling, such as machine learning techniques. The method is straightforward and adaptable, + +Batscanseevia +Theelephantsneezed +Wondering,sheopened +echolocation.Seethe +atthesightofpotatoes. +thedoortothestudio. +batsightsneeze!12 | P a g e + +and it may be used to extract information from documents in a variety of ways. A bag of +words is a text representation of the words that appear within a document. +It involves two ways: +1. A vocabulary of known words. +2. The measure of the presence of known words should be considered. +It's dubbed a bag of words because all information in the text about the sequence or +arrangement of words is discarded. The model simply considers where recognized phrases +appear in the document, not where they exist in the document. The histogram of the words +inside the text is examined in this method. The bag of words approach (BOW), which +considers each word count as a function, is a popular role extraction method for sentences +and texts. The assumption is that documents are similar if they have comparable content. +Furthermore, we can deduce something about the document's significance solely from its +content. You can make the word bag as simple or as complex as you desire. The difficulty +arises from deciding how to create the lexicon of knows words (or tokens) as well as how to +rate the presence of such words. +4.9 Term Frequency-Inverse Document Frequency +TF-IDF stands for term frequency-inverse document frequency. It's a statistical metric for +determining how important a word in a corpus of documents is to a text. This importance is +related to the number of times a word appears in the text, but is offset by the number of +documents in the corpus that contain that word. +4.10 Count Vectorization +The number of occurrences in a document for each word (i.e. independent text, such as an +article, book, or simply a paragraph!) can be tallied using Count Vectorization. The Sci-kit +learning library in Python offers a tool called CountVectorizer that can help with this. "The +weather was nice today, so I went outdoors to enjoy the gorgeous and sunny weather," for +example. The words "the," "weather," and "and" all appeared twice in the performance below, +whereas other words only appeared once. Count Vectorization achieves this goal. +4.11 Distributional Similarity Based Representations- Word Embedding’s + +13 | P a g e + +A word embedding may be a sort of learnt text illustration (real valued vectors) during which +words with connected meanings square measure drawn equally, like the favored "King-Man ++ girl = Queen" example. For every word, the idea of employing a dense, distributed +illustration is central to the approach. Every term may be a real-value vector with tens or +many dimensions. In distinction, distributed word representations, like one-hot encryption, +need thousands or countless dimensions. The 2 most well liked word embedding’s square +measure Word2Vec and Glove: +4.12. Word2Vec: +There square measure primarily 2 versions of Word2Vec — Skip-Gram and Continuous Bag +of Words (CBOW). The embedding is learned by the CBOW model anticipating the present +word supported its context (surrounding words). Given a current word, the Skip-Gram model +learns by predicting the encompassing words (context). Word2vec represents every separate +word with a selected list of numbers referred to as a vector, because the name suggests. The +vectors square measure a basic function (the similarity of the trigonometric function between +the vectors) is employed. Indicates the amount of linguistics similarity between the words +drawn by those vectors. Skip-gram seeks to predict a given word's immediate neighbors. We +tend to take a central word and a context window (neighbors) terms, and take a few window +step sizes round the central word, we tend to attempt to predict the context words. So, our +model can describe a distribution of chance, i.e. the chance of a word occurring in context +given a central word, and to maximize the chance, we'll select our vector representations. +We tend to take away the output layer and use the hidden layer to urge our word vectors till +we are able to predict the encompassing words with an affordable degree of exactitude. We +tend to begin with little random data formatting of vectors of terms. By minimizing the loss +perform, our prognosticative model learns the vectors. In Word2vec, this happens with a +feed-forward neural network with a language modeling task (predict next word) and +improvement techniques like random gradient descent. This measures the options that I +actually have used for machine learning algorithms like SVM, Naïve Thomas Bayes, and +Random Forest. + +14 | P a g e + + +Fig. 6: Word2Vec CBOW vs Skip-gram +4.13. Gensim Model for DOC2VEC used in LSTM +Gensim is paid as a kit of natural language processing that 'Topic Modeling for +Humans' does. But it's even more than that, technically. I used Doc2Vec for LSTM, +meaning the model of a deep learning algorithm, so clarify Doc2Vec first. I have +already mentioned the Word2Vec I used in machine learning algorithms since I used +algorithms for machine learning as well as deep learning algorithms. So, for machine +learning, I used Word2Vec and I used Doc2Vec for the LSTM model. The aim of +doc2vec, regardless of its length, is to construct a numeric representation of a text, as +mentioned. But records do not come in logical forms like words and unlike words, so another +technique needs to be found. Because the Doc2vec model is an unsupervised process, it +should be slightly adjusted to "participate" in this challenge. Fortunately, as in most +circumstances, we can utilize a couple tricks: remember how we introduced another +document vector in fig 3 that was unique to each document? If you think about it, you can +add more vectors that don't have to be unique: for example, if we have tags for our documents +(which we have), we can add them and receive their representation as vectors. +Additionally, they don’t have to be unique. This way, we can add to the unique document tag +one of our 17 tags, and create a doc2vec representation for them as well! See below: + + +INPUT +PROJECTION +OUTPUT +INPUT +PROJECTION +OUTPUT +w(t-2) +w(t-2) +w(t-1) +w(t-1) +SUM +w(t) +w(t) +w(t+1) +w(t+1) +w(t+2) +w(t+2) +CBOW +Skip-gram15 | P a g e + + +Figure. 8: Doc2Vec model with tag Vector +4.14. Training +From the training set both label and tweet are required by the classifier. The training set in +this example refers to the set of tweets that must be processed before being fed into a +classifier. The set of tweets must be transformed to a vector representation for further +processing. The set of labels corresponding to each tweet is also provided into the classifier +as a vector. +4.15. Saving the classifier and the count vectorizer object +Since training needs to be done once, it is necessary to load the trained classifier object into +a pickle file. Same is applicable with the count Vectorizer Object. It count the number of +words present in the pickle file using the TF-IDF method. And then the testing phase takes +place for unseen data. +4.16. Testing +Loading of saved models: Qualified classification models are loaded from the pickle file to +be used for test dataset prediction (Individual tweet profiling data). The test dataset is +preprocessed in a way that is similar to the data from the training for test tweets class +prediction. Each tweet is graded into a class that is depressed or non-depressed. We calculate +the confusion matrix for the assessment of classification results based on the values of true +or false positives and negatives. Confusion Matrix is used for how much our classifier shows +accurate results as well as false results. +V. RESULTS +The depression detection system that we designed uses emotional artificial intelligence for +the detection of depression using online social profiles. We used supervised learning models + +Classifier +on +Average/Concatenate +冒 +冒 +T ++ +T +D +w +tag +document +the +cat +sat +vector +vector16 | P a g e + +and Deep Learning algorithm i.e Long Short Term Memory (LSTM) were training of the +classifier was given using the emotion dataset. The dataset has 15,000 tweets manually +annotated by us. The highest accuracy of the classifier is around 70% for LSTM and for +SVM the highest accuracy is 81.79%. I have trained the classifier on the datasets that are +widely used in literature for the emotion mining task. The results are presented using the +classification metrics like accuracy and confusion matrix. +Confusion Matrix: A confusion matrix is a machine learning classification performance +evaluation methodology. It's a type of table that allows you to see how well a classification +model performs on a set of test data for which the true values are known. True Positive (TP), +True Negative (TN), False Positive (FP), False Negative (FN) these given four parameters +of the confusion matrix are used to calculate the Accuracy, Recall, Precision and FI-Score, +with the help of these four classes we compute all values. The results are evaluated on the +basis of F1 score and accuracy. The F1 score is the primary performance measure and +accuracy is the secondary measure with the help of confusion matrix we call Precision, +Recall, F1 Score and Accuracy it simply shows how much our classifier shows accurate +results or with the help of given formulas: + +A. Precision: TP/TP+FP, where TP is true positive, FP is false positive. +B. Recall: TP/TP+FN. +C.FI Score: 2* (Precision*Recall)/ Precision + Recall +The dataset was balanced having 7,500 tweets in each data class that is depressive and non- +depressive. SVM classifier its accuracy is more than other classifiers. We choose SVM +classifier for unseen data. We trained our system using LSTM (Deep Learning Classifier), +SVM, Naïve Bayes and Random Forest classifier (Machine Learning Classifier). The +classifiers accuracy is described in the below table: +Classifier +Precision +Recall +F1 Score +Accuracy +LSTM +0.88 +0.60 +0.71 +70.00% +Multinomial NB +0.79 +0.78 +0.78 +79.32% +SVM +0.79 +0.82 +0.79 +81.79% +Random Forest +0.75 +0.82 +0.78 +80.37% + + +17 | P a g e + +Table 1 Results of different Classifier performance +Figure 9, 10, 11 and 12 show the performance of the system using confusion matrices: + + + +Fig. 9: Confusion Matrix of SVM Fig.10: Confusion Matrix of LSTM + + +Fig.10: Multinomial Naïve Bayes Fig. 12: Random Forest + +V1. CONCLUSION AND FUTURE WORK +The analytics performed on the chosen dataset offer some insight into the analysis issues. +The subsequent may be an outline of our findings: What is depression, and what area unit +the foremost current causes of depression? whereas we have a tendency to all feel irritable, +unhappy, or low from time to time, some individuals experience these feelings on an + +Confusion Matrix +1200 +0 +12B0 +323 +1000 +Tue label +800 +600 +1 +254 +1361 +400 +1 +Predicted labelConfusion Matrix +1200 +0 +1368 +182 +1000 +True label +800 +600 +1 +880 +738 +400 +-200 +0 +1 +Predicted labelConfusionMatrix +1200 +1231 +322 +1000 +Tuelabel +800 +1 +EEE +1282 +600 +400 +0 +1 +PredictedlabelConfusion Matrix +1200 +0 +374 +1000 +True label +800 +600 +1 +248 +1367 +400 +0 +1 +Predicted label18 | P a g e + +everyday basis, over long periods of your time (weeks, months, or perhaps years), and in +some cases for no obvious cause. Despondence is over simply a nasty mood; it is a state of +affairs that has a bearing on an individuals physical and mental well-being. Any one can be +suffering from depression at any time. However, bound stages or circumstances in modern +day world expose people to anxiety and depression. During entire life the changes around or +inside an individual all end in a surge of emotions which will contribute to unhappiness in +introvert and socially separated people. We conducted a comparison of state-of-art deep +learning models to pre-detect depression from tweets at the user level. We ran our models +on a manually preprocessed dataset and discovered that SVM created higher results. In the +future, we can use a special methodology to extract paraphrases from a wider set of +emotional qualities. We will additionally check the quality and potency of our models with +lot more datasets. + REFERENCES +[1] Mander Deshpande and Vignesh Rao, “Depression Detection using Emotion Artificial +Intelligence”, IEEE, 2017. +[2] Marcel Trotzek, Sven Koitka and Christoph M. Friedrich,”Utilizing Neural Networks +and Linguistic Metadata for Early Detection of Depression Indications in Text Sequences”, +IEEE, 2018. +[3] Adyan Marendra Ramadhani and Hong Soon Goo, “Twitter Sentiment Analysis using +Deep Learning Methods”, 7th International Annual Engineering Seminar (InAES), +Yogyakarta Indonesia, 2017. +[4] Wei Tong Mok, Rachael Sing, Xiuting Jiang and Swee See, “Investigation of Social +Media on Depression, IEEE, 2014. +[5] Niko Colneric and Janez Demsar, “Emotion Recognition on Twitter: Comparative Study +and Training a Unison Model”, IEEE, 2018. +[6] Mohammad Jabreel and Antonio Moreno, “A deep Learning-Based Approach for Multi- +label Emotion Classification in Tweets”, applied science, MDPI, 2019. +[7] Hakak Nida, Kirmani Mahira, Mohd. Mudasir, Muttoo Mudasir Ahmad and Mohd. +Mohsin, “Automatic Emotion Classifier”, Springer, 2019. + +19 | P a g e + +[8] Krishna Shrestha, “Machine Learning for Depression Diagnosis using Twitter data”, +International Journal of Computer Engineering in Research Trends using Twitter data +(IJCERT), 2018. +[9] Nida Manzoor Hakak, Mohsin Mohd, Mahira Kirmani and Mudasir Mohd, “Emotion +Analysis: A survey”, IEEE, 2017. +[10] Hemanthkumar M, Latha, “Depression Detection with Sentiment Analysis of Tweets”, +International Research Journal of Engineering and Technology (IRJET), 2019. +[11] Md. Rafiqul Islam, Muhammad Ashad Kabir, Ashir Ahmed, Abu Raihan M. Kamal, +Hua Wang and Anwaar Ulhaq, ”Depression detection from social network data using +machine learning techniques”, Springer, 2018. +[12] Aswathy K S, Rafeeque P C and Reena Murali, “Deep Learning Approach for the +Detection of Depression in Twitter”, International conference on systems energy and +environment, 2019. + + + + + + + + + + + + + + + +20 | P a g e + + + + + + + + diff --git a/itE1T4oBgHgl3EQffwSj/content/tmp_files/load_file.txt b/itE1T4oBgHgl3EQffwSj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4259d0572c5f96e30e0cf4213b3109e6ddfcf20f --- /dev/null +++ b/itE1T4oBgHgl3EQffwSj/content/tmp_files/load_file.txt @@ -0,0 +1,388 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf,len=387 +page_content='1 | P a g e Machine Learning Algorithms for Depression Detection and Their Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Danish Muzafar1, Furqan Yaqub Khan2, Mubashir Qayoom3 1Department of Information Technology, Central University of Kashmir, J&K, India 2Department of CSE, IIT Patna 3Department of Physics, University of Kashmir, J&K, India Email: furkaan309@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='com Abstract –Textual emotional intelligence is playing a ubiquitously important role in leveraging human emotions on social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Social media platforms are privileged with emotional contents and are leveraged for various purposes like opinion mining, emotion mining, and sentiment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This data analysis is also levered for prevention of online bullying, suicide prevention and depression detection among social media users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In this article we have designed an automatic depression detection of online social media users by analyzing their social media behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The designed depression detection classification can be effectively used to mine user’s social media interactions and one can determine whether a social media user is suffering from depression or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The underlying classifier is made using state-of-art technology in emotional artificial intelligence which includes LSTM (Long Short Term Memory) and other machine learning classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The highest accuracy of the classifier is around 70% of LSTM and for SVM the highest accuracy is 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='79%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We trained the classifier on the datasets that are widely used in literature for the emotion mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Confusion matrix of results is also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Keyword---Emotion Artificial Intelligence, Deep Learning, Machine Learning Algorithms, Word Embedding’s, Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' INTRODUCTION Depression triggers sorrow, frustration, lack of interest, annoyance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' It will contribute to variety of emotional and physical problems and might cut back the power of a person to operate at full potential [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The matter of mental state in humans is incredibly serious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' At its worst, depression is the main cause of death for 80000 human beings leading them to suicide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Teenagers are most prone to depression as a result of they often need Psychotherapy, 2 | P a g e Teenagers limitlessly use mobile phones and social media and spend less time in real world and more in virtuality which causes a gap between their internal state of mind and outward world, causing a strong sense of loneliness which in-turn leads to depression[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Bullying via social media based on factors like social status, standard of language, race, ethnicity, immigration standards, health, behavioral ethics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' are the main drawbacks of social media which cause depression to user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [4] The definition of problem dealt in this research article is to find whether the person under consideration does have symptoms of depression or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' With the use of neural networks we have used users tweets to diagnose whether the person under consideration is behaving positively or negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' If the person is shows particular standard of positivity in his tweets then we can predict him or her to be depression free, otherwise he or she is suffering from depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The standard used in this article for a person to be considered under depression is that if 80% of his tweets are classified negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We do the analysis of user feeds posted on a twitter account via state-of-art machine learning and deep learning techniques based on scientifically accepted 80% criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='[6][7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Some Initial Symptoms of depression are listed below: o Feeling crazy, sad or in a depressing mood o Weight loss o Increase fatigue and sleep difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' o Loss of energy and increased fatigue with urge to suicide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' o Thinking very difficultly, concentrating, or making decisions based on the assumption based on the idea of not liking anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' o Frequent and continuous of death or suicide Symptoms must last at least two weeks or longer than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Depression may have a different effect on the sexes and shows that men with depression can have signs such as irritability, escape or dangerous behavior, or that everyone is really angry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Symptoms must last at least two weeks or even more for a diagnosis of depression [1] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Risk Factors for depression- Depression can affect anyone, even a person who seems to be living in reasonably ideal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Depression can be caused by a variety of circumstances: o Biochemistry: Harmonal differences may contribute to symptoms of depression in certain cases due to chemicals imbalance in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 3 | P a g e o Genetics: Genetics also plays an essential role in co-depression for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' if one identical twin has depression, for instance, the other has a 70 percent risk of developing the disorder sometime in life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' o Personality: People with low self-confidence are more likely to experience depression, causing them to be speechless by stress, or who are generally pessimistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' o Environmental factors: Continuous exposure to violence, negligence, abuse or poverty can make some individuals more susceptible to depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Role of Social Media and Depression Social networks are becoming an important part of the lives of individuals, they mirror the personal life of the user on social media, people like to share happiness, joy and sorrow [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Researchers are using these platforms to classify and detect the causes of depression, reading an old article on the News website about how Twitter can tell if you're depressed and the possibility of developing an artificial intelligence model that can scan your Twitter feed and tell you if an individual is at risk of depression, or accepting notifications from third parties, such as those advising you to seek help based on an automatic scan of your tweets [1]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The day has come to an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Early detection of depression can be a big step toward addressing the mental illness and providing help to those who are suffering from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Various approaches for sentiment analysis in Machine Learning, including decision-based systems, Bayesian classifiers, support vector machines, neural networks, and sample-based methods can be used to detect symptoms of depression before it is too late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Depression and Emotion Intelligence If a person is attached with someone closely then it is clear the person is emotionally attached to that person and if this emotional dependence is too high that the decisions of attached persons are dependent on each other this is called emotion bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This is when these persons take emotions seriously sometimes their emotions are not positive in nature hence leading to depression in one or all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Emotional intelligence is commonly defined by four attributes: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Self-management – You can control impulsive emotions and actions, handle your emotions in healthy ways, take action, meet commitments, and respond to changing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4 | P a g e b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Self-awareness – You recognize your own feelings and how your thoughts and conduct are affected by surrounding conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' You are aware of your strengths and weaknesses, and have faith and confidence in ones self.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Social awareness – You've got empathy." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" You should understand other people's thoughts, desires, and concerns, pick up on emotional signals, socially feel relaxed, and know the dynamics of power in a group or organization." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Relationship management – You know how to build and maintain good relationships, clearly communicate, inspire and influence others, work well as a team, and resolve conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' DISCUSSION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Dataset We have developed a depression detection dataset by manually annotating the twitter data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We used two classes for this research one is depressive and other non-depressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The Tweets were downloads from the twitter Platform and some from the kaggle data and initial dataset had 25,000 tweets from which foreign language tweets and re-tweets were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This dataset contains three columns: First Column contains tweet ID, second column contains text and third column contain labels these columns are features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The dataset was then annotated by three annotators and majority voting was kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Thus if a tweet was annotated depressive by more than one annotator only then it was given a depressive label likewise was done for non-depressive tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In the final dataset we had 7,500 depressive tweets and 7,500 non-depressive tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Different algorithms are used we discuss one by one: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Support Vector Machine (SVM) Support Vector Machine(SVM) is a supervised algorithm for machine learning that can be used for problems with classification and regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In classification issues, however, it is mostly used [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In the SVM algorithm, we depict each data item as a point in n-dimensional space (where n is the number of characteristics you have), with the value of each 5 | P a g e characteristic being the coordinate value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Then, we carry out classification by determining the hyper-plane that clearly distinguishes the two classes [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The Support Vector Machine is a discriminative classifier that is formally described by a distinct hyperplane (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In other words, given labelled training data, the algorithm generates an ideal hyperplane that categorizes fresh examples (supervised learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This hyperplane is a line in two- dimensional space that divides a plane into two sections, with each class on either side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The learning of the hyperplane in linear SVM is finished by changing the issue using some linear algebra [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This is when the kernel comes into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In our experiment we gave the linear SVM using train test split of 80-20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We employed scikit-learn kernels for the classification purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Random Forest Classifier The Random Forest Classifier an ensemble algorithm, produces a set of decision trees from the training set's randomly selected subset." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' To evaluate the final class of the test object, it then aggregates the votes from various decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Different decision trees can create subsets that overlapping in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" To deal with this weighting principle is used to consider the impact of any decision tree's result." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Trees with a high mistake rate are given a low weight value, whereas trees with a low mistake rate are given a high weight value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Low error-rate trees would have a greater impact on decision-making as a result of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The total number of trees to be constructed, are as well the important characteristics for the decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The minimum split, split criteria, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=', are all basic Random Forest Classifier parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We used Random Forest classifier with 40 estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The training and testing sets as inputs to classifier were divided into 80-20 train test splits of total data entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We employed scikit- learn kernels for the classification purpose as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Multinomial Naïve Bayes Naive Bayes is a family of algorithms based on applying the Bayes theorem with the strong (naive) assumption that each feature is independent of the others in order to predict the category of a given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Because they are probabilistic classifiers, the Bayes theorem will be used to compute the likelihood of each category, and the category with the highest probability will be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Naive Bayes classifiers have been used successfully in a variety of fields, including Natural Language Processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='Other options, such as Support Vector Machines (SVM) and neural networks, are available when dealing with NLP issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 6 | P a g e Their simplicity of Naive Bayes classifiers, on the other hand, makes them particularly desirable for such classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Furthermore, they have been proved to be fast, reliable, and exact in a variety of NLP applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We use a non-naive Bayes approach to look at sentences in their whole, thus if a sentence does not appear in the training set, we will get a zero probability, making further computations impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' However, in the case of Naive Bayes, each word is assumed to be independent of the others [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We now examine individual words as a phrase rather than the complete sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='5 Long Short Term Memory (LSTM) LSTMs have proven to be effective in tasks involving sequential knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" The input vectors used for the other classifiers, such as the TF-IDF weights vector, do not, however, maintain any information on the sequential relationship between each document's terms and phrases." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Thus, the more appropriate input for LSTMs are sequences of word embedding vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' in fact, word embedding are the preferred option for textual representation in modern deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We tuned the number of memory units, number of epochs, batch size, and input and recurring dropout rates of LSTM by using the training and tuning set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The number of epochs is the number of times in the neural network the entire training set is moved forward and backward [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The network can be fit and the resulting neuron weights may be far from optimal if the number of epochs is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Before each change to the weights, the batch size is simply the number of samples fed into the network and helps to make training more effective by reducing the memory requirements and the number of iterations needed to achieve the optimum weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Finally, in an attempt to minimize overfitting, the dropout rate is simply the likelihood that an input or recurrent node will be omitted from consideration during a weight update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Input to the classifier in the 80-20 train test format was given to the training and testing set respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 7 | P a g e Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 1: Mechanism of LSTM Classifier A cell, an input gate, an output gate, and a forget gate make up a standard LSTM unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The cell recalls values across arbitrary time intervals, and the three gates govern the flow of information into and out of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Because there might be lags of undetermined duration between critical occurrences in a time series, LSTM networks are well suited to categorizing, processing, and making predictions based on time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' LSTMs were created to deal with the vanishing gradient problem that might occur when training regular RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The relative insensitivity to gap length of LSTM over RNNs, hidden Markov models, and other sequence learning approaches provides a benefit a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' LSTM architecture: Multiple architectures of LSTM units exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" The standard architecture consists of a cell (the LSTM unit's memory portion) and three information flow regulators inside the LSTM unit, typically referred to as gates: an input gate, an output gate, and a forgotten gate." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Any of the versions of the LSTM device may not have one or both of these gates, or even other gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Gated Recurrent Units (GRUs) do not, for example, have an output gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" The input gate determines how much a new value enters the cell, the forgotten gate determines how much the cell value remains, and the output gate determines how much the cell value is used to generate the LSTM unit's output activation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The activation function of the LSTM gates is also the logistic sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 2: Process of LSTM B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Word Embedding A word embedding is a group of ways for representing words and texts that use a dense vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Instead, in an embedding, words are represented by dense vectors, Input Feed1-hotvectorsonaString SequenceLength:15 Embed 0 Reduce Order S66 sa LSTM Featur 0 Do Magic Output 0 Predict H 0 877 sa pappin reversed 0 S unordered 0 0 966 H 08 | P a g e each of which reflects the word's projection into a continuous vector space [2]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The placement of a word is based on the words that surround it when it is used and is learned from text inside the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The embedding of a word is its position in the learned vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Two common examples of learning methods for embedding words from text include: • Word2Vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' • GloVe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In this Research article Word2Vec and Doc2Vec are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Keras Embedding layer Keras provides an embedding layer which can be used on text data for neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This includes integer encoding of the input data, so that each word is represented by a unique integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The Tokenizer API that is also supplied with Keras can be used to perform this data preparation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' All of the words in the training dataset will learn an embedding since the Embedding layer is initialized with random weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" It's a versatile layer that can be applied in a variety of ways, like it can be used on its own to teach a word integration that can then be stored and reused in a different model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' It can be utilized as part of a deep learning model where the embedding is taught alongside the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' It is a type of transfer learning, and can be used to load a pre-trained word embedding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Weights are learned in the Embedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' If you save your model as a file, the weights for the Embedding layer will be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" An embedding layer's output is a 2D vector in the input word sequence (input document), with one embedding for each word." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' You must first flatten the 2D output matrix to a 1D vector if you want to link a dense layer directly to an embedded layer using the Flatten layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Finally we train the LSTM Classifier for training and testing data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' METHOLODOLOGY In this section we discuss the overall methodology that is used for depression detection using online social profiles of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We employ emotional artificial intelligence for the depression detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We create a depression classifier using Deep-Learning framework and machine learning and use that classifier for classifying tweets from an online social 9 | P a g e profile for automatic depression detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Figure 5 shows the overall methodology used for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 3: Overall methodology of the depression detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The system receives as input an emotion labeled dataset which is used for the training of supervised deep-learning classifier or machine learning classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Once the classifier is trained on emotion labeled dataset we use the tweets from online social profiles for the depression detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' If the tweets from the profile contain more than 75% tweets classified as depressive, our system classifies gives the red flag for the profile that it may be having depression symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Figure 6 shows the components of the system in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4: Architecture of the system The system has 5 modules and uses emotion labeled dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Data Normalization TESTDATASET DATASETWISENTIMENT X Downioad DataFile Feeddatato TwitterAP!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Saveastxt Tweet Training& Downloader Testing PREDICTANDSHOWACCURACT NaiveBayesian DTree SVM ForTesting Ra fomForestData Data Dataset Normalization Preprocessing Supervised YES Classification Deep-Learning >75% Classifier Depressive NO OnlineSocial Profile Non-Depressive10 | P a g e As part of data preparation for machine learning, standardization is a technique sometimes used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The aim of adjustment is to adjust the numeric column values in the dataset to use a common scale, without distorting disparities in the value ranges or losing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' You also deleted all retweets, all non-alphanumeric characters, URLs, and @mentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Besides, all stop words were abolished, with the exception of first, second and third person pronouns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' About 40% work done by this through NLTK automatically and all libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Data Preprocessing Preprocessing each tweet before passing it to the automated classifier is how the preprocessing module prepares it for the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Slangs and abbreviations are eliminated by the use of English directories from the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' If a term is identified that does not have a meaningful definition, all of the words are passed into the dictionary module to be looked up, and then into the word replacer module to be replaced with the right word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In the word replacer module, we employed the SMS dictionary, Netlingo, and the urban dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Preprocessing stages are made up of the following steps: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Tokenization Is performed using Stanford Core NLP package to break tweets into sentences and words, such as units of words, sentences, or themes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The first column of the csv file containing the tweet is extracted and converted to individual tokens in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Stemming Stemming is the process of reducing words to their simplest form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This would allow us to group terms that are similar together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Poster Stemmer is utilized for implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Stop Words Removal Because they are of little use in training, the widely used words known as stop words need to be omitted and may also lead to incorrect results if not ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' There is a collection of stop words in the NLTK library which can be used as a guide for removing stop words from the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Stop words that are deleted from tweets using the Tf-Idf, Stanford, and wiki features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' POS Tagger 11 | P a g e The tokenized text is allocated to the respective sections of the speech by using a POS tagger to improve the quality of the remaining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Since other parts of speech are not of much importance, this can be used to remove only adjectives, nouns and adverbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Example: 'I LOVE CODING' is extracted-'I 'is a pronoun, rest is omitted." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Lemmatization It's used to apply stem words to their stems." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Lemmatization refers to performing things correctly utilizing a vocabulary and morphological word analysis, mainly aimed at merely removing inflectional endings and restoring the root or dictionary form of a word known as the lemma, utilizing the Stanford core NLP kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' A bag of words is generated after all of these pre-processing stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The number of occurrences of each word is calculated in a bag of words, which is subsequently utilized as a feature to train a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Bag of words or One Hot Encoding Each part of the vector corresponds to one word or n-gram (token) within the corpus vocabulary during this methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Then if the token at a selected index exists within the document, that part is marked as one, else it’s marked as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='5: BOW Representation In the above example, our corpus consists of every unique word across the three documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' A bag of Words model, or BOW for short, is a method of extracting text properties for use in modeling, such as machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The method is straightforward and adaptable, Batscanseevia Theelephantsneezed Wondering,sheopened echolocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='Seethe atthesightofpotatoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' thedoortothestudio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' batsightsneeze!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='12 | P a g e and it may be used to extract information from documents in a variety of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' A bag of words is a text representation of the words that appear within a document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' It involves two ways: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' A vocabulary of known words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The measure of the presence of known words should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" It's dubbed a bag of words because all information in the text about the sequence or arrangement of words is discarded." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The model simply considers where recognized phrases appear in the document, not where they exist in the document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The histogram of the words inside the text is examined in this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The bag of words approach (BOW), which considers each word count as a function, is a popular role extraction method for sentences and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The assumption is that documents are similar if they have comparable content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Furthermore, we can deduce something about the document's significance solely from its content." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' You can make the word bag as simple or as complex as you desire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The difficulty arises from deciding how to create the lexicon of knows words (or tokens) as well as how to rate the presence of such words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='9 Term Frequency-Inverse Document Frequency TF-IDF stands for term frequency-inverse document frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" It's a statistical metric for determining how important a word in a corpus of documents is to a text." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This importance is related to the number of times a word appears in the text, but is offset by the number of documents in the corpus that contain that word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='10 Count Vectorization The number of occurrences in a document for each word (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' independent text, such as an article, book, or simply a paragraph!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=') can be tallied using Count Vectorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The Sci-kit learning library in Python offers a tool called CountVectorizer that can help with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' "The weather was nice today, so I went outdoors to enjoy the gorgeous and sunny weather," for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The words "the," "weather," and "and" all appeared twice in the performance below, whereas other words only appeared once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Count Vectorization achieves this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='11 Distributional Similarity Based Representations- Word Embedding’s 13 | P a g e A word embedding may be a sort of learnt text illustration (real valued vectors) during which words with connected meanings square measure drawn equally, like the favored "King-Man + girl = Queen" example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' For every word, the idea of employing a dense, distributed illustration is central to the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Every term may be a real-value vector with tens or many dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In distinction, distributed word representations, like one-hot encryption, need thousands or countless dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The 2 most well liked word embedding’s square measure Word2Vec and Glove: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Word2Vec: There square measure primarily 2 versions of Word2Vec — Skip-Gram and Continuous Bag of Words (CBOW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The embedding is learned by the CBOW model anticipating the present word supported its context (surrounding words).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Given a current word, the Skip-Gram model learns by predicting the encompassing words (context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Word2vec represents every separate word with a selected list of numbers referred to as a vector, because the name suggests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The vectors square measure a basic function (the similarity of the trigonometric function between the vectors) is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Indicates the amount of linguistics similarity between the words drawn by those vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Skip-gram seeks to predict a given word's immediate neighbors." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We tend to take a central word and a context window (neighbors) terms, and take a few window step sizes round the central word, we tend to attempt to predict the context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' So, our model can describe a distribution of chance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" the chance of a word occurring in context given a central word, and to maximize the chance, we'll select our vector representations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We tend to take away the output layer and use the hidden layer to urge our word vectors till we are able to predict the encompassing words with an affordable degree of exactitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We tend to begin with little random data formatting of vectors of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' By minimizing the loss perform, our prognosticative model learns the vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In Word2vec, this happens with a feed-forward neural network with a language modeling task (predict next word) and improvement techniques like random gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This measures the options that I actually have used for machine learning algorithms like SVM, Naïve Thomas Bayes, and Random Forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 14 | P a g e Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 6: Word2Vec CBOW vs Skip-gram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" Gensim Model for DOC2VEC used in LSTM Gensim is paid as a kit of natural language processing that 'Topic Modeling for Humans' does." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" But it's even more than that, technically." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' I used Doc2Vec for LSTM, meaning the model of a deep learning algorithm, so clarify Doc2Vec first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' I have already mentioned the Word2Vec I used in machine learning algorithms since I used algorithms for machine learning as well as deep learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' So, for machine learning, I used Word2Vec and I used Doc2Vec for the LSTM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The aim of doc2vec, regardless of its length, is to construct a numeric representation of a text, as mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' But records do not come in logical forms like words and unlike words, so another technique needs to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Because the Doc2vec model is an unsupervised process, it should be slightly adjusted to "participate" in this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Fortunately, as in most circumstances, we can utilize a couple tricks: remember how we introduced another document vector in fig 3 that was unique to each document?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" If you think about it, you can add more vectors that don't have to be unique: for example, if we have tags for our documents (which we have), we can add them and receive their representation as vectors." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Additionally, they don’t have to be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' This way, we can add to the unique document tag one of our 17 tags, and create a doc2vec representation for them as well!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' See below: INPUT PROJECTION OUTPUT INPUT PROJECTION OUTPUT w(t-2) w(t-2) w(t-1) w(t-1) SUM w(t) w(t) w(t+1) w(t+1) w(t+2) w(t+2) CBOW Skip-gram15 | P a g e Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 8: Doc2Vec model with tag Vector 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Training From the training set both label and tweet are required by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The training set in this example refers to the set of tweets that must be processed before being fed into a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The set of tweets must be transformed to a vector representation for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The set of labels corresponding to each tweet is also provided into the classifier as a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Saving the classifier and the count vectorizer object Since training needs to be done once, it is necessary to load the trained classifier object into a pickle file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Same is applicable with the count Vectorizer Object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' It count the number of words present in the pickle file using the TF-IDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' And then the testing phase takes place for unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Testing Loading of saved models: Qualified classification models are loaded from the pickle file to be used for test dataset prediction (Individual tweet profiling data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The test dataset is preprocessed in a way that is similar to the data from the training for test tweets class prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Each tweet is graded into a class that is depressed or non-depressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We calculate the confusion matrix for the assessment of classification results based on the values of true or false positives and negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Confusion Matrix is used for how much our classifier shows accurate results as well as false results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' RESULTS The depression detection system that we designed uses emotional artificial intelligence for the detection of depression using online social profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We used supervised learning models Classifier on Average/Concatenate 冒 冒 T + T D w tag document the cat sat vector vector16 | P a g e and Deep Learning algorithm i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='e Long Short Term Memory (LSTM) were training of the classifier was given using the emotion dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The dataset has 15,000 tweets manually annotated by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The highest accuracy of the classifier is around 70% for LSTM and for SVM the highest accuracy is 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='79%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' I have trained the classifier on the datasets that are widely used in literature for the emotion mining task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The results are presented using the classification metrics like accuracy and confusion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Confusion Matrix: A confusion matrix is a machine learning classification performance evaluation methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=" It's a type of table that allows you to see how well a classification model performs on a set of test data for which the true values are known." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' True Positive (TP), True Negative (TN), False Positive (FP), False Negative (FN) these given four parameters of the confusion matrix are used to calculate the Accuracy, Recall, Precision and FI-Score, with the help of these four classes we compute all values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The results are evaluated on the basis of F1 score and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The F1 score is the primary performance measure and accuracy is the secondary measure with the help of confusion matrix we call Precision, Recall, F1 Score and Accuracy it simply shows how much our classifier shows accurate results or with the help of given formulas: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Precision: TP/TP+FP, where TP is true positive, FP is false positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Recall: TP/TP+FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='FI Score: 2* (Precision*Recall)/ Precision + Recall The dataset was balanced having 7,500 tweets in each data class that is depressive and non- depressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' SVM classifier its accuracy is more than other classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We choose SVM classifier for unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We trained our system using LSTM (Deep Learning Classifier), SVM, Naïve Bayes and Random Forest classifier (Machine Learning Classifier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The classifiers accuracy is described in the below table: Classifier Precision Recall F1 Score Accuracy LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='71 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='00% Multinomial NB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='78 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='32% SVM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='79 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='79% Random Forest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='78 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='37% 17 | P a g e Table 1 Results of different Classifier performance Figure 9, 10, 11 and 12 show the performance of the system using confusion matrices: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 9: Confusion Matrix of SVM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='10: Confusion Matrix of LSTM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content='10: Multinomial Naïve Bayes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 12: Random Forest V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK The analytics performed on the chosen dataset offer some insight into the analysis issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' The subsequent may be an outline of our findings: What is depression, and what area unit the foremost current causes of depression?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' whereas we have a tendency to all feel irritable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' unhappy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' or low from time to time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' some individuals experience these feelings on an Confusion Matrix 1200 0 12B0 323 1000 Tue label 800 600 1 254 1361 400 1 Predicted labelConfusion Matrix 1200 0 1368 182 1000 True label 800 600 1 880 738 400 -200 0 1 Predicted labelConfusionMatrix 1200 1231 322 1000 Tuelabel 800 1 EEE 1282 600 400 0 1 PredictedlabelConfusion Matrix 1200 0 374 1000 True label 800 600 1 248 1367 400 0 1 Predicted label18 | P a g e everyday basis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' over long periods of your time (weeks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' months,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' or perhaps years),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' and in some cases for no obvious cause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Despondence is over simply a nasty mood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' it is a state of affairs that has a bearing on an individuals physical and mental well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Any one can be suffering from depression at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' However, bound stages or circumstances in modern day world expose people to anxiety and depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' During entire life the changes around or inside an individual all end in a surge of emotions which will contribute to unhappiness in introvert and socially separated people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We conducted a comparison of state-of-art deep learning models to pre-detect depression from tweets at the user level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We ran our models on a manually preprocessed dataset and discovered that SVM created higher results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' In the future, we can use a special methodology to extract paraphrases from a wider set of emotional qualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' We will additionally check the quality and potency of our models with lot more datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' REFERENCES [1] Mander Deshpande and Vignesh Rao, “Depression Detection using Emotion Artificial Intelligence”, IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [2] Marcel Trotzek, Sven Koitka and Christoph M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Friedrich,”Utilizing Neural Networks and Linguistic Metadata for Early Detection of Depression Indications in Text Sequences”, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [3] Adyan Marendra Ramadhani and Hong Soon Goo, “Twitter Sentiment Analysis using Deep Learning Methods”, 7th International Annual Engineering Seminar (InAES), Yogyakarta Indonesia, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [4] Wei Tong Mok, Rachael Sing, Xiuting Jiang and Swee See, “Investigation of Social Media on Depression, IEEE, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [5] Niko Colneric and Janez Demsar, “Emotion Recognition on Twitter: Comparative Study and Training a Unison Model”, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [6] Mohammad Jabreel and Antonio Moreno, “A deep Learning-Based Approach for Multi- label Emotion Classification in Tweets”, applied science, MDPI, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [7] Hakak Nida, Kirmani Mahira, Mohd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Mudasir, Muttoo Mudasir Ahmad and Mohd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Mohsin, “Automatic Emotion Classifier”, Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 19 | P a g e [8] Krishna Shrestha, “Machine Learning for Depression Diagnosis using Twitter data”, International Journal of Computer Engineering in Research Trends using Twitter data (IJCERT), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [9] Nida Manzoor Hakak, Mohsin Mohd, Mahira Kirmani and Mudasir Mohd, “Emotion Analysis: A survey”, IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [10] Hemanthkumar M, Latha, “Depression Detection with Sentiment Analysis of Tweets”, International Research Journal of Engineering and Technology (IRJET), 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [11] Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Rafiqul Islam, Muhammad Ashad Kabir, Ashir Ahmed, Abu Raihan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' Kamal, Hua Wang and Anwaar Ulhaq, ”Depression detection from social network data using machine learning techniques”, Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' [12] Aswathy K S, Rafeeque P C and Reena Murali, “Deep Learning Approach for the Detection of Depression in Twitter”, International conference on systems energy and environment, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} +page_content=' 20 | P a g e' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE1T4oBgHgl3EQffwSj/content/2301.03222v1.pdf'} diff --git a/ldE2T4oBgHgl3EQfIwbb/content/tmp_files/2301.03685v1.pdf.txt b/ldE2T4oBgHgl3EQfIwbb/content/tmp_files/2301.03685v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..80589222385b2b78eba37169f0798bcc22872cff --- /dev/null +++ b/ldE2T4oBgHgl3EQfIwbb/content/tmp_files/2301.03685v1.pdf.txt @@ -0,0 +1,290 @@ +arXiv:2301.03685v1 [math.AT] 9 Jan 2023 +EILENBERG–MACLANE SPACES AND STABILISATION +IN HOMOTOPY TYPE THEORY +DAVID W¨ARN +Abstract. In this note, we study the delooping of spaces and maps in homotopy type theory. +We show that in some cases, spaces have a unique delooping, and give a simple description of +the delooping in these cases. We explain why some maps, such as group homomorphisms, have a +unique delooping. We discuss some applications to Eilenberg–MacLane spaces and cohomology. +1. Introduction +The loop space functor Ω is an operation on pointed types and pointed maps between them. In +this note, we study the delooping of types and maps: given a pointed type X, when can we find a +pointed type whose loop space is equivalent to X? And given a pointed map f : ΩA →pt ΩB, when +can we find a map A →pt B whose looping equals f? The general answer is rather complicated, +involving group operations and an infinite tower of coherences, but according to the stabilisation +theorem [BDR18], the answer becomes much simpler if we put some connectivity and truncation +assumptions on A and B. The purpose of this note is to give a direct, type-theoretic account +of these simple special cases. We also explain how to use these results to set up the theory of +Eilenberg–MacLane spaces and cohomology operations. We assume only basic familiarity with +homotopy type theory, as developed in [Uni13]. +We will not need to assume the Freudenthal +suspension theorem, nor will we make use of any higher inductive types other than propositional +truncation. +Notation. As in [Uni13], we write a = b for the type of identifications between a and b, refla : a = a +for the reflexivity identification, +: (a = b) → (b = c) → (a = c) for path concatenation, +apf : (a = b) → (f a = f b) for the action of a function on paths, U for a univalent universe, and +∥A∥ for propositional truncation. We write (a : A) → B a for the Π-type Πa:AB a, and (a : A)×B a +for the Σ-type Σa:AB a. We write Upt for the type (X : U) × X of pointed types. For A : Upt, +we will write |A| : U for its underlying type, and ptA : |A| for its point. For A, B : Upt, we write +A →pt B for the type (f : |A| → |B|) × (f ptA = ptB) of pointed functions. For f : A →pt B, we +write |f| : |A| → |B| for the underlying function, and ptf : |f| ptA = ptB for the proof that it is +pointed. For A : Upt, we write ΩA : Upt for the loop space (ptA = ptA, reflptA). For f : A →pt B +we write Ωf : ΩA →pt ΩB for the action on loops, p : ptA = ptA �→ pt−1 +f +ap|f|p ptf : ptB = ptB. +We write A ≃pt B for the type (f : A ≃ B) × f ptA = ptB of pointed equivalences. +Acknowledgements. We thank Thierry Coquand for his support throughout the project, as +well as Felix Cherubini, Louise Leclerc, Jarl G. Taxer˚as Flaten, and Axel Ljungstr¨om for fruitful +discussions. +2. Delooping types +Let X : Upt be a pointed type, and suppose we want – without further inputs – to construct a +delooping of X. That is, we want to find a pointed type whose loop space is equivalent to X. One +way would be to use the suspension ΣX [LF14], which is freely generated by a map X →pt ΩΣX +1 + +2 +DAVID W¨ARN +and so necessarily maps to any delooping of X. Instead, we will use a cofree construction, which +necessarily has a map from any delooping of X. Similar ideas are discussed in [Buc+23]. +Definition 1. For X : Upt, the type TX of X-torsors is given by +TX := (Y : U) × ∥Y ∥ × (y : Y ) → X ≃pt (Y, y).1 +Intuitively, an X-torsor is a type which looks like X at every point, and merely has a point, +even though we might not have access to any particular point. If only we knew that the type +((Y, h, µ) : TX) × Y of pointed X-torsors were contractible, it would follow that TX is pointed +and, by the fundamental theorem of identity types [Rij22], a delooping of X. The following lemma +provides an alternative description of this type of pointed X-torsors, which will make it feasible +to determine when it is contractible. +Lemma 2. We have an equivalence of types +((Y, h, µ) : TX) × Y ≃ (µ : (x : |X|) → X ≃pt (|X|, x)) × (µ ptX = idX). +The right-hand side is roughly the type of coherent H-space structures on X, but note that we +asymmetrically require invertibility on one side. +Proof. We have +((Y, h, µ) : TX) × Y ≃ (Y : U) × ∥Y ∥ × ((y : Y ) → X ≃pt (Y, y)) × Y +≃ (Z : Upt) × (µ : (z : |Z|) → X ≃pt (|Z|, z)) +≃ (Z : Upt) × (µ : (z : |Z|) → X ≃pt (|Z|, z)) × (p : X ≃pt Z) × (µ ptZ = p) +≃ (µ : (x : |X|) → X ≃pt (|X|, x)) × (µ ptX = idX). +In the first line, we simply unfold the definition of TX, and in the second line we do some simple +rearrangement, dropping the redundant assumption ∥Y ∥. In the third line, we use contractibility +of singletons to add two redundant fields p : X ≃pt Z and µ ptZ = p. And in the final line, we use +univalence and contractibility of singletons to remove two redundant fields Z and p. +□ +The following lemma will be our main tool to determine when types are contractible. It is a +special case of Lemma 8.6.1 from [Uni13], and has a direct proof by induction. +Lemma 3. If A : Upt is an n-connected2 pointed type, B : |A| → U is a family of (n + m + 1)- +truncated types, and ptB : B ptA, then the type of ‘pointed sections of B’, +(f : (a : |A|) → B a) × (f ptA = ptB), +is m-truncated. +Corollary 4. If A : Upt is n-connected and B : Upt is (n + m + 1)-truncated, then A →pt B is +m-truncated. If A and B are both n-connected and (n + m + 1)-truncated, then A ≃pt B is also +m-truncated. +1A priori, since U is a large type, so is T X. However, we could just as well quantify over Y : BAut |X| in the +definition of T X, where BAut |X| ≃ (Y : U) × ∥Y ≃ |X|∥. It is reasonable to assume that BAut |X| is small, either +by the replacement principle from [Rij17], or by simply postulating the existence of enough small univalent type +families. In the rest of the note we ignore universe issues and assume T X : U. +2While there are several equivalent definitions of connectedness, this note is most easily understood with a +recursive definition: every type is (−2)-connected, and a type is (n + 1)-connected if it is merely inhabited and its +identity types are n-connected. + +EILENBERG–MACLANE SPACES AND STABILISATION IN HOMOTOPY TYPE THEORY +3 +Proof. The first claim is a direct consequence of Lemma 3. For the second, we have an equivalence +between A ≃pt B and the type (f : A →pt B) × (g h : B →pt A) × (f ◦ g = idB) × (h ◦ f = idA) of +biinvertible pointed maps. This is m-truncated since m-truncated types are closed under Σ and +identity types. +□ +Corollary 5. If X is n-connected and (2n + m + 2)-truncated, then the type of pointed X-torsors +is m-truncated. +Proof. Combining Lemma 2, Lemma 3, and Corollary 4. +□ +Corollary 6. If X is n-connected and 2n-truncated, then TX is a delooping of X. +Proof. In this case, the type of pointed X-torsors is (−2)-truncated, i.e. contractible. In particular +we have a pointed X-torsor, and by the fundamental theorem of identity types its loop space is +equivalent to X. +□ +Corollary 7. If X is n-connected and (2n + 1)-truncated and TX is merely inhabited, then TX +is a delooping of X. +Proof. In this case the type of pointed X-torsors is (−1)-truncated, i.e. a proposition. Since we +assume TX is merely inhabited, there also merely exists a pointed X-torsor. A merely inhabited +proposition is contractible, so we may proceed as in the previous case. +□ +3. Delooping maps +Suppose A, B : Upt are pointed types, and f : ΩA →pt ΩB is a pointed map on loop spaces. +When can we find F : A →pt B such that f = ΩF? More precisely, we want a useful description +of the type Ω−1f := (F : A →pt B) × (f = ΩF). For example, it is necessary that we have +f(p q) = f(p) f(q). +Lemma 8. We have an equivalence of types +Ω−1f ≃ (a : |A|) → (b : |B|) × C a b +where C : |A| → |B| → U is given by +C a b := (h : (a = ptA) → (b = ptB)) × ((p : a = ptA) → f = F(h, p)) +and we define F(h, p) : ΩA →pt ΩB by +|F(h, p)|(q) = (h p)−1 h(p q), +pointed in the obvious way. +We can think of C as a proof-relevant relation approximating a function F : |A| → |B|; it would +be a function if only (b : |B|) × C a b were contractible for all a : |A|. +Proof. We have +Ω−1f ≃ (F : |A| → |B|) × (ptF : F ptA = ptB)) × (f = Ω(F, ptF)) +(a : |A|) → (b : |B|) × C a b ≃ (F : |A| → |B|) × (a : |A|) → C a (F a). +So it suffices to show that for F : |A| → |B|, we have +(ptF : F ptA = ptB) × (f = Ω(F, ptF)) ≃ (a : |A|) → C a (F a). +Now by path induction and type-theoretic choice, we have +(a : |A|) → C a (F a) ≃ (h : (a : |A|) → (a = ptA) → (F a = ptB)) × (f = F(h, reflptA)). + +4 +DAVID W¨ARN +Again by path induction, we have ((a : |A|) → (a = ptA) → (F a = ptB)) ≃ (F ptA = ptB). It +suffices to show that if h corresponds to ptF under this equivalence, then F(h, reflptA) = Ω(F, ptF). +This holds essentially by definition. +□ +Corollary 9. Suppose |A| is n-connected and |B| is (2n + m + 2)-truncated, where n ≥ 0 and +m ≥ −2. Then Ω−1f is m-truncated. +Proof. It suffices to show that, for any a : |A|, the type (b : |B|) × C a b is m-truncated. Since to +be truncated is a proposition and |A| is at least 0-connected, it suffices to consider the case where +a is ptA. In this case we have +(b : |B|) × C ptA b ≃ (b : |B|) × ((h, t) : C ptA b) × (q : b = ptB) × (h reflptA = q) +≃ (h : ΩA →pt ΩB) × (p : ptA = ptA) → (f = F(|h|, p)), +by first adding two redundant singleton fields, and then removing another pair of singleton fields. +One can prove G(h) : h = F(|h|, reflptA) using unit laws, so we further have +(b : |B|) × C ptA b ≃ (t : (p : ptA = ptA) → f = F(|f|, p)) × (t reflptA = G(f)). +This is the type of pointed sections of a pointed type family over ptA = ptA. The fibres are identity +types in ΩA →pt ΩB, which is (n+m+1)-truncated. Since the fibres are (n+m)-truncated and the +base ptA = ptA is (n − 1)-connected, the type of pointed sections is m-truncated as claimed. +□ +Corollary 10. If |A| is n-connected and |B| is 2n-truncated, then Ω is an equivalence +(A →pt B) ≃ (ΩA →pt ΩB). +Corollary 11. If |A| is n-connected and |B| is (2n+1)-truncated, then Ω identifies A →pt B with +the subtype of ΩA →pt ΩB consisting of f : ΩA →pt ΩB such that (b : |B|) × C ptA b, which is +logically equivalent to C ptA ptB, and hence to (p q : ptA = ptA) → f(p q) = f(p) f(q). +4. Applications +In homotopy type theory, we define the ordinary cohomology group Hn(X; G) of a type X with +coefficients in a an abelian group G as the set-truncation ∥X → K(G, n)∥0, where K(G, n) is an +Eilenberg–MacLane space. The algebraic structure of these cohomology groups comes from various +operations at the level of Eilenberg–MacLane spaces, which we now discuss. +4.1. K(G, n). Let G be a group, so that in particular G is a 0-truncated type. One can define a +0-connected pointed type K(G, 1) : Upt with ΩK(G, 1) ≃grp G as a type of torsors, similar to our +TX [Bez+]. Note that K(G, 1) is necessarily 1-truncated. By Corollary 11, we have that if B is +1-truncated, then (K(G, 1) →pt B) ≃ (G →grp ΩB); we think of this as an elimination principle +for K(G, 1). From this elimination principle, it follows that if X : Upt is another 0-connected, +1-truncated pointed type, then (K(G, 1) ≃pt X) ≃ (G ≃grp ΩX). +When can we find K(G, 2) : Upt with ΩK(G, 2) ≃pt K(G, 1)? By Corollary 7, it suffices to have +(µ : (x : |K(G, 1)|) → K(G, 1) ≃pt (|K(G, 1)|, x)) × (µ pt = id), +or equivalently +(µ : (x : |K(G, 1)|) → G ∼=grp (x = x)) × (µ pt = id). +Given a dependent elimination principle for K(G, 1), we could analyse this type of pointed sections +directly. +Alternatively, we can think of pointed sections as pointed maps into a Σ-type with +extra structure, and apply our non-dependent elimination principle. The loop space of the Σ-type +(x : |K(G, 1)|) × G ∼=grp (x = x) is the centre Z(G) of G, and so we are left to ask when the +inclusion Z(G) →grp G has a section. This happens precisely when G is abelian. So K(G, 1) has a +delooping if and only if G is abelian, in which case the delooping is unique. As soon as we have + +EILENBERG–MACLANE SPACES AND STABILISATION IN HOMOTOPY TYPE THEORY +5 +K(G, 2), Corollary 6 gives K(G, n) : Upt for every n with ΩK(G, n+1) ≃pt K(G, n). We also get an +elimination principle by repeated application of Corollary 10: for any n ≥ 1 and any n-truncated +type B, we have (K(G, n) →pt B) ≃ (G →grp ΩnB). One can check, combining the definition of +TX with the elimination principle for K(G, n), that for n ≥ 0 we have +K(G, n + 2) ≃ (Y : U) × n−connected(Y ) × (y : Y ) → G ≃grp Ωn+1(Y, y). +4.2. πn(Sn). While we have systematically avoided talking about higher inductive types, we can +still say something about them. Recall that the n-sphere Sn : Upt is defined as a pointed type with +(Sn →pt B) ≃ ΩnB. If B is n-truncated for n ≥ 1, we have ΩnB ≃ (Z →grp ΩnB), since Z is the +free group on one generator, which as we’ve seen is equivalent to K(Z, n) →pt B. By the Yoneda +lemma, we get that K(Z, n) is the n-truncation of Sn. In particular, πn(Sn) ≃grp Ωn(∥Sn∥n) ≃grp +Ωn(K(Z, n)) ≃grp Z, and πk(Sn) = 0 for k < n. +More generally, this argument shows that A ≃pt ∥ΣΩA∥2n when A is n-connected and 2n- +truncated. Applying the same fact to the delooping TA of A, we get that TA ≃pt ∥ΣA∥2n+1. +Taking loop spaces of both sidse, we get A ≃pt ∥ΩΣA∥2n, which is part of the Freudenthal suspen- +sion theorem. +4.3. The cup product. We now give some sketches on how to define cohomology operations. +Given a bilinear map L →grp M →grp N, we define a cup-product +⌣ : K(L, n) →pt K(M, m) →pt K(N, n + m), +similar to the definition in [BLM22]. Note that we ask for the cup product to respect pointing, +corresponding to 0 ⌣ y = x ⌣ 0 = 0; without this extra piece of specification, the definition +would not work. Indeed, K(M, m) →pt K(N, n + m) is n-truncated, so the elimination principle +applies:3 +K(L, n) →pt K(M, m) →pt K(N, n + m) ≃ L →grp Ωn(K(M, m) →pt K(N, n + m)) +≃ L →grp K(M, m) →pt ΩnK(N, m + n) +≃ L →grp K(M, m) →pt K(N, m) +≃ L →grp M →grp N. +The forward maps in this composite are given explicitly by iterated looping, so we arrive at a +definition of the cup product as the unique bi-pointed map whose looping gives back the bilinear +map we started with. With this characterisation, we expect that algebraic properties of the cup +product follow from analogous properties of looping. For example, one can prove that the following +square anti-commutes, and this corresponds to graded commutativity of the cup product. +A →pt B →pt C +ΩA →pt B →pt ΩC +A →pt ΩB →pt ΩC +ΩA →pt ΩB →pt Ω2C +3Formally this argument assumes m, n ≥ 1, but it can be adapted to cover all m, n ≥ 0. + +6 +DAVID W¨ARN +4.4. Steenrod squares. Let us now use Corollary 11 to construct Steenrod squares as ‘stable +cohomology operations’ Sqi +n : K(Z/2, n) →pt K(Z/2, n + i) with Sqi +n corresponding to ΩSqi +n+1. We +first define Sqi +i as the cup product square x �→ x ⌣ x. To deloop this to Sqi +i+1, we need to show +(x + y) ⌣ (x + y) = x ⌣ x + y ⌣ y, +which follows from distributivity and graded commutativity since we are working mod 2. Given +Sqi +i+1 we can define Sqi +n for all n using Corollary 10, by looping and delooping as appropriate. +5. Concluding remarks +Our Lemma 8 can be compared with the construction of functors out of a Rezk comple- +tion in [Uni13, Theorem 9.9.4] and the construction of maps K(G, 1) →pt K(H, 1) in [Bez+, +Lemma 4.10.1]. Variants of the relation C a b are used in all cases. The idea can be understood as +a type-theoretic analogue of the arguments in [Del91, Sections 5.2-5.3]. +The arguments in this note are well-suited to formalisation. Indeed, many parts have already +been formalised twice: first by Louise Leclerc [Lec22], and later by Axel Ljungstr¨om in order to +develop the theory of Steenrod squares. +In upcoming work, we take the ideas of this note much further to give an exact, infinitary de- +scription of higher groups – as well as higher equivalence relations more generally – and morphisms +between them. In fact the description of morphisms is in a precise sense obtained mechanically +from the descriptions of objects, explaining the similarity between the second and third sections +of this note (compare for example Corollaries 5, 6, 7 with Corollaries 9, 10, 11). +References +[BDR18] +Ulrik Buchholtz, Floris van Doorn, and Egbert Rijke. “Higher groups in homotopy +type theory”. In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in +Computer Science. 2018, pp. 205–214. +[Bez+] +Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, and Daniel R. Grayson. +Symmetry. https://github.com/UniMath/SymmetryBook. +[BLM22] +Guillaume Brunerie, Axel Ljungstr¨om, and Anders M¨ortberg. “Synthetic cohomology +theory in cubical agda”. In: Computer Science Logic (CSL’22). 2022. +[Buc+23] +Ulrik Buchholtz, J. Daniel Christensen, Jarl G. Taxer˚as Flaten, and Egbert Rijke. +Central H-spaces and banded types. 2023. doi: 10.48550/ARXIV.2301.02636. url: +https://arxiv.org/abs/2301.02636. +[Del91] +Pierre Deligne. “Le symbole mod´er´e”. In: Publications Math´ematiques de l’Institut des +Hautes ´Etudes Scientifiques 73 (1991), pp. 147–181. +[Lec22] +Louise Leclerc. A formalisation of Eilenberg-Maclane spaces using HoTT-Agda. https://github.com/luyise/EM-spaces. +2022. +[LF14] +Daniel R. Licata and Eric Finster. “Eilenberg-MacLane Spaces in Homotopy Type The- +ory”. In: Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Con- +ference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE +Symposium on Logic in Computer Science (LICS). CSL-LICS ’14. Vienna, Austria: As- +sociation for Computing Machinery, 2014. isbn: 9781450328869. doi: 10.1145/2603088.2603153. +url: https://doi.org/10.1145/2603088.2603153. +[Rij17] +Egbert Rijke. The join construction. 2017. doi: 10.48550/ARXIV.1701.07538. url: +https://arxiv.org/abs/1701.07538. +[Rij22] +Egbert Rijke. Introduction to Homotopy Type Theory. 2022. doi: 10.48550/ARXIV.2212.11082. +url: https://arxiv.org/abs/2212.11082. + +REFERENCES +7 +[Uni13] +The Univalent Foundations Program. Homotopy Type Theory: Univalent Foundations of +Mathematics. Institute for Advanced Study: https://homotopytypetheory.org/book, +2013. +Email address: warnd@chalmers.se + diff --git a/ldE2T4oBgHgl3EQfIwbb/content/tmp_files/load_file.txt b/ldE2T4oBgHgl3EQfIwbb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5645a4a8c49333ab3630f603a3fa852566126dcf --- /dev/null +++ b/ldE2T4oBgHgl3EQfIwbb/content/tmp_files/load_file.txt @@ -0,0 +1,259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf,len=258 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='03685v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='AT] 9 Jan 2023 EILENBERG–MACLANE SPACES AND STABILISATION IN HOMOTOPY TYPE THEORY DAVID W¨ARN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In this note, we study the delooping of spaces and maps in homotopy type theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We show that in some cases, spaces have a unique delooping, and give a simple description of the delooping in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We explain why some maps, such as group homomorphisms, have a unique delooping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We discuss some applications to Eilenberg–MacLane spaces and cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Introduction The loop space functor Ω is an operation on pointed types and pointed maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In this note, we study the delooping of types and maps: given a pointed type X, when can we find a pointed type whose loop space is equivalent to X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' And given a pointed map f : ΩA →pt ΩB, when can we find a map A →pt B whose looping equals f?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The general answer is rather complicated, involving group operations and an infinite tower of coherences, but according to the stabilisation theorem [BDR18], the answer becomes much simpler if we put some connectivity and truncation assumptions on A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The purpose of this note is to give a direct, type-theoretic account of these simple special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We also explain how to use these results to set up the theory of Eilenberg–MacLane spaces and cohomology operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We assume only basic familiarity with homotopy type theory, as developed in [Uni13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We will not need to assume the Freudenthal suspension theorem, nor will we make use of any higher inductive types other than propositional truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' As in [Uni13], we write a = b for the type of identifications between a and b, refla : a = a for the reflexivity identification, : (a = b) → (b = c) → (a = c) for path concatenation, apf : (a = b) → (f a = f b) for the action of a function on paths, U for a univalent universe, and ∥A∥ for propositional truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We write (a : A) → B a for the Π-type Πa:AB a, and (a : A)×B a for the Σ-type Σa:AB a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We write Upt for the type (X : U) × X of pointed types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For A : Upt, we will write |A| : U for its underlying type, and ptA : |A| for its point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For A, B : Upt, we write A →pt B for the type (f : |A| → |B|) × (f ptA = ptB) of pointed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For f : A →pt B, we write |f| : |A| → |B| for the underlying function, and ptf : |f| ptA = ptB for the proof that it is pointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For A : Upt, we write ΩA : Upt for the loop space (ptA = ptA, reflptA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For f : A →pt B we write Ωf : ΩA →pt ΩB for the action on loops, p : ptA = ptA �→ pt−1 f ap|f|p ptf : ptB = ptB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We write A ≃pt B for the type (f : A ≃ B) × f ptA = ptB of pointed equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We thank Thierry Coquand for his support throughout the project, as well as Felix Cherubini, Louise Leclerc, Jarl G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Taxer˚as Flaten, and Axel Ljungstr¨om for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Delooping types Let X : Upt be a pointed type, and suppose we want – without further inputs – to construct a delooping of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' That is, we want to find a pointed type whose loop space is equivalent to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' One way would be to use the suspension ΣX [LF14], which is freely generated by a map X →pt ΩΣX 1 2 DAVID W¨ARN and so necessarily maps to any delooping of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Instead, we will use a cofree construction, which necessarily has a map from any delooping of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Similar ideas are discussed in [Buc+23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For X : Upt, the type TX of X-torsors is given by TX := (Y : U) × ∥Y ∥ × (y : Y ) → X ≃pt (Y, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1 Intuitively, an X-torsor is a type which looks like X at every point, and merely has a point, even though we might not have access to any particular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If only we knew that the type ((Y, h, µ) : TX) × Y of pointed X-torsors were contractible, it would follow that TX is pointed and, by the fundamental theorem of identity types [Rij22], a delooping of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The following lemma provides an alternative description of this type of pointed X-torsors, which will make it feasible to determine when it is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We have an equivalence of types ((Y, h, µ) : TX) × Y ≃ (µ : (x : |X|) → X ≃pt (|X|, x)) × (µ ptX = idX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The right-hand side is roughly the type of coherent H-space structures on X, but note that we asymmetrically require invertibility on one side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We have ((Y, h, µ) : TX) × Y ≃ (Y : U) × ∥Y ∥ × ((y : Y ) → X ≃pt (Y, y)) × Y ≃ (Z : Upt) × (µ : (z : |Z|) → X ≃pt (|Z|, z)) ≃ (Z : Upt) × (µ : (z : |Z|) → X ≃pt (|Z|, z)) × (p : X ≃pt Z) × (µ ptZ = p) ≃ (µ : (x : |X|) → X ≃pt (|X|, x)) × (µ ptX = idX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In the first line, we simply unfold the definition of TX, and in the second line we do some simple rearrangement, dropping the redundant assumption ∥Y ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In the third line, we use contractibility of singletons to add two redundant fields p : X ≃pt Z and µ ptZ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' And in the final line, we use univalence and contractibility of singletons to remove two redundant fields Z and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ The following lemma will be our main tool to determine when types are contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' It is a special case of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1 from [Uni13], and has a direct proof by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If A : Upt is an n-connected2 pointed type, B : |A| → U is a family of (n + m + 1)- truncated types, and ptB : B ptA, then the type of ‘pointed sections of B’, (f : (a : |A|) → B a) × (f ptA = ptB), is m-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If A : Upt is n-connected and B : Upt is (n + m + 1)-truncated, then A →pt B is m-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If A and B are both n-connected and (n + m + 1)-truncated, then A ≃pt B is also m-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 1A priori, since U is a large type, so is T X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' However, we could just as well quantify over Y : BAut |X| in the definition of T X, where BAut |X| ≃ (Y : U) × ∥Y ≃ |X|∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' It is reasonable to assume that BAut |X| is small, either by the replacement principle from [Rij17], or by simply postulating the existence of enough small univalent type families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In the rest of the note we ignore universe issues and assume T X : U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 2While there are several equivalent definitions of connectedness, this note is most easily understood with a recursive definition: every type is (−2)-connected, and a type is (n + 1)-connected if it is merely inhabited and its identity types are n-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' EILENBERG–MACLANE SPACES AND STABILISATION IN HOMOTOPY TYPE THEORY 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The first claim is a direct consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For the second, we have an equivalence between A ≃pt B and the type (f : A →pt B) × (g h : B →pt A) × (f ◦ g = idB) × (h ◦ f = idA) of biinvertible pointed maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' This is m-truncated since m-truncated types are closed under Σ and identity types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If X is n-connected and (2n + m + 2)-truncated, then the type of pointed X-torsors is m-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Combining Lemma 2, Lemma 3, and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If X is n-connected and 2n-truncated, then TX is a delooping of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In this case, the type of pointed X-torsors is (−2)-truncated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In particular we have a pointed X-torsor, and by the fundamental theorem of identity types its loop space is equivalent to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If X is n-connected and (2n + 1)-truncated and TX is merely inhabited, then TX is a delooping of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In this case the type of pointed X-torsors is (−1)-truncated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' a proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Since we assume TX is merely inhabited, there also merely exists a pointed X-torsor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' A merely inhabited proposition is contractible, so we may proceed as in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Delooping maps Suppose A, B : Upt are pointed types, and f : ΩA →pt ΩB is a pointed map on loop spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' When can we find F : A →pt B such that f = ΩF?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' More precisely, we want a useful description of the type Ω−1f := (F : A →pt B) × (f = ΩF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For example, it is necessary that we have f(p q) = f(p) f(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We have an equivalence of types Ω−1f ≃ (a : |A|) → (b : |B|) × C a b where C : |A| → |B| → U is given by C a b := (h : (a = ptA) → (b = ptB)) × ((p : a = ptA) → f = F(h, p)) and we define F(h, p) : ΩA →pt ΩB by |F(h, p)|(q) = (h p)−1 h(p q), pointed in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We can think of C as a proof-relevant relation approximating a function F : |A| → |B|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' it would be a function if only (b : |B|) × C a b were contractible for all a : |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We have Ω−1f ≃ (F : |A| → |B|) × (ptF : F ptA = ptB)) × (f = Ω(F, ptF)) (a : |A|) → (b : |B|) × C a b ≃ (F : |A| → |B|) × (a : |A|) → C a (F a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' So it suffices to show that for F : |A| → |B|, we have (ptF : F ptA = ptB) × (f = Ω(F, ptF)) ≃ (a : |A|) → C a (F a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Now by path induction and type-theoretic choice, we have (a : |A|) → C a (F a) ≃ (h : (a : |A|) → (a = ptA) → (F a = ptB)) × (f = F(h, reflptA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 4 DAVID W¨ARN Again by path induction, we have ((a : |A|) → (a = ptA) → (F a = ptB)) ≃ (F ptA = ptB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' It suffices to show that if h corresponds to ptF under this equivalence, then F(h, reflptA) = Ω(F, ptF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' This holds essentially by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Suppose |A| is n-connected and |B| is (2n + m + 2)-truncated, where n ≥ 0 and m ≥ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Then Ω−1f is m-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' It suffices to show that, for any a : |A|, the type (b : |B|) × C a b is m-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Since to be truncated is a proposition and |A| is at least 0-connected, it suffices to consider the case where a is ptA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In this case we have (b : |B|) × C ptA b ≃ (b : |B|) × ((h, t) : C ptA b) × (q : b = ptB) × (h reflptA = q) ≃ (h : ΩA →pt ΩB) × (p : ptA = ptA) → (f = F(|h|, p)), by first adding two redundant singleton fields, and then removing another pair of singleton fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' One can prove G(h) : h = F(|h|, reflptA) using unit laws, so we further have (b : |B|) × C ptA b ≃ (t : (p : ptA = ptA) → f = F(|f|, p)) × (t reflptA = G(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' This is the type of pointed sections of a pointed type family over ptA = ptA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The fibres are identity types in ΩA →pt ΩB, which is (n+m+1)-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Since the fibres are (n+m)-truncated and the base ptA = ptA is (n − 1)-connected, the type of pointed sections is m-truncated as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' □ Corollary 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If |A| is n-connected and |B| is 2n-truncated, then Ω is an equivalence (A →pt B) ≃ (ΩA →pt ΩB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If |A| is n-connected and |B| is (2n+1)-truncated, then Ω identifies A →pt B with the subtype of ΩA →pt ΩB consisting of f : ΩA →pt ΩB such that (b : |B|) × C ptA b, which is logically equivalent to C ptA ptB, and hence to (p q : ptA = ptA) → f(p q) = f(p) f(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Applications In homotopy type theory, we define the ordinary cohomology group Hn(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' G) of a type X with coefficients in a an abelian group G as the set-truncation ∥X → K(G, n)∥0, where K(G, n) is an Eilenberg–MacLane space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The algebraic structure of these cohomology groups comes from various operations at the level of Eilenberg–MacLane spaces, which we now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' K(G, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Let G be a group, so that in particular G is a 0-truncated type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' One can define a 0-connected pointed type K(G, 1) : Upt with ΩK(G, 1) ≃grp G as a type of torsors, similar to our TX [Bez+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Note that K(G, 1) is necessarily 1-truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' By Corollary 11, we have that if B is 1-truncated, then (K(G, 1) →pt B) ≃ (G →grp ΩB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' we think of this as an elimination principle for K(G, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' From this elimination principle, it follows that if X : Upt is another 0-connected, 1-truncated pointed type, then (K(G, 1) ≃pt X) ≃ (G ≃grp ΩX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' When can we find K(G, 2) : Upt with ΩK(G, 2) ≃pt K(G, 1)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' By Corollary 7, it suffices to have (µ : (x : |K(G, 1)|) → K(G, 1) ≃pt (|K(G, 1)|, x)) × (µ pt = id), or equivalently (µ : (x : |K(G, 1)|) → G ∼=grp (x = x)) × (µ pt = id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Given a dependent elimination principle for K(G, 1), we could analyse this type of pointed sections directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Alternatively, we can think of pointed sections as pointed maps into a Σ-type with extra structure, and apply our non-dependent elimination principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The loop space of the Σ-type (x : |K(G, 1)|) × G ∼=grp (x = x) is the centre Z(G) of G, and so we are left to ask when the inclusion Z(G) →grp G has a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' This happens precisely when G is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' So K(G, 1) has a delooping if and only if G is abelian, in which case the delooping is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' As soon as we have EILENBERG–MACLANE SPACES AND STABILISATION IN HOMOTOPY TYPE THEORY 5 K(G, 2), Corollary 6 gives K(G, n) : Upt for every n with ΩK(G, n+1) ≃pt K(G, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We also get an elimination principle by repeated application of Corollary 10: for any n ≥ 1 and any n-truncated type B, we have (K(G, n) →pt B) ≃ (G →grp ΩnB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' One can check, combining the definition of TX with the elimination principle for K(G, n), that for n ≥ 0 we have K(G, n + 2) ≃ (Y : U) × n−connected(Y ) × (y : Y ) → G ≃grp Ωn+1(Y, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' πn(Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' While we have systematically avoided talking about higher inductive types, we can still say something about them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Recall that the n-sphere Sn : Upt is defined as a pointed type with (Sn →pt B) ≃ ΩnB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' If B is n-truncated for n ≥ 1, we have ΩnB ≃ (Z →grp ΩnB), since Z is the free group on one generator, which as we’ve seen is equivalent to K(Z, n) →pt B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' By the Yoneda lemma, we get that K(Z, n) is the n-truncation of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In particular, πn(Sn) ≃grp Ωn(∥Sn∥n) ≃grp Ωn(K(Z, n)) ≃grp Z, and πk(Sn) = 0 for k < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' More generally, this argument shows that A ≃pt ∥ΣΩA∥2n when A is n-connected and 2n- truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Applying the same fact to the delooping TA of A, we get that TA ≃pt ∥ΣA∥2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Taking loop spaces of both sidse, we get A ≃pt ∥ΩΣA∥2n, which is part of the Freudenthal suspen- sion theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The cup product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We now give some sketches on how to define cohomology operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Given a bilinear map L →grp M →grp N, we define a cup-product ⌣ : K(L, n) →pt K(M, m) →pt K(N, n + m), similar to the definition in [BLM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Note that we ask for the cup product to respect pointing, corresponding to 0 ⌣ y = x ⌣ 0 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' without this extra piece of specification, the definition would not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Indeed, K(M, m) →pt K(N, n + m) is n-truncated, so the elimination principle applies:3 K(L, n) →pt K(M, m) →pt K(N, n + m) ≃ L →grp Ωn(K(M, m) →pt K(N, n + m)) ≃ L →grp K(M, m) →pt ΩnK(N, m + n) ≃ L →grp K(M, m) →pt K(N, m) ≃ L →grp M →grp N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The forward maps in this composite are given explicitly by iterated looping, so we arrive at a definition of the cup product as the unique bi-pointed map whose looping gives back the bilinear map we started with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' With this characterisation, we expect that algebraic properties of the cup product follow from analogous properties of looping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' For example, one can prove that the following square anti-commutes, and this corresponds to graded commutativity of the cup product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' A →pt B →pt C ΩA →pt B →pt ΩC A →pt ΩB →pt ΩC ΩA →pt ΩB →pt Ω2C 3Formally this argument assumes m, n ≥ 1, but it can be adapted to cover all m, n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 6 DAVID W¨ARN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Steenrod squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Let us now use Corollary 11 to construct Steenrod squares as ‘stable cohomology operations’ Sqi n : K(Z/2, n) →pt K(Z/2, n + i) with Sqi n corresponding to ΩSqi n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' We first define Sqi i as the cup product square x �→ x ⌣ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' To deloop this to Sqi i+1, we need to show (x + y) ⌣ (x + y) = x ⌣ x + y ⌣ y, which follows from distributivity and graded commutativity since we are working mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Given Sqi i+1 we can define Sqi n for all n using Corollary 10, by looping and delooping as appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Concluding remarks Our Lemma 8 can be compared with the construction of functors out of a Rezk comple- tion in [Uni13, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='4] and the construction of maps K(G, 1) →pt K(H, 1) in [Bez+, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Variants of the relation C a b are used in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The idea can be understood as a type-theoretic analogue of the arguments in [Del91, Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The arguments in this note are well-suited to formalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Indeed, many parts have already been formalised twice: first by Louise Leclerc [Lec22], and later by Axel Ljungstr¨om in order to develop the theory of Steenrod squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In upcoming work, we take the ideas of this note much further to give an exact, infinitary de- scription of higher groups – as well as higher equivalence relations more generally – and morphisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In fact the description of morphisms is in a precise sense obtained mechanically from the descriptions of objects, explaining the similarity between the second and third sections of this note (compare for example Corollaries 5, 6, 7 with Corollaries 9, 10, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' References [BDR18] Ulrik Buchholtz, Floris van Doorn, and Egbert Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' “Higher groups in homotopy type theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 205–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [Bez+] Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, and Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Grayson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='com/UniMath/SymmetryBook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [BLM22] Guillaume Brunerie, Axel Ljungstr¨om, and Anders M¨ortberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' “Synthetic cohomology theory in cubical agda”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In: 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [Del91] Pierre Deligne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' “Le symbole mod´er´e”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In: Publications Math´ematiques de l’Institut des Hautes ´Etudes Scientifiques 73 (1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 147–181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [Lec22] Louise Leclerc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' A formalisation of Eilenberg-Maclane spaces using HoTT-Agda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='com/luyise/EM-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [LF14] Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Licata and Eric Finster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' “Eilenberg-MacLane Spaces in Homotopy Type The- ory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' In: Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Con- ference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' CSL-LICS ’14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Vienna, Austria: As- sociation for Computing Machinery, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' isbn: 9781450328869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1145/2603088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='2603153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1145/2603088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='2603153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [Rij17] Egbert Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' The join construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='07538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='org/abs/1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='07538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' [Rij22] Egbert Rijke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Introduction to Homotopy Type Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='11082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='11082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' REFERENCES 7 [Uni13] The Univalent Foundations Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Homotopy Type Theory: Univalent Foundations of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Institute for Advanced Study: https://homotopytypetheory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='org/book, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content=' Email address: warnd@chalmers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} +page_content='se' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfIwbb/content/2301.03685v1.pdf'} diff --git 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sha256:41887d5f1c250d56d6e5502a249a2971028651d1d1e56609557e3f6a27701ad0 +size 56980 diff --git a/o9A0T4oBgHgl3EQfKP9c/content/tmp_files/2301.02101v1.pdf.txt b/o9A0T4oBgHgl3EQfKP9c/content/tmp_files/2301.02101v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0c7d38841e6298af2a3ec21f194a6f39ee6696f --- /dev/null +++ b/o9A0T4oBgHgl3EQfKP9c/content/tmp_files/2301.02101v1.pdf.txt @@ -0,0 +1,968 @@ +MNRAS 00, 1 (2022) +https://doi.org/10.1093/mnras/stac3807 +Disco v ery of a resolved white dwarf–brown dwarf binary with a small +projected separation: SDSS J222551.65 + 001637.7AB +Jenni R. French , 1 ‹ Sarah L. Casewell, 1 Trent J. Dupuy, 2 John H. Debes, 3 Elena Manjavacas, 3 +Emily C. Martin 4 and Siyi Xu( �� � ) +5 +1 School of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom +2 Royal Observatory Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, United Kingdom +3 AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, Maryland 21218, USA +4 Department of Astronomy & Astrophysics, University of California Santa Cruz, 1156 High Street, Santa Cruz, California 95064, USA +5 Gemini Observatory/NSF’s NOIRLab, 670 N. A’ohoku Place, Hilo, Hawaii, 96720, USA +Accepted 2022 December 21. Received 2022 December 20; in original form 2022 October 26 +A B S T R A C T +We present the confirmation of SDSS J222551.65 + 001637.7AB as a closely separated, resolved, white dwarf–brown dwarf +binary. We have obtained spectroscopy from GNIRS and seeing-limited K s -band imaging from NIRI on Gemini North. The +target is spatially resolved into its constituent components: a 10926 ± 246 K white dwarf, with log g = 8.214 ± 0.168 and +a mass of 0.66 + 0 . 11 +−0 . 06 M ⊙, and an L4 brown dwarf companion, which are separated by 0.9498 ± 0.0022 arcsec. We derive the +fundamental properties of the companion from the Sonora–Bobcat evolutionary models, finding a mass of 25–53 M Jup and a +radius of 0.101–0.128 R ⊙ for the brown dwarf, at a confidence level of 1 σ. We use WDWARFDATE to determine the age of the +binary as 1 . 97 + 4 . 41 +−0 . 76 Gyr. A kinematic analysis shows that this binary is likely a member of the thick disc. The distance to the +binary is 218 + 14 +−13 pc, and hence the projected separation of the binary is 207 + 13 +−12 au. Whilst the white dwarf progenitor was on the +main sequence the binary separation would have been 69 ± 5 au. SDSS J222551.65 + 001637.7AB is the third closest spatially +resolved white dwarf–brown dwarf binary after GD 165AB and PHL 5038AB. +K ey words: bro wn dwarfs – white dwarfs – binaries. +1 INTRODUCTION +The prolific disco v eries of field brown dwarfs hav e been enabled by +infrared (IR) and optical surv e ys such as the Two Micron All Sky +Surv e y (2MASS), UKIRT Infrared Deep Sky Survey (UKIDSS), +and Wide-field Infrared Surv e y Explorer (WISE) (e.g. Kirkpatrick +et al. 1998 ; Pinfield et al. 2008 ; Aberasturi, Solano & Mart ´ın 2011 ). +Despite o v er a thousand known field brown dwarfs, it is often +challenging to determine their masses, ages, and luminosities as +these parameters are degenerate due to the lack of fusion in the +cores of brown dwarfs (Burrows & Liebert 1993 ; Auddy, Basu & +Valluri 2016 ). Consequently, estimating brown dwarf parameters +relies on models which, despite recent impro v ements, are sensitiv e +to atmospheric processes that remain poorly understood, such as +non-equilibrium chemistry (Tremblin et al. 2015 ) and the formation +and presence of atmospheric clouds (Morley et al. 2014 ), and any +resultant variability. +‘Benchmark’ brown dwarfs, those for which physical proper- +ties can be independently determined, can test and impro v e the +current evolutidaonary and atmospheric models for brown dwarfs. +Double-lined eclipsing brown dwarfs are ideal benchmark ob- +jects, but they are extremely rare, with only two known to date, +⋆ E-mail: jf328@leicester.ac.uk +2MASS J05352184-0546085 (Stassun, Mathieu & Valenti 2006 ) +and 2MASSW J1510478-281817 (Triaud et al. 2020 ). Since binary +systems are formed from the same matter, companions can be +assumed to have the same ages and metallicities as their primary +stars (e.g. Moe & Di Stef ano 2017 ). Therefore, brown dw arfs that +are members of a binary system can have their physical properties +estimated with evolutionary models using the age constraints of the +primary (Pinfield et al. 2006 ), provided that the primary’s age is well +calibrated. These brown dwarfs can then be used as benchmarks to +test and refine current brown dwarf models. +Widely separated binaries with a white dwarf primary and a +brown dwarf companion are particularly valuable for identifying +benchmark brown dwarfs. The white dwarf cooling age provides a +lower limit on the age of the binary, which can be used to calculate +the physical properties of the brown dwarf companion (Fontaine, +Brassard & Bergeron 2001 ). Where the binary system is resolvable, +meaning it is wide ( ≳ 50 au) or a common proper motion pair +(ultra-wide, ≳ 500 au, Zhang et al. 2020 ), it is unlikely that either +the brown dwarf was affected during the evolution of the white +dwarf progenitor, or that the white dwarf evolution was truncated +by the presence of a brown dwarf companion (Meisner et al. 2020 ). +Close white dwarf–brown dwarf binaries are not appropriate for +identifying benchmark brown dwarfs because they undergo a period +of common-envelope evolution instead of each component evolving +separately (e.g. Casewell et al. 2018 ). Wide white dwarf–brown +© 2022 The Author(s) +Published by Oxford University Press on behalf of Royal Astronomical Society + +2 +J. R. French et al. +MNRAS 00, 1 (2022) +Table 1. Parameters of the brown dwarf companions in the confirmed wide, +comoving white dwarf–brown dwarf binaries. +System +Spectral +Type +Separation +(au) +Total Age +(Gyr) +Reference +GD 165AB +L4 +123 ± 12 +1.2–5.5 +1, 2, 3 +PHL 5038AB +L8 +69 ± 1 +1.9–2.7 +4, 5 +LSPM 1459 + 0857AB +T4.5 +16500- +26500 +> 4.8 +5, 6 +WD 0806-661AB +Y1 +2504 ± 4 +1.5–2.7 +5, 7, 8 +LSPM J0241 + 2553AB +L1 +2380 ± 36 +< 10 +5, 9 +COCONUTS-1AB +T4 +1290 ± 13 +7 . 3 + 2 . 8 +−1 . 6 +5, 10 +LSPM J0055 + 5948AB +T8 +402 ± 2 +10 ± 3 +5, 11 +Note. References are 1: Becklin & Zuckerman ( 1988 ); 2: Kirkpatrick, Henry +& Liebert ( 1993 ); 3: Kirkpatrick et al. ( 1999 ); 4: Steele et al. ( 2009 ); 5: Gaia +Collaboration ( 2020 ); 6: Day-Jones et al. ( 2011 ); 7: Luhman, Burgasser & +Bochanski ( 2011 ); 8: Rodriguez et al. ( 2011 ), 9: Deacon et al. ( 2014 ); 10: +Zhang et al. ( 2020 ); 11: Meisner et al. ( 2020 ). +dwarf binaries are thus ideal systems to identify benchmark brown +dwarfs to impro v e model calibrations and flag for follow-up with +instruments such as NIRSpec on JWST . +Despite several all-sky surveys that have identified thousands of +white dwarfs (e.g. Girven et al. 2011 ; Gentile Fusillo et al. 2019 ), +and studies searching for ultracool companions to white dwarfs (e.g. +Steele et al. 2010 ; Debes et al. 2011 ; Hogg et al. 2020 ), only ∼0.1– +0.5 percent of white dwarfs are predicted to have a brown dwarf +companion (Farihi, Becklin & Zuckerman 2005 ; Steele et al. 2011 ; +Rebassa-Mansergas et al. 2019 ). There are currently only 7 known +wide, comoving white dwarf–brown dwarf binaries, which are listed +in Table 1 . Their companion spectral types range from L1 to Y1 +with separations ranging from 69 au to 16500–26500 au. The larger +separations correspond to T-dwarf companions but this is likely a +selection effect. In addition to these 7 confirmed white dwarf–brown +dwarf binaries, there are also several candidates that have been identi- +fied using photometry b ut ha ve not been spectroscopically confirmed +(e.g. Kiwy et al. 2022 : NSC J053232.31-512450.75AB, WD + L3, +a = 508 au; NSC J130527.23-224728.44AB, WD + L3, a = 716 au), +or that need more-IR data to confirm them as white dwarf–brown +dwarf binaries (e.g. Meisner et al. 2020 : LSR J0002 + 6357AB, +WD + mid/late T, a = 8695 au). +In this paper, we present a new white dwarf–brown dwarf binary, +SDSS J222551.65 + 001637.7AB, which hosts an important bench- +mark brown dwarf. SDSS J222551.65 + 001637.7AB is the third +closest separated resolved white dwarf–brown dwarf binary after +GD 165AB and PHL 5038AB and understanding its evolution will +provide insights into the formation of these rare wide white dwarf– +brown dwarf binaries. Section 2 discusses this system, Section 3 +describes our observations and data reduction, Section 4 presents +our results, and Section 5 discusses our analysis. +2 SDSS J22255 1.65 + 001637.7AB +SDSS +J222551.65 + 001637.7AB +(henceforth +SDSS J22255 + 0016AB) was first reported by Eisenstein et al. +( 2006 ) where SDSS J22255 + 0016A was identified as a hydrogen- +rich DA white dwarf as part of a catalogue of spectroscopically +confirmed white dwarfs from the Sloan Digital Sk y Surv e y (SDSS). +IR observations were taken as part of the UKIDSS Large Area +Surv e y (La wrence et al. 2007 ). Girv en et al. ( 2011 ) identified +SDSS J22255 + 0016AB as having a potential photometric near-IR +excess, with Steele et al. ( 2011 ) suggesting the excess is indicative of +Figure 1. SOFI K -band image of SDSS J222551.65 + 001637.7AB showing +the elongation of the central star before the components were resolvable. +Reproduced from fig. 11 of Steele et al. ( 2011 ). +a partially resolved L-dwarf companion. Fig. 1 , which is reproduced +from Steele et al. ( 2011 ), shows that in the SOFI K -band image +SDSS J22255 + 0016AB is considerably elongated with a full width +at half-maximum (FWHM) of 1.8 arcsec, suggesting a predicted +separation of a < 350 au at their adopted distance of 190 ± 20 pc. +Eisenstein et al. ( 2006 ) fit the SDSS spectrum of +SDSS J22255 + 0016A to a grid of model atmospheres, determining +T eff = 10640 ± 94 K and log g = 8.16 ± 0.09. Anguiano et al. ( 2017 ) +refined these measurements to T eff = 10926 ± 246 K and log g = +8.214 ± 0.168, with a derived distance of 226 ± 41 pc using updated +model spectra and a correction for the 3D dependence of convection. +Gentile Fusillo et al. ( 2021 ) performed a photometric fit to the Gaia +eDR3 magnitudes obtaining T eff = 9370 ± 765 K and log g = 7.738 +± 0.276. This method of fitting is less sensitive to log g and relies +on the Gaia magnitudes alone. Jim ´enez-Esteban et al. ( 2018 ) also +performed a photometric fit of the white dwarf, and they found T eff = +9000 ± 125 K and log g = 6.5 ± 0.25 when considering the spectral +energy distribution from the SDSS u -band to the UKIDSS K -band. +Although these two photometric fits yield white dwarf parameters +that are consistent with each other, neither of these fits account for +extinction. Jim ´enez-Esteban et al. ( 2018 ) state that for objects at +∼200 pc, extinction should be accounted for. SDSS J22255 + 0016A +is at a distance of 218 + 14 +−13 pc, and has an extinction of A g = 0.323 +(Eisenstein et al. 2006 ), which is significant and should thus be +considered. +Fig. 2 depicts a comparison of hydrogen-rich DA white dwarf +models (Koester 2010 ) for both the derived spectroscopic and photo- +metric parameters, alongside dereddened photometry measurements, +and the SDSS spectrum of SDSS J22255 + 0016A. Considering the +dereddened photometry, the white dwarf model that uses spectro- +scopic parameters ( T eff = 11000 K, log g = 8.25) is consistent with +the data, whereas the white dwarf model which uses photometric +parameters ( T eff = 9250 K, log g = 7.75) is not. This is particularly +evident at shorter wavelengths, where the photometry measurements +are unlikely to be influenced by the flux of the brown dwarf. The +residuals in Fig. 2 for the photometric parameters are 12 percent +higher than those for the spectroscopic parameters. The white dwarf +model using spectroscopic parameters also better fits the depth of the +hydrogen lines. +Our comparison of two Koester DA white dwarf models with +the dereddened photometry measurements indicates that the spectro- + +A close resolved white dwarf–brown dwarf binary +3 +MNRAS 00, 1 (2022) +Figure 2. SDSS spectrum of SDSS J222551.65 + 001637.7A with dered- +dened SDSS, PAN-STARRS and UKIDSS photometry (blue). The Koester +DA white dwarf model with T eff = 9250 K and log g = 7.75, deriving from +the photometric fits, is shown in red. The Koester DA white dwarf model +with T eff = 11000 K and log g = 8.25, deriving from the spectroscopic fits, is +shown in light blue. The residuals between the SDSS spectrum and the white +dwarf models are shown in the bottom panel. +scopic parameters derived by Anguiano et al. ( 2017 ) are the most +appropriate. Furthermore, Groenewegen ( 2020 ) found that effective +temperatures calculated from photometric fits were consistently +underestimated compared to those derived from spectroscopy, with +discrepancies of 2000 K in some cases. Since the Anguiano et al. +( 2017 ) parameters are derived from a spectroscopic fit, are consistent +with the photometry and SDSS spectrum of SDSS J22255 + 0016A, +and their derived distance of 226 ± 41 pc is within 1 σ of the Gaia +eDR3 distance of 218 + 14 +−13 pc, we adopt these parameters of T eff = +10926 ± 246 K and log g = 8.214 ± 0.168 for the white dwarf. +Although we start to see the contributions of the brown dwarf from +the r -band, the brown dwarf emits mainly in the IR, and thus will not +contribute significantly to the measured T eff and log g of the white +dwarf. +3 OBSERVATIONS AND DATA REDUCTION +3.1 GNIRS spectroscopy +We observed SDSS J22255 + 0016AB using the cross-dispersed spec- +trograph GNIRS on Gemini North (Elias et al. 2006 ) on 2020 July 8th +and July 10th UTC, as a part of programme GN-2020A-Q-322 (PI: +John H. Debes). Spectra were taken using the short blue camera with +the 32 l/mm grating and a slit width of 1.0 arcsec, giving a resolution +of ( λ / �λ) ∼ 500 across the entire wavelength range of 0.8–2.5 µm. +We nodded the observations, taking 300 s exposures at each nod +point, totalling 20 exposures. Arc lamp and flat-field calibration +frames were taken immediately after the science observations. All +exposures across three individual hours of observation were then +combined during data reduction. Both the white dwarf and the brown +dwarf were in the slit during our observations. The data were then +reduced using a version of SPEXTOOL 4.1 (Cushing, Vacca & Rayner +2004 ) which has been adapted for use with GNIRS data (K. Allers, +pri v ate communication). Two A0V standard stars, HIP115119 and +HIP110963, were observed using the same settings with four frames +taken for each hour of observation with an exposure time of 1.0 s. +Two stars were observed to ensure that for each hour of science +Figure +3. GNIRS +H -band +acquisition +image +of +SDSS J222551.65 + 001637.7AB showing the resolved white dwarf +and brown dwarf components of the binary. The position angle of the +acquisition image is 144.49 ◦. Both the white dwarf and the brown dwarf +were in the slit during our observations. +Figure 4. NIRI K s -band image of SDSS J222551.65 + 001637.7AB showing +we have clearly resolved the white and brown dwarf components of the binary. +A is the white dwarf, and B is the brown dwarf. +observation, the standard star observed was at a similar airmass. +The telluric correction was performed during reduction using the +XTELLCOR package (Vacca, Cushing & Rayner 2003 ) in SPEXTOOL +using our observed standard stars. +The acquisition images from GNIRS revealed that the system +is in fact spatially resolved (Fig. 3 ). The images were taken with +3 s exposures in the H -band using the same settings as the science +observations. +3.2 NIRI imaging +We imaged SDSS J22255 + 0016AB on the 2021 June 13th UTC with +NIRI in the K s -band as part of programme GN-2021A-FT-207 (PI: +Elena Manjavacas). We obtained 13 60 s exposures at airmass 1.09 +with the f/6 camera providing a pixel scale of 117.1 mas pixel −1 . + +N +PA = 144.49° +BN +E +A +B4 +J. R. French et al. +MNRAS 00, 1 (2022) +Figure 5. GNIRS spectrum of SDSS J222551.65 + 001637.7AB with SDSS +and UKIDSS photometry (blue) and the Koester DA white dwarf model. +The combined white dwarf + L3 template spectra, combined white dwarf + L4 +template spectra and combined white dwarf + L5 template spectra are also +shown. The residual spectra for L3–L5 spectral types are shown in the bottom +panel. The GNIRS spectrum has been binned to a resolution of 1.4 Å to +increase signal to noise, and the telluric line-dominated section between +18000 Å and 19000 Å has been masked for clarity. +We reduced the data using the DRAGONS software (Labrie et al. +2019 ) provided by the Gemini observatory, reducing using flat-field +and dark frames provided as part of the calibration set and creating a +bad pixel mask using 10 s dark frames. The 13 images were reduced +and stacked using stars in the image as references. This is depicted +in Fig 4. and shows that we have clearly resolved the white dwarf +and brown dwarf components. +4 RESULTS +4.1 Spectral type +Although SDSS J22255 + 0016AB is resolved in the acquisition +image, both of the components were within the slit during obser- +vations. To confirm the spectral type of the secondary, we created +composite DA white dwarf + cool L-dwarf templates similar to +those in Steele et al. ( 2011 ) and Casewell, Geier & Lodieu ( 2017 ). +We used a hydrogen-rich DA white dwarf model (Koester 2010 ) +with T eff = 11 000 K and log g = 8.25 to best match the parameters +of SDSS J22255 + 0016A. We took L-dwarf template spectra from +the SpeX Prism Library (Burgasser 2014 ) for spectral types L3-L5, +which are reported in Burgasser et al. ( 2010 ). We combined the +white dwarf model with the L-dwarf template spectra by setting both +to 10 pc and then combining them. To set the white dwarf model to +10 pc, we normalized using the Gaia eDR3 distance of 218 + 14 +−13 pc and +the broad-band photometry measurement in the SDSS r -band (Alam +et al. 2015 ). To normalize the brown dwarf template spectra to 10 pc, +we used the mean absolute J -band magnitudes for each spectral +type from Dupuy & Liu ( 2012 ). Once combined, we normalized +the composite white dwarf + L-dwarf models to the broad-band r - +band photometry, and we normalized the GNIRS spectrum to the +broad-band UKIDSS J -band photometry (Girven et al. 2011 ). We +then compare the GNIRS spectrum to the composite white dwarf ++ L-dwarf models to determine the presence of an IR excess and +the nature of the companion. Fig. 5 shows the GNIRS spectrum for +SDSS J22255 + 0016AB alongside photometry measurements and +our composite white dwarf + brown dwarf models. From this, we +determine the spectral type of the brown dwarf as L4 ± 1. +4.2 Relati v e astrometry +From our stacked NIRI image (Fig. 4) , we measured relative +astrometry for SDSS J22255 + 0016AB. We fitted an analytic point +spread function model to each component, where the model used +three concentric 2D Gaussians with different amplitudes, standard +deviations, ellipticities, and angles for the ellipticities. This approach +is based on previous work with adaptive optics imaging of low-mass +binaries (e.g. Liu et al. 2006 ; Mann et al. 2019 ). We converted the +pixel positions of the two components into sky coordinates using +the WCS information in the FITS header. Given how well resolved +SDSS J22255 + 0016AB is, the errors on its relative astrometry are +dominated by the astrometric calibration of NIRI, with a fractional +uncertainty of 0.23 percent in pixel scale and an uncertainty of 0.1 ◦ in +parallactic angle (Mann et al. 2019 ). This results in a separation of +949.8 ± 2.2 mas and a parallactic angle of 194.6 ± 0.1 ◦ between the +two components, measured as the position of B from A. Our binary +fit also provides a measurement of the relative photometry of K s , B − +K s , A = −0.99 ± 0.02 mag. Here, we have defined the more massive +white dwarf as the A component and its lower mass ultracool dwarf +companion, which is brighter in the K s -band, as the B component. +We do not deri ve relati ve astrometry from the GNIRS acquisition +images, as it is not astrometrically well-calibrated. We note, ho we ver, +that there does not appear to be any astrometric motion relative to +the NIRI imaging. As the two images were taken 1 year apart, this is +not surprising. +The two binary components are separated by 0.9498 ± 0.0022 +arcsec and a parallactic angle of 194.6 ± 0.1 ◦ and have a magnitude +difference of 0.990 mags in the K s -band. This magnitude difference +is consistent with the difference in absolute magnitudes of the two +components as predicted by the models of Tremblay, Bergeron & +Gianninas ( 2011 ) for DA white dwarfs and the absolute magnitude +spectral type relations of Dupuy & Liu ( 2012 ) for an L4 brown dwarf. +Using the Gaia eDR3 distance of 218 + 14 +−13 pc and our separation of +0.9498 ± 0.0022 arcsec, we calculate the projected separation of +SDSS J22255 + 0016AB as 207 + 13 +−12 au. +5 DISCUSSION +We determined the spectral type of SDSS J22255 + 0016B as L4 ±1 +by comparing template brown dwarf spectra to the GNIRS spectrum. +This spectral type is consistent with the difference in magnitude for +the components measured from the NIRI image. Although the K - +band photometry and spectrum are brighter than that of the white +dwarf + L4 combined model, it is consistent within the errors, +which are dominated by the absolute magnitudes in Dupuy & Liu +( 2012 ). The offset between the models and the spectrum at 10000 Å +is due to the SpeX template dwarf spectra not extending very far into +optical wavelengths. Our spectral type is consistent with the L-dwarf +companion proposed by Steele et al. ( 2011 ) based on the UKIDSS +photometry. Our projected separation of 207 + 13 +−12 au agrees with their +prediction of a < 350 au. +5.1 Age of the system +To determine the age of SDSS J22255 + 0016AB, we used WDWARF- +DATE , which estimates the age of a white dwarf, as well as its +final mass and initial mass, from T eff and log g using a Bayesian +framework (Kiman et al. 2022 ). The cooling age and mass of the + +A close resolved white dwarf–brown dwarf binary +5 +MNRAS 00, 1 (2022) +Table 2. System parameters for SDSS J222551.65 + 001637.7AB derived +using WDWARFDATE . +Parameter +Value +Cooling Age (Gyr) +0 . 58 + 0 . 17 +−0 . 08 +Final Mass (M ⊙) +0 . 66 + 0 . 11 +−0 . 06 +Initial Mass (M ⊙) +1 . 97 + 1 . 14 +−0 . 76 +Main Sequence Age (Gyr) +1 . 40 + 4 . 48 +−0 . 98 +Total System Age (Gyr) +1 . 97 + 4 . 41 +−0 . 76 +white dwarf are determined from the evolutionary models of the +Montreal White Dwarf Group (B ´edard et al. 2020 ), and an initial- +final mass relationship is used to calculate the initial mass of the white +dwarf progenitor. The progenitor lifetime, also referred to as the main +sequence (MS) age, is then determined using the MIST isochrones +(Choi et al. 2016 ; Dotter 2016 ). The total lifetime of the white dwarf is +calculated as the sum of the cooling age and the progenitor’s MS age +(T able 2 ). W e utilized the initial-final mass relationship of Cummings +et al. ( 2018 ) and assumed solar metallicity and v / v crit = 0 for the fit, +where v / v crit quantifies stellar rotation (Sun et al. 2021 ). The white +dwarf mass of 0 . 66 + 0 . 11 +−0 . 06 M ⊙ is within 1 σ of the Anguiano et al. +( 2017 ) white dwarf mass of 0 . 72 + 0 . 10 +−0 . 10 M ⊙. Our cooling age of the +white dwarf gives the minimum age of the system as 0 . 58 + 0 . 17 +−0 . 08 Gyr. +We estimate the total age of the system as 1 . 97 + 4 . 41 +−0 . 76 Gyr; ho we ver, +this value is particularly sensitive to uncertainties in the choice of +initial-final mass relationship and the MS age of the white dwarf +progenitor. +To further constrain the age of SDSS J22255 + 0016AB, we used +the Gaia eDR3 proper motions and the radial velocity measured +by Anguiano et al. ( 2017 ) to undertake a kinematic analysis. +We calculate the UVW space velocities with respect to the local +standard of rest as: U = −9.52 ± 7 km s −1 , V = 54.5 ± 13 km s −1 , +W = −71.8 ± 15 km s −1 . Here, U is positive towards the Galactic cen- +tre, V is positive in the direction of Galactic rotation, and W is positive +towards the North Galactic Pole. Following the method of Bensby, +Feltzing & Oey ( 2014 ) with their observed fractions of thick disc, thin +disc, and halo populations in the solar neighbourhood, we determine +the relative probabilities for SDSS J22255 + 0016AB belonging to +each of these populations. We find that SDSS J22255 + 0016AB is +495 times more likely to belong to the thick disc than the thin disc +and 461 times more likely to belong to the thick disc than the stellar +halo. It is thus likely that SDSS J22255 + 0016AB is a member of the +thick disc. The thick disc has an age of ∼10 Gyr (Kilic et al. 2017 ), +meaning that if SDSS J22255 + 0016AB is indeed a member of the +thick disc, the total system age is likely closer to the upper uncertainty +of the age we determine with WDWARFDATE . In their analysis of white +dwarfs in the thin and thick discs, Raddi et al. ( 2022 ) find that the total +age distribution of white dwarfs peaks at 2 Gyr, which may explain +why the total age of SDSS J22255 + 0016AB is young for a thick +disc object. Additionally, Torres et al. ( 2021 ) find that 13 percent of +halo white dwarfs in Gaia DR2 are younger than expected compared +to the average halo white dwarf age. This indicates the presence of +younger white dwarfs in both disc and halo populations, of which +SDSS J22255 + 0016A may be one; ho we ver, the origin of these +younger objects is unclear. We note that older age is derived if we +use the photometric parameters of the white dwarf; ho we ver, this has +larger uncertainties, and the photometric parameters are less reliable +due to their lack of reddening. Despite the large uncertainty in total +age, which is dominated by uncertainties in the initial-final mass +Table 3. Absolute and apparent magnitudes for each component of +SDSS J222551.65 + 001637.7AB. +Star +Absolute K s +Magnitude +Apparent K s +Magnitude +White Dwarf +12.26 ± 0.20 +18.96 ± 0.15 +Brown Dwarf +11.27 ± 0.18 +17.97 ± 0.13 +relationship, SDSS J22255 + 0016B is an important member in the +small population of wide white dwarf–brown dwarf binaries. +As a member of the thick disc, it is unlikely that +SDSS J22255 + 0016AB is extremely metal-poor in comparison +to objects residing in the stellar halo. There is no evidence of +photospheric metal pollution in the SDSS optical spectrum of the +white dwarf that would indicate accretion from a tidally disrupted +asteroid or another companion (Zuckerman et al. 2003 ; Debes 2006 ). +SDSS J22255 + 0016A has a low effective temperature, and if it were +polluted, absorption features would be easily detectable in the optical +spectrum. Since we do not detect any pollution, it is thus likely that +SDSS J22255 + 0016AB has no other companions. Ho we ver, we note +that the Ca II line can appear weak in white dwarf spectra, and a high +resolution echelle spectrum of the white dwarf would be required to +place definitive limits on any potential pollution (Zuckerman et al. +2003 ). +5.2 SDSS J222551.65 + 001637.7B +As discussed in Section 4.2 , our observed absolute magnitude +difference between the white dwarf and the brown dwarf is consistent +with predictions from theoretical models. Using this magnitude +difference and taking our observed UKIDSS K -band magnitude of +17.6 ± 0.13 as a proxy for the observed K s -band magnitude, we +calculate the apparent magnitudes of both the white dwarf and the +brown dwarf. Using the Gaia distance of 218 + 14 +−13 pc, we calculate +the absolute magnitudes in the K s -band for both the white dwarf +and the brown dwarf. These magnitudes are presented in Table 3 . +We find that absolute K s -band magnitude of the brown dwarf is +11.27 ± 0.18. This is consistent with the mean absolute K s -band +magnitude of 11.55 ± 0.28, which Dupuy & Liu ( 2012 ) report for +L4 companions. Our absolute K s -band magnitude for the white dwarf +is 12.26 ± 0.18. This is consistent with that predicted by synthetic +photometry calculated from the Tremblay et al. ( 2011 ) white dwarf +models (Holberg & Bergeron 2006 ; Kowalski & Saumon 2006 ). 1 +The estimated L4 companion spectral type provides a consis- +tent theoretical and observed absolute magnitude, indicating that +SDSS J22255 + 0016B is indeed an L4 ± 1 companion. We esti- +mate the ef fecti ve temperature of SDSS J22255 + 0016B as T eff = +1800 + 70 +−60 K for our spectral type of L4 ± 1 as this is the mean ef fecti ve +temperature of an L4 dwarf determined from the analysis of M, L, +and T dwarfs performed by Vrba et al. ( 2004 ). We then compare +our estimated ef fecti ve temperature and our K s -band magnitude for +an L4 spectral type to the Sonora–Bobcat models, assuming solar +metallicity (Marley et al. 2021 ). From these models, a brown dwarf +with T eff = 1800 K and K s = 11.27 would have a mass of 43.88 M Jup +and a radius of 0.1071 R ⊙. This mass estimate is consistent with the +mass of 47 ± 3 M Jup determined by Steele et al. ( 2011 ) using the +Lyon group models. +With our K s -band magnitude, we use the relations of +Dupuy & Liu ( 2017 ) to calculate the bolometric luminosity of +1 http://www.astr o.umontr eal.ca/ ∼ber geron/CoolingModels + +6 +J. R. French et al. +MNRAS 00, 1 (2022) +SDSS J22255 + 0016B as log ( L bol /L ⊙) = −3.92 ± 0.11. We also +utilize their Lyon T eff relation to impro v e our temperature estimate +to T eff = 1817 ± 90 K. We compare our bolometric luminosity and +ef fecti ve temperature of the brown dwarf with the Sonora–Bobcat +models, assuming solar metallicity (Marley et al. 2021 ). From these +models, a brown dwarf with log ( L bol /L ⊙) = −3.92 ± 0.11 and +T eff = 1817 ± 90 K would have a mass of 25–53 M Jup and a radius +of 0.101–0.128 R ⊙. We find that the most appropriate model for +our bolometric luminosity and ef fecti v e temperature pro vides an +age estimate and a K s -band magnitude that are consistent with our +results. +5.3 Evolution of the system +During the evolution of the MS progenitor of SDSS J22255 + 0016A, +the orbital separation would have increased by a maximum factor of +M MS / M WD = 2.99 (Burleigh, Clarke & Hodgkin 2002 ). The initial +projected separation would therefore have been > 69 au, confirming +that this is not a post-common envelope binary. Burleigh et al. ( 2002 ) +state that white dwarfs will retain their planetary companions if the +initial separation from the MS progenitor star is > 5 au, as is the case +for SDSS J22255 + 0016AB. +Since the initial separation of SDSS J22255 + 0016AB is too +wide to be a post-common envelope system, it will have evolved +differently to close white dwarf–brown dwarf binaries (e.g. Maxted +et al. 2006 ; Casewell et al. 2018 ). The two components will have +evolved separately, and the brown dwarf will not have truncated +the white dwarf’s e volution. Ho we ver, the bro wn dwarf may have +been affected by stellar winds from the primary, with the angular +momentum lost by the white dwarf causing the separation to increase +(Schrøder et al. 2021 ). During the evolution of the MS progenitor +of the white dwarf, the star undergoes a phase of evolution on the +Asymptotic Giant Branch (AGB) before reaching its end stage as a +white dwarf (Iben & Renzini 1983 ). In the AGB phase, the mass- +loss increases until the envelope is fully ejected, which causes stellar +winds that can affect the substellar companion. Mayer et al. ( 2014 ) +found dust-enriched winds of v w = 5–20 km s −1 and H ¨ofner & +Olofsson ( 2018 ) report outflowing winds between v w = 3–30 km s −1 , +affecting companions at separations on the order of ∼100 au. It is +possible for the presence of the companion to shape these winds, +morphing spherical AGB stars into non-spherical planetary nebulae, +but at these wide separations ( ≳ 50 au), this is unlikely to alter the +white dwarf progenitor’s evolution (Decin et al. 2020 ). +5.4 Orbit +Using the white dwarf mass, the brown dwarf mass we esti- +mate from the Sonora–Bobcat models, and our projected orbital +separation of 207 + 13 +−12 au, we calculate the likely orbital period of +SDSS J22255 + 0016AB as P = 3560 ± 383 yr. This is a minimum +period assuming a circular orbit; ho we v er, man y brown dwarfs +are in eccentric orbits. Ma & Ge ( 2014 ) report that the eccentric- +ity distribution of brown dwarfs changes at a threshold mass of +42.5 M Jup , with brown dwarfs below this mass having eccentricities +similar to massive planets and brown dwarfs abo v e this mass having +eccentricities consistent with binaries. This suggests two distinct +formation mechanisms for brown dwarfs: protoplanetary discs and +stellar binary-like formation. SDSS J22255 + 0016B resides near +this mass boundary, and further investigations such as continuous +monitoring to calculate its dynamical mass and observations to +obtain an uncontaminated spectrum of the brown dwarf and a C/O +ratio measurement, would enable us to determine its formation +mechanism. The mass ratio of SDSS J22255 + 0016AB is q = +M BD / M MS = 0.012–0.048. Bowler, Blunt & Nielsen ( 2020 ) state +that for binary mass ratios > 0.01 stellar binary-like formation is +fa v oured, which also indicates a higher eccentricity than systems in +which the brown dwarf formed via planet-like formation. +Fig. 6 depicts the currently known white dw arf–brown dw arf bina- +ries as well as directly imaged brown dwarfs and exoplanets around +MS stars. The white dwarf–brown dwarf binaries are colour coded +according to the spectral type of the brown dwarf. The outlined star +represents SDSS J22255 + 0016AB in its evolved form as it is now. +The outlined pentagon represents SDSS J22255 + 0016AB whilst the +white dwarf progenitor was still on the MS, with M WD = 1.97 M ⊙ and +a binary separation of 69 au. We identify four directly imaged brown +dwarfs around MS stars that are similar to SDSS J22255 + 0016AB +before the primary star evolved into a white dwarf. These systems are +outlined in black and are, top to bottom, HD 19467AB, HD 33632AB, +HR 3549AB, and GJ 758AB. A comparison of these four systems +with the progenitor of SDSS J22255 + 0016AB is made in Table 4 . +These four systems all have similar mass ratios, separations and +companion masses to SDSS J22255 + 0016AB before the white +dwarf progenitor evolved into the white dwarf, increasing the +separation of the brown dwarf as it evolved. It is therefore likely +that SDSS J22255 + 0016AB formed via a similar mechanism to +these binaries, which all formed in stellar-like or stellar binary-like +mechanisms (Vigan et al. 2016 ; Currie et al. 2020 ; Maire et al. 2020 ). +Additionally, when the MS stars in HD 19467AB, HD 33632AB, +HR 3549AB, and GJ 758AB evolve into a white dwarf, their +evolved forms will resemble SDSS J22255 + 0016AB. In particular, +HR 3549AB will be most comparable to SDSS J22255 + 0016AB +once it has evolved. HR 3549A has a white dwarf mass within +1 σ of the mass of SDSS J22255 + 0016A and a brown dwarf +mass within the mass range of SDSS J22255 + 0016B. Of the four +objects highlighted here, HR 3549AB has a separation most akin +to the estimated initial separation of SDSS J222551.65 + 001637.7. +Furthermore, HR 3549AB is younger than SDSS J22255 + 0016AB, +and the brown dwarf has an earlier spectral type, meaning it could +concei v ably e volv e into an e xtremely similar system o v er time. +With a projected separation of 207 + 13 +−12 au between the two compo- +nents, SDSS J22255 + 0016AB is the third closest separated spatially +resolved wide white dw arf–brown dw arf binary after GD 165AB +(Becklin & Zuckerman 1988 ) and PHL 5038AB (Steele et al. 2009 ). +These 3 systems comprise a subset of wide, but not ultra-wide, +white dwarf–brown dwarf binaries which are spatially resolved, +as opposed to the other 5 ultra-wide, como ving, resolv ed systems +currently known. Table 5 details the parameters of these 3 systems. +SDSS J22255 + 0016AB is most similar to GD 165AB, with compara- +ble white dwarf masses, ef fecti ve temperatures and surface gravities, +as well as total ages. Ho we ver, GD 165B has a higher mass and +smaller physical separation than SDSS J22255 + 0016B. Although +the brown dwarf in PHL 5038AB is a later spectral type, and its white +dwarf primary is cooler than SDSS J22255 + 0016A, these binaries +are still akin to each other, with separations on the order of 100 au, +and white dwarf masses and surface gravities within 1 σ of each other. +The dominant factor influencing the evolution of these binaries is the +white dwarf mass and the separation, since at wide separations the +brown dwarf is not massive enough to affect the evolution of the +binary. It is thus likely that these three resolved systems all evolved +in the same manner. + +A close resolved white dwarf–brown dwarf binary +7 +MNRAS 00, 1 (2022) +Figure 6. Distribution of known white dwarf–brown dwarf binaries alongside directly imaged exoplanets and binaries around main sequence stars. Exoplanets +are in orange and brown dwarfs are colour coded by their spectral type. Directly imaged objects are represented by circles and the white dwarf–brown dwarf +binaries are represented by squares. Point size is proportional to the mass of the primary star. The star represents SDSS J222551.65 + 001637.7AB at present. +The pentagon represents SDSS J222551.65 + 001637.7AB whilst the white dwarf progenitor was still on the main sequence. The outlined circles are the four +systems most similar to the progenitor of SDSS J222551.65 + 001637.7AB: HD 19467AB, HD 33632AB, HR 3549AB, and GJ 758AB. +Table 4. Comparison of the four directly imaged main sequence-brown dwarf binaries most similar to +SDSS J222551.65 + 001637.7AB: HD 19467, HD 33632, HR 3549, and GJ 758. SDSS J222551.65 + 001637.7AB is reported as it +was when the white dwarf progenitor was still on the main sequence. +Binary +M MS (M ⊙) +Age (Gyr) +BD Spectral +Type +M BD ( M Jup ) Separation (au) +Ref +HD 19467AB +0.953 ± 0.022 +5.4 + 1 . 9 +−1 . 3 +T5.5 +65 . 4 + 5 . 9 +−4 . 6 +51.1 ± 0.1 +3, 4, 5, 6 +HD 33632AB +1.1 ± 0.1 +1.7 ± 0.4 +L9.5 +50 . 0 + 5 . 6 +−5 . 0 +23 . 6 + 3 . 2 +−4 . 5 +4, 7 +HR 3549AB +2.3 ± 0.2 +0.10 − 0.15 +M9.5 +45 ± 5 +80.0 ± 2.0 +1, 2 +GJ 758AB +0.96 ± 0.3 +8.3 + 2 . 7 +−2 . 1 +T8 +38.0 ± 0.8 +33.0 ± 6.0 +4, 8, 9, 10 +SDSS J222551.65 + 001637.7AB +1.97 + 1 . 14 +−0 . 76 +1 . 40 + 4 . 48 +−0 . 98 +L4 +25–53 +69 ± 5 +This Work +Note . References are 1: Ma wet et al. ( 2015 ); 2: Mesa et al. ( 2016 ); 3: Maire et al. ( 2020 ); 4: Brandt et al. ( 2021 ); 5: Crepp et al. +( 2015 ); 6: Jensen-Clem et al. ( 2016 ); 7: Currie et al. ( 2020 ); 8: Takeda ( 2007 ); 9: Vigan et al. ( 2016 ); 10: Brandt, Dupuy & Bowler +( 2019 ). +Table 5. Comparison of the three closest separated resolved white dwarf–brown dwarf binary systems, GD 165AB, PHL 5038AB, and +SDSS J222551.65 + 001637.7AB. +Binary +M WD (M ⊙) +T eff (K) +log g +M BD ( M Jup ) +Spectral Type Separation (au) +Age (Gyr) +Ref +GD 165AB +0.64 ± 0.02 +12130 ± 450 +8.052 ± 0.035 +62.58 ± 15.57 +L4 +123 ± 12 +1.2–5.5 +1, 2, 3, 4, 5 +PHL 5038AB +0.72 ± 0.15 +8000 ± 100 +8.2 ± 0.1 +60 +L8 +69 ± 1 +1.9–2.7 +6 +SDSS J222551.65 + 001637.7AB +0.66 + 0 . 11 +−0 . 06 +10926 ± 246 +8.214 ± 0.168 +25–53 +L4 +207 + 13 +−12 +1.2–6.4 +7, This Work +Note. References are 1: Giammichele et al. ( 2016 ); 2: Filippazzo et al. ( 2015 ); 3: Becklin & Zuckerman ( 1988 ); 4: Kirkpatrick et al. ( 1993 ); 5: Kirkpatrick et al. +( 1999 ); 6: Steele et al. ( 2009 ); 7: Anguiano et al. ( 2017 ). +6 CONCLUSIONS +We confirm SDSS J222551.65 + 001637.7AB as a wide, comoving +white dw arf–brown dw arf binary, which has now become resolved. +Alongside the photometry measurements, the near-IR spectrum +taken by GNIRS shows an IR excess that indicates a brown dwarf +companion of spectral type L4 ± 1. We determine the absolute +K s -band magnitude of the brown dwarf as 11.27 ± 0.18, which is +consistent with an L4 ± 1 spectral type. We calculate the white dwarf +mass as 0 . 66 + 0 . 11 +−0 . 06 M ⊙ and the total system age as 1 . 97 + 4 . 41 +−0 . 76 Gyr. We +use the Sonora–Bobcat evolutionary models to estimate the mass +of the companion as 25–53 M Jup and its radius as 0.101–0.128 R ⊙, +confirming that it is a brown dwarf. The white dwarf shows no metal- +line pollution that would indicate the presence of another companion. +The acquisition image from the GNIRS spectrum and subsequent +NIRI imaging confirm that SDSS J222551.65 + 001637.7AB is spa- +tially resolved with an angular separation of 0.9498 ± 0.0022 arcsec, + +8 +J. R. French et al. +MNRAS 00, 1 (2022) +which corresponds to a projected separation of 207 + 13 +−12 au at the Gaia +eDR3 distance of 218 + 14 +−13 pc. We calculate UVW space velocities +to demonstrate that this system is likely a member of the thick +disc. We estimate the minimum orbital period of this binary as +P = 3560 ± 383 yr. Due to the wide separation, it is unlikely +that the brown dwarf companion altered the primary progenitor’s +evolution. This system is only the 8th confirmed wide comoving +white dw arf–brown dw arf binary and constitutes the third closest +separated resolved system after GD 165AB (Becklin & Zuckerman +1988 ) and PHL 5038AB (Steele et al. 2009 ). +ACKNOWLEDGEMENTS +The authors would like to thank Kathleen Labrie of Noirlab for +her help in reducing the data using the Dragons pipeline. JRF +acknowledges support of a University of Leicester College of Science +and Engineering PhD studentship. SLC acknowledges the support +of a Science and Technology Facilities Council Ernest Rutherford +Fello wship (ST/R003726/1). ECM ackno wledges the support of the +Heising Simons Foundation 51 Pegasi b Fellowship (#21-0684). +This work is based on observations obtained at the international +Gemini Observatory, a program of the National Science Foundation’s +National Optical-Infared Astronomy Research Laboratory, which +is managed by the Association of Universities for Research in +Astronomy (AURA) under a cooperative agreement with the National +Science Foundation on behalf of the Gemini Observatory partner- +ship: the National Science Foundation (United States), National +Research Council (Canada), Agencia Nacional de Investigaci ´on +y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog ´ıa e Inno- +vaci ´on (Argentina), Misit ´erio da Ci ˆ encia, Tecnologia, Inova c ¸˜ oes e +Comunica c ¸˜ oes (Brazil), and Korea Astronomy and Space Science +Institute (Republic of Korea). This w ork w as enabled by observations +made from the Gemini North telescope, located within the Maunakea +Science Reserve and adjacent to the summit of Maunakea. We are +grateful for the privilege of observing the Universe from a place that +is unique in both its astronomical quality and its cultural significance. +This research has made use of data obtained from or tools provided +by the portal exoplanet.eu of The Extrasolar Planets Encyclopaedia. +We thank the anonymous re vie wer for their helpful comments which +impro v ed the manuscript. +DATA AVAILABILITY +All data in this paper are publicly available in the Gemini archive. +REFERENCES +Aberasturi M., Solano E., Mart ´ın E. L., 2011, A&A, 534, L7 +Alam S. et al., 2015, ApJS, 219, 12 +Anguiano B., Rebassa-Mansergas A., Garc ´ıa-Berro E., Torres S., Freeman +K. C., Zwitter T., 2017, MNRAS, 469, 2102 +Auddy S., Basu S., Valluri S. R., 2016, Adv. Astron., 2016, 574327 +Becklin E. 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N., Hunsch M., 2003, ApJ, 596, 477 +This paper has been typeset from a T E X/L +A T E X file prepared by the author. + diff --git a/o9A0T4oBgHgl3EQfKP9c/content/tmp_files/load_file.txt b/o9A0T4oBgHgl3EQfKP9c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d67b9cb2acf7194be88efa8ef85fc99f575b3d17 --- /dev/null +++ b/o9A0T4oBgHgl3EQfKP9c/content/tmp_files/load_file.txt @@ -0,0 +1,1252 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf,len=1251 +page_content='MNRAS 00, 1 (2022) https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1093/mnras/stac3807 Disco v ery of a resolved white dwarf–brown dwarf binary with a small projected separation: SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB Jenni R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' French , 1 ‹ Sarah L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Casewell, 1 Trent J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Dupuy, 2 John H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Debes, 3 Elena Manjavacas, 3 Emily C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Martin 4 and Siyi Xu( �� � ) 5 1 School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' University of Leicester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' University Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Leicester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' LE1 7RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' United Kingdom 2 Royal Observatory Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Blackford Hill,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' EH9 3HJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' United Kingdom 3 AURA for the European Space Agency (ESA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Space Telescope Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Maryland 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' USA 4 Department of Astronomy & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' University of California Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 1156 High Street,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' California 95064,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' USA 5 Gemini Observatory/NSF’s NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 670 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' A’ohoku Place, Hilo, Hawaii, 96720, USA Accepted 2022 December 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Received 2022 December 20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' in original form 2022 October 26 A B S T R A C T We present the confirmation of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB as a closely separated, resolved, white dwarf–brown dwarf binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We have obtained spectroscopy from GNIRS and seeing-limited K s -band imaging from NIRI on Gemini North.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The target is spatially resolved into its constituent components: a 10926 ± 246 K white dwarf, with log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='168 and a mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='66 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 06 M ⊙, and an L4 brown dwarf companion, which are separated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='9498 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0022 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We derive the fundamental properties of the companion from the Sonora–Bobcat evolutionary models, finding a mass of 25–53 M Jup and a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='101–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='128 R ⊙ for the brown dwarf, at a confidence level of 1 σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We use WDWARFDATE to determine the age of the binary as 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 97 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 41 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 76 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' A kinematic analysis shows that this binary is likely a member of the thick disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The distance to the binary is 218 + 14 −13 pc, and hence the projected separation of the binary is 207 + 13 −12 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Whilst the white dwarf progenitor was on the main sequence the binary separation would have been 69 ± 5 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB is the third closest spatially resolved white dwarf–brown dwarf binary after GD 165AB and PHL 5038AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' K ey words: bro wn dwarfs – white dwarfs – binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 1 INTRODUCTION The prolific disco v eries of field brown dwarfs hav e been enabled by infrared (IR) and optical surv e ys such as the Two Micron All Sky Surv e y (2MASS), UKIRT Infrared Deep Sky Survey (UKIDSS), and Wide-field Infrared Surv e y Explorer (WISE) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 1998 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Pinfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2008 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Aberasturi, Solano & Mart ´ın 2011 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Despite o v er a thousand known field brown dwarfs, it is often challenging to determine their masses, ages, and luminosities as these parameters are degenerate due to the lack of fusion in the cores of brown dwarfs (Burrows & Liebert 1993 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Auddy, Basu & Valluri 2016 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Consequently, estimating brown dwarf parameters relies on models which, despite recent impro v ements, are sensitiv e to atmospheric processes that remain poorly understood, such as non-equilibrium chemistry (Tremblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2015 ) and the formation and presence of atmospheric clouds (Morley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2014 ), and any resultant variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ‘Benchmark’ brown dwarfs, those for which physical proper- ties can be independently determined, can test and impro v e the current evolutidaonary and atmospheric models for brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Double-lined eclipsing brown dwarfs are ideal benchmark ob- jects, but they are extremely rare, with only two known to date, ⋆ E-mail: jf328@leicester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='uk 2MASS J05352184-0546085 (Stassun, Mathieu & Valenti 2006 ) and 2MASSW J1510478-281817 (Triaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Since binary systems are formed from the same matter, companions can be assumed to have the same ages and metallicities as their primary stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Moe & Di Stef ano 2017 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Therefore, brown dw arfs that are members of a binary system can have their physical properties estimated with evolutionary models using the age constraints of the primary (Pinfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2006 ), provided that the primary’s age is well calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' These brown dwarfs can then be used as benchmarks to test and refine current brown dwarf models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Widely separated binaries with a white dwarf primary and a brown dwarf companion are particularly valuable for identifying benchmark brown dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The white dwarf cooling age provides a lower limit on the age of the binary, which can be used to calculate the physical properties of the brown dwarf companion (Fontaine, Brassard & Bergeron 2001 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Where the binary system is resolvable, meaning it is wide ( ≳ 50 au) or a common proper motion pair (ultra-wide, ≳ 500 au, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ), it is unlikely that either the brown dwarf was affected during the evolution of the white dwarf progenitor, or that the white dwarf evolution was truncated by the presence of a brown dwarf companion (Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Close white dwarf–brown dwarf binaries are not appropriate for identifying benchmark brown dwarfs because they undergo a period of common-envelope evolution instead of each component evolving separately (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Casewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2018 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Wide white dwarf–brown © 2022 The Author(s) Published by Oxford University Press on behalf of Royal Astronomical Society 2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' MNRAS 00, 1 (2022) Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Parameters of the brown dwarf companions in the confirmed wide, comoving white dwarf–brown dwarf binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' System Spectral Type Separation (au) Total Age (Gyr) Reference GD 165AB L4 123 ± 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 1, 2, 3 PHL 5038AB L8 69 ± 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='9–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7 4, 5 LSPM 1459 + 0857AB T4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 16500- 26500 > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='8 5, 6 WD 0806-661AB Y1 2504 ± 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7 5, 7, 8 LSPM J0241 + 2553AB L1 2380 ± 36 < 10 5, 9 COCONUTS-1AB T4 1290 ± 13 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3 + 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 8 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6 5, 10 LSPM J0055 + 5948AB T8 402 ± 2 10 ± 3 5, 11 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' References are 1: Becklin & Zuckerman ( 1988 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2: Kirkpatrick, Henry & Liebert ( 1993 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3: Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 1999 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4: Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2009 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5: Gaia Collaboration ( 2020 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6: Day-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 7: Luhman, Burgasser & Bochanski ( 2011 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 8: Rodriguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ), 9: Deacon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2014 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 10: Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2020 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11: Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' dwarf binaries are thus ideal systems to identify benchmark brown dwarfs to impro v e model calibrations and flag for follow-up with instruments such as NIRSpec on JWST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Despite several all-sky surveys that have identified thousands of white dwarfs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Girven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2011 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2019 ), and studies searching for ultracool companions to white dwarfs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2010 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Debes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2011 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ), only ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 percent of white dwarfs are predicted to have a brown dwarf companion (Farihi, Becklin & Zuckerman 2005 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2011 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Rebassa-Mansergas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2019 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' There are currently only 7 known wide, comoving white dwarf–brown dwarf binaries, which are listed in Table 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Their companion spectral types range from L1 to Y1 with separations ranging from 69 au to 16500–26500 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The larger separations correspond to T-dwarf companions but this is likely a selection effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' In addition to these 7 confirmed white dwarf–brown dwarf binaries, there are also several candidates that have been identi- fied using photometry b ut ha ve not been spectroscopically confirmed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Kiwy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2022 : NSC J053232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='31-512450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='75AB, WD + L3, a = 508 au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' NSC J130527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='23-224728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='44AB, WD + L3, a = 716 au), or that need more-IR data to confirm them as white dwarf–brown dwarf binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Meisner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 : LSR J0002 + 6357AB, WD + mid/late T, a = 8695 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' In this paper, we present a new white dwarf–brown dwarf binary, SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB, which hosts an important bench- mark brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB is the third closest separated resolved white dwarf–brown dwarf binary after GD 165AB and PHL 5038AB and understanding its evolution will provide insights into the formation of these rare wide white dwarf– brown dwarf binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Section 2 discusses this system, Section 3 describes our observations and data reduction, Section 4 presents our results, and Section 5 discusses our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2 SDSS J22255 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB (henceforth SDSS J22255 + 0016AB) was first reported by Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2006 ) where SDSS J22255 + 0016A was identified as a hydrogen- rich DA white dwarf as part of a catalogue of spectroscopically confirmed white dwarfs from the Sloan Digital Sk y Surv e y (SDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' IR observations were taken as part of the UKIDSS Large Area Surv e y (La wrence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2007 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Girv en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ) identified SDSS J22255 + 0016AB as having a potential photometric near-IR excess, with Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ) suggesting the excess is indicative of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SOFI K -band image of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB showing the elongation of the central star before the components were resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Reproduced from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11 of Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' a partially resolved L-dwarf companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 1 , which is reproduced from Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ), shows that in the SOFI K -band image SDSS J22255 + 0016AB is considerably elongated with a full width at half-maximum (FWHM) of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='8 arcsec, suggesting a predicted separation of a < 350 au at their adopted distance of 190 ± 20 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2006 ) fit the SDSS spectrum of SDSS J22255 + 0016A to a grid of model atmospheres, determining T eff = 10640 ± 94 K and log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Anguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2017 ) refined these measurements to T eff = 10926 ± 246 K and log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='168, with a derived distance of 226 ± 41 pc using updated model spectra and a correction for the 3D dependence of convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Gentile Fusillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2021 ) performed a photometric fit to the Gaia eDR3 magnitudes obtaining T eff = 9370 ± 765 K and log g = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='738 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This method of fitting is less sensitive to log g and relies on the Gaia magnitudes alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Jim ´enez-Esteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2018 ) also performed a photometric fit of the white dwarf, and they found T eff = 9000 ± 125 K and log g = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='25 when considering the spectral energy distribution from the SDSS u -band to the UKIDSS K -band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Although these two photometric fits yield white dwarf parameters that are consistent with each other, neither of these fits account for extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Jim ´enez-Esteban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2018 ) state that for objects at ∼200 pc, extinction should be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J22255 + 0016A is at a distance of 218 + 14 −13 pc, and has an extinction of A g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='323 (Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2006 ), which is significant and should thus be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2 depicts a comparison of hydrogen-rich DA white dwarf models (Koester 2010 ) for both the derived spectroscopic and photo- metric parameters, alongside dereddened photometry measurements, and the SDSS spectrum of SDSS J22255 + 0016A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Considering the dereddened photometry, the white dwarf model that uses spectro- scopic parameters ( T eff = 11000 K, log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='25) is consistent with the data, whereas the white dwarf model which uses photometric parameters ( T eff = 9250 K, log g = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='75) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This is particularly evident at shorter wavelengths, where the photometry measurements are unlikely to be influenced by the flux of the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The residuals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2 for the photometric parameters are 12 percent higher than those for the spectroscopic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The white dwarf model using spectroscopic parameters also better fits the depth of the hydrogen lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Our comparison of two Koester DA white dwarf models with the dereddened photometry measurements indicates that the spectro- A close resolved white dwarf–brown dwarf binary 3 MNRAS 00, 1 (2022) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS spectrum of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7A with dered- dened SDSS, PAN-STARRS and UKIDSS photometry (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The Koester DA white dwarf model with T eff = 9250 K and log g = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='75, deriving from the photometric fits, is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The Koester DA white dwarf model with T eff = 11000 K and log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='25, deriving from the spectroscopic fits, is shown in light blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The residuals between the SDSS spectrum and the white dwarf models are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' scopic parameters derived by Anguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2017 ) are the most appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Furthermore, Groenewegen ( 2020 ) found that effective temperatures calculated from photometric fits were consistently underestimated compared to those derived from spectroscopy, with discrepancies of 2000 K in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Since the Anguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2017 ) parameters are derived from a spectroscopic fit, are consistent with the photometry and SDSS spectrum of SDSS J22255 + 0016A, and their derived distance of 226 ± 41 pc is within 1 σ of the Gaia eDR3 distance of 218 + 14 −13 pc, we adopt these parameters of T eff = 10926 ± 246 K and log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='168 for the white dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Although we start to see the contributions of the brown dwarf from the r -band, the brown dwarf emits mainly in the IR, and thus will not contribute significantly to the measured T eff and log g of the white dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3 OBSERVATIONS AND DATA REDUCTION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 GNIRS spectroscopy We observed SDSS J22255 + 0016AB using the cross-dispersed spec- trograph GNIRS on Gemini North (Elias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2006 ) on 2020 July 8th and July 10th UTC, as a part of programme GN-2020A-Q-322 (PI: John H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Debes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Spectra were taken using the short blue camera with the 32 l/mm grating and a slit width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 arcsec, giving a resolution of ( λ / �λ) ∼ 500 across the entire wavelength range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='8–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We nodded the observations, taking 300 s exposures at each nod point, totalling 20 exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Arc lamp and flat-field calibration frames were taken immediately after the science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' All exposures across three individual hours of observation were then combined during data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Both the white dwarf and the brown dwarf were in the slit during our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The data were then reduced using a version of SPEXTOOL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 (Cushing, Vacca & Rayner 2004 ) which has been adapted for use with GNIRS data (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Allers, pri v ate communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Two A0V standard stars, HIP115119 and HIP110963, were observed using the same settings with four frames taken for each hour of observation with an exposure time of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Two stars were observed to ensure that for each hour of science Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' GNIRS H -band acquisition image of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB showing the resolved white dwarf and brown dwarf components of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The position angle of the acquisition image is 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='49 ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Both the white dwarf and the brown dwarf were in the slit during our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' NIRI K s -band image of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB showing we have clearly resolved the white and brown dwarf components of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' A is the white dwarf, and B is the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' observation, the standard star observed was at a similar airmass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The telluric correction was performed during reduction using the XTELLCOR package (Vacca, Cushing & Rayner 2003 ) in SPEXTOOL using our observed standard stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The acquisition images from GNIRS revealed that the system is in fact spatially resolved (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The images were taken with 3 s exposures in the H -band using the same settings as the science observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 NIRI imaging We imaged SDSS J22255 + 0016AB on the 2021 June 13th UTC with NIRI in the K s -band as part of programme GN-2021A-FT-207 (PI: Elena Manjavacas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We obtained 13 60 s exposures at airmass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='09 with the f/6 camera providing a pixel scale of 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 mas pixel −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' N PA = 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='49° BN E A B4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' MNRAS 00, 1 (2022) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' GNIRS spectrum of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB with SDSS and UKIDSS photometry (blue) and the Koester DA white dwarf model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The combined white dwarf + L3 template spectra, combined white dwarf + L4 template spectra and combined white dwarf + L5 template spectra are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The residual spectra for L3–L5 spectral types are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The GNIRS spectrum has been binned to a resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='4 Å to increase signal to noise, and the telluric line-dominated section between 18000 Å and 19000 Å has been masked for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We reduced the data using the DRAGONS software (Labrie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2019 ) provided by the Gemini observatory, reducing using flat-field and dark frames provided as part of the calibration set and creating a bad pixel mask using 10 s dark frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The 13 images were reduced and stacked using stars in the image as references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This is depicted in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' and shows that we have clearly resolved the white dwarf and brown dwarf components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4 RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 Spectral type Although SDSS J22255 + 0016AB is resolved in the acquisition image, both of the components were within the slit during obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' To confirm the spectral type of the secondary, we created composite DA white dwarf + cool L-dwarf templates similar to those in Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ) and Casewell, Geier & Lodieu ( 2017 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We used a hydrogen-rich DA white dwarf model (Koester 2010 ) with T eff = 11 000 K and log g = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='25 to best match the parameters of SDSS J22255 + 0016A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We took L-dwarf template spectra from the SpeX Prism Library (Burgasser 2014 ) for spectral types L3-L5, which are reported in Burgasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2010 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We combined the white dwarf model with the L-dwarf template spectra by setting both to 10 pc and then combining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' To set the white dwarf model to 10 pc, we normalized using the Gaia eDR3 distance of 218 + 14 −13 pc and the broad-band photometry measurement in the SDSS r -band (Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2015 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' To normalize the brown dwarf template spectra to 10 pc, we used the mean absolute J -band magnitudes for each spectral type from Dupuy & Liu ( 2012 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Once combined, we normalized the composite white dwarf + L-dwarf models to the broad-band r - band photometry, and we normalized the GNIRS spectrum to the broad-band UKIDSS J -band photometry (Girven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2011 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We then compare the GNIRS spectrum to the composite white dwarf + L-dwarf models to determine the presence of an IR excess and the nature of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5 shows the GNIRS spectrum for SDSS J22255 + 0016AB alongside photometry measurements and our composite white dwarf + brown dwarf models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' From this, we determine the spectral type of the brown dwarf as L4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 Relati v e astrometry From our stacked NIRI image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4) , we measured relative astrometry for SDSS J22255 + 0016AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We fitted an analytic point spread function model to each component, where the model used three concentric 2D Gaussians with different amplitudes, standard deviations, ellipticities, and angles for the ellipticities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This approach is based on previous work with adaptive optics imaging of low-mass binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2006 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2019 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We converted the pixel positions of the two components into sky coordinates using the WCS information in the FITS header.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Given how well resolved SDSS J22255 + 0016AB is, the errors on its relative astrometry are dominated by the astrometric calibration of NIRI, with a fractional uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='23 percent in pixel scale and an uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 ◦ in parallactic angle (Mann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2019 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This results in a separation of 949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 mas and a parallactic angle of 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 ◦ between the two components, measured as the position of B from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Our binary fit also provides a measurement of the relative photometry of K s , B − K s , A = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='02 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Here, we have defined the more massive white dwarf as the A component and its lower mass ultracool dwarf companion, which is brighter in the K s -band, as the B component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We do not deri ve relati ve astrometry from the GNIRS acquisition images, as it is not astrometrically well-calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We note, ho we ver, that there does not appear to be any astrometric motion relative to the NIRI imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' As the two images were taken 1 year apart, this is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The two binary components are separated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='9498 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0022 arcsec and a parallactic angle of 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 ◦ and have a magnitude difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='990 mags in the K s -band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This magnitude difference is consistent with the difference in absolute magnitudes of the two components as predicted by the models of Tremblay, Bergeron & Gianninas ( 2011 ) for DA white dwarfs and the absolute magnitude spectral type relations of Dupuy & Liu ( 2012 ) for an L4 brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Using the Gaia eDR3 distance of 218 + 14 −13 pc and our separation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='9498 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0022 arcsec, we calculate the projected separation of SDSS J22255 + 0016AB as 207 + 13 −12 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5 DISCUSSION We determined the spectral type of SDSS J22255 + 0016B as L4 ±1 by comparing template brown dwarf spectra to the GNIRS spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This spectral type is consistent with the difference in magnitude for the components measured from the NIRI image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Although the K - band photometry and spectrum are brighter than that of the white dwarf + L4 combined model, it is consistent within the errors, which are dominated by the absolute magnitudes in Dupuy & Liu ( 2012 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The offset between the models and the spectrum at 10000 Å is due to the SpeX template dwarf spectra not extending very far into optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Our spectral type is consistent with the L-dwarf companion proposed by Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ) based on the UKIDSS photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Our projected separation of 207 + 13 −12 au agrees with their prediction of a < 350 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 Age of the system To determine the age of SDSS J22255 + 0016AB, we used WDWARF- DATE , which estimates the age of a white dwarf, as well as its final mass and initial mass, from T eff and log g using a Bayesian framework (Kiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2022 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The cooling age and mass of the A close resolved white dwarf–brown dwarf binary 5 MNRAS 00, 1 (2022) Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' System parameters for SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB derived using WDWARFDATE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Parameter Value Cooling Age (Gyr) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 58 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 17 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 08 Final Mass (M ⊙) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 66 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 06 Initial Mass (M ⊙) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 97 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 14 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 76 Main Sequence Age (Gyr) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 40 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 48 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 98 Total System Age (Gyr) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 97 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 41 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 76 white dwarf are determined from the evolutionary models of the Montreal White Dwarf Group (B ´edard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ), and an initial- final mass relationship is used to calculate the initial mass of the white dwarf progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The progenitor lifetime, also referred to as the main sequence (MS) age, is then determined using the MIST isochrones (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Dotter 2016 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The total lifetime of the white dwarf is calculated as the sum of the cooling age and the progenitor’s MS age (T able 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' W e utilized the initial-final mass relationship of Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2018 ) and assumed solar metallicity and v / v crit = 0 for the fit, where v / v crit quantifies stellar rotation (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2021 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The white dwarf mass of 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 66 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 06 M ⊙ is within 1 σ of the Anguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2017 ) white dwarf mass of 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 72 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 10 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 10 M ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Our cooling age of the white dwarf gives the minimum age of the system as 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 58 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 17 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 08 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We estimate the total age of the system as 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 97 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 41 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 76 Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ho we ver, this value is particularly sensitive to uncertainties in the choice of initial-final mass relationship and the MS age of the white dwarf progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' To further constrain the age of SDSS J22255 + 0016AB, we used the Gaia eDR3 proper motions and the radial velocity measured by Anguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2017 ) to undertake a kinematic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We calculate the UVW space velocities with respect to the local standard of rest as: U = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='52 ± 7 km s −1 , V = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 ± 13 km s −1 , W = −71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='8 ± 15 km s −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Here, U is positive towards the Galactic cen- tre, V is positive in the direction of Galactic rotation, and W is positive towards the North Galactic Pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Following the method of Bensby, Feltzing & Oey ( 2014 ) with their observed fractions of thick disc, thin disc, and halo populations in the solar neighbourhood, we determine the relative probabilities for SDSS J22255 + 0016AB belonging to each of these populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We find that SDSS J22255 + 0016AB is 495 times more likely to belong to the thick disc than the thin disc and 461 times more likely to belong to the thick disc than the stellar halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' It is thus likely that SDSS J22255 + 0016AB is a member of the thick disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The thick disc has an age of ∼10 Gyr (Kilic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2017 ), meaning that if SDSS J22255 + 0016AB is indeed a member of the thick disc, the total system age is likely closer to the upper uncertainty of the age we determine with WDWARFDATE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' In their analysis of white dwarfs in the thin and thick discs, Raddi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2022 ) find that the total age distribution of white dwarfs peaks at 2 Gyr, which may explain why the total age of SDSS J22255 + 0016AB is young for a thick disc object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Additionally, Torres et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2021 ) find that 13 percent of halo white dwarfs in Gaia DR2 are younger than expected compared to the average halo white dwarf age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This indicates the presence of younger white dwarfs in both disc and halo populations, of which SDSS J22255 + 0016A may be one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ho we ver, the origin of these younger objects is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We note that older age is derived if we use the photometric parameters of the white dwarf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ho we ver, this has larger uncertainties, and the photometric parameters are less reliable due to their lack of reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Despite the large uncertainty in total age, which is dominated by uncertainties in the initial-final mass Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Absolute and apparent magnitudes for each component of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Star Absolute K s Magnitude Apparent K s Magnitude White Dwarf 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='20 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='15 Brown Dwarf 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='18 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='13 relationship, SDSS J22255 + 0016B is an important member in the small population of wide white dwarf–brown dwarf binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' As a member of the thick disc, it is unlikely that SDSS J22255 + 0016AB is extremely metal-poor in comparison to objects residing in the stellar halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' There is no evidence of photospheric metal pollution in the SDSS optical spectrum of the white dwarf that would indicate accretion from a tidally disrupted asteroid or another companion (Zuckerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2003 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Debes 2006 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J22255 + 0016A has a low effective temperature, and if it were polluted, absorption features would be easily detectable in the optical spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Since we do not detect any pollution, it is thus likely that SDSS J22255 + 0016AB has no other companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Ho we ver, we note that the Ca II line can appear weak in white dwarf spectra, and a high resolution echelle spectrum of the white dwarf would be required to place definitive limits on any potential pollution (Zuckerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2003 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7B As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 , our observed absolute magnitude difference between the white dwarf and the brown dwarf is consistent with predictions from theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Using this magnitude difference and taking our observed UKIDSS K -band magnitude of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='13 as a proxy for the observed K s -band magnitude, we calculate the apparent magnitudes of both the white dwarf and the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Using the Gaia distance of 218 + 14 −13 pc, we calculate the absolute magnitudes in the K s -band for both the white dwarf and the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' These magnitudes are presented in Table 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We find that absolute K s -band magnitude of the brown dwarf is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This is consistent with the mean absolute K s -band magnitude of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='28, which Dupuy & Liu ( 2012 ) report for L4 companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Our absolute K s -band magnitude for the white dwarf is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This is consistent with that predicted by synthetic photometry calculated from the Tremblay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ) white dwarf models (Holberg & Bergeron 2006 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Kowalski & Saumon 2006 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 1 The estimated L4 companion spectral type provides a consis- tent theoretical and observed absolute magnitude, indicating that SDSS J22255 + 0016B is indeed an L4 ± 1 companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We esti- mate the ef fecti ve temperature of SDSS J22255 + 0016B as T eff = 1800 + 70 −60 K for our spectral type of L4 ± 1 as this is the mean ef fecti ve temperature of an L4 dwarf determined from the analysis of M, L, and T dwarfs performed by Vrba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2004 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We then compare our estimated ef fecti ve temperature and our K s -band magnitude for an L4 spectral type to the Sonora–Bobcat models, assuming solar metallicity (Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2021 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' From these models, a brown dwarf with T eff = 1800 K and K s = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='27 would have a mass of 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='88 M Jup and a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1071 R ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This mass estimate is consistent with the mass of 47 ± 3 M Jup determined by Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2011 ) using the Lyon group models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' With our K s -band magnitude, we use the relations of Dupuy & Liu ( 2017 ) to calculate the bolometric luminosity of 1 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='astr o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='umontr eal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='ca/ ∼ber geron/CoolingModels 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' MNRAS 00, 1 (2022) SDSS J22255 + 0016B as log ( L bol /L ⊙) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We also utilize their Lyon T eff relation to impro v e our temperature estimate to T eff = 1817 ± 90 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We compare our bolometric luminosity and ef fecti ve temperature of the brown dwarf with the Sonora–Bobcat models, assuming solar metallicity (Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2021 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' From these models, a brown dwarf with log ( L bol /L ⊙) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='11 and T eff = 1817 ± 90 K would have a mass of 25–53 M Jup and a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='101–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='128 R ⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We find that the most appropriate model for our bolometric luminosity and ef fecti v e temperature pro vides an age estimate and a K s -band magnitude that are consistent with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='3 Evolution of the system During the evolution of the MS progenitor of SDSS J22255 + 0016A, the orbital separation would have increased by a maximum factor of M MS / M WD = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='99 (Burleigh, Clarke & Hodgkin 2002 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The initial projected separation would therefore have been > 69 au, confirming that this is not a post-common envelope binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Burleigh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2002 ) state that white dwarfs will retain their planetary companions if the initial separation from the MS progenitor star is > 5 au, as is the case for SDSS J22255 + 0016AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Since the initial separation of SDSS J22255 + 0016AB is too wide to be a post-common envelope system, it will have evolved differently to close white dwarf–brown dwarf binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Maxted et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2006 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Casewell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2018 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The two components will have evolved separately, and the brown dwarf will not have truncated the white dwarf’s e volution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Ho we ver, the bro wn dwarf may have been affected by stellar winds from the primary, with the angular momentum lost by the white dwarf causing the separation to increase (Schrøder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2021 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' During the evolution of the MS progenitor of the white dwarf, the star undergoes a phase of evolution on the Asymptotic Giant Branch (AGB) before reaching its end stage as a white dwarf (Iben & Renzini 1983 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' In the AGB phase, the mass- loss increases until the envelope is fully ejected, which causes stellar winds that can affect the substellar companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Mayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2014 ) found dust-enriched winds of v w = 5–20 km s −1 and H ¨ofner & Olofsson ( 2018 ) report outflowing winds between v w = 3–30 km s −1 , affecting companions at separations on the order of ∼100 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' It is possible for the presence of the companion to shape these winds, morphing spherical AGB stars into non-spherical planetary nebulae, but at these wide separations ( ≳ 50 au), this is unlikely to alter the white dwarf progenitor’s evolution (Decin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='4 Orbit Using the white dwarf mass, the brown dwarf mass we esti- mate from the Sonora–Bobcat models, and our projected orbital separation of 207 + 13 −12 au, we calculate the likely orbital period of SDSS J22255 + 0016AB as P = 3560 ± 383 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This is a minimum period assuming a circular orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ho we v er, man y brown dwarfs are in eccentric orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Ma & Ge ( 2014 ) report that the eccentric- ity distribution of brown dwarfs changes at a threshold mass of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 M Jup , with brown dwarfs below this mass having eccentricities similar to massive planets and brown dwarfs abo v e this mass having eccentricities consistent with binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This suggests two distinct formation mechanisms for brown dwarfs: protoplanetary discs and stellar binary-like formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J22255 + 0016B resides near this mass boundary, and further investigations such as continuous monitoring to calculate its dynamical mass and observations to obtain an uncontaminated spectrum of the brown dwarf and a C/O ratio measurement, would enable us to determine its formation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The mass ratio of SDSS J22255 + 0016AB is q = M BD / M MS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='012–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Bowler, Blunt & Nielsen ( 2020 ) state that for binary mass ratios > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='01 stellar binary-like formation is fa v oured, which also indicates a higher eccentricity than systems in which the brown dwarf formed via planet-like formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6 depicts the currently known white dw arf–brown dw arf bina- ries as well as directly imaged brown dwarfs and exoplanets around MS stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The white dwarf–brown dwarf binaries are colour coded according to the spectral type of the brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The outlined star represents SDSS J22255 + 0016AB in its evolved form as it is now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The outlined pentagon represents SDSS J22255 + 0016AB whilst the white dwarf progenitor was still on the MS, with M WD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='97 M ⊙ and a binary separation of 69 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We identify four directly imaged brown dwarfs around MS stars that are similar to SDSS J22255 + 0016AB before the primary star evolved into a white dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' These systems are outlined in black and are, top to bottom, HD 19467AB, HD 33632AB, HR 3549AB, and GJ 758AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' A comparison of these four systems with the progenitor of SDSS J22255 + 0016AB is made in Table 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' These four systems all have similar mass ratios, separations and companion masses to SDSS J22255 + 0016AB before the white dwarf progenitor evolved into the white dwarf, increasing the separation of the brown dwarf as it evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' It is therefore likely that SDSS J22255 + 0016AB formed via a similar mechanism to these binaries, which all formed in stellar-like or stellar binary-like mechanisms (Vigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2016 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2020 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Additionally, when the MS stars in HD 19467AB, HD 33632AB, HR 3549AB, and GJ 758AB evolve into a white dwarf, their evolved forms will resemble SDSS J22255 + 0016AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' In particular, HR 3549AB will be most comparable to SDSS J22255 + 0016AB once it has evolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' HR 3549A has a white dwarf mass within 1 σ of the mass of SDSS J22255 + 0016A and a brown dwarf mass within the mass range of SDSS J22255 + 0016B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Of the four objects highlighted here, HR 3549AB has a separation most akin to the estimated initial separation of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Furthermore, HR 3549AB is younger than SDSS J22255 + 0016AB, and the brown dwarf has an earlier spectral type, meaning it could concei v ably e volv e into an e xtremely similar system o v er time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' With a projected separation of 207 + 13 −12 au between the two compo- nents, SDSS J22255 + 0016AB is the third closest separated spatially resolved wide white dw arf–brown dw arf binary after GD 165AB (Becklin & Zuckerman 1988 ) and PHL 5038AB (Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2009 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' These 3 systems comprise a subset of wide, but not ultra-wide, white dwarf–brown dwarf binaries which are spatially resolved, as opposed to the other 5 ultra-wide, como ving, resolv ed systems currently known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Table 5 details the parameters of these 3 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J22255 + 0016AB is most similar to GD 165AB, with compara- ble white dwarf masses, ef fecti ve temperatures and surface gravities, as well as total ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Ho we ver, GD 165B has a higher mass and smaller physical separation than SDSS J22255 + 0016B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Although the brown dwarf in PHL 5038AB is a later spectral type, and its white dwarf primary is cooler than SDSS J22255 + 0016A, these binaries are still akin to each other, with separations on the order of 100 au, and white dwarf masses and surface gravities within 1 σ of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The dominant factor influencing the evolution of these binaries is the white dwarf mass and the separation, since at wide separations the brown dwarf is not massive enough to affect the evolution of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' It is thus likely that these three resolved systems all evolved in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' A close resolved white dwarf–brown dwarf binary 7 MNRAS 00, 1 (2022) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Distribution of known white dwarf–brown dwarf binaries alongside directly imaged exoplanets and binaries around main sequence stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Exoplanets are in orange and brown dwarfs are colour coded by their spectral type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Directly imaged objects are represented by circles and the white dwarf–brown dwarf binaries are represented by squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Point size is proportional to the mass of the primary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The star represents SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The pentagon represents SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB whilst the white dwarf progenitor was still on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The outlined circles are the four systems most similar to the progenitor of SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB: HD 19467AB, HD 33632AB, HR 3549AB, and GJ 758AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Comparison of the four directly imaged main sequence-brown dwarf binaries most similar to SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB: HD 19467, HD 33632, HR 3549, and GJ 758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB is reported as it was when the white dwarf progenitor was still on the main sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Binary M MS (M ⊙) Age (Gyr) BD Spectral Type M BD ( M Jup ) Separation (au) Ref HD 19467AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='953 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='022 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='4 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 9 −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3 T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 65 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4 + 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 9 −4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 3, 4, 5, 6 HD 33632AB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='4 L9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 0 + 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6 −5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 0 23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6 + 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2 −4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5 4, 7 HR 3549AB 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='10 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='15 M9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 45 ± 5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 1, 2 GJ 758AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='3 + 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 7 −2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 1 T8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='8 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0 4, 8, 9, 10 SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='97 + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 14 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 76 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 40 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 48 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 98 L4 25–53 69 ± 5 This Work Note .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' References are 1: Ma wet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2015 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2: Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2016 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3: Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2020 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4: Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2021 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5: Crepp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2015 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6: Jensen-Clem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2016 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 7: Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2020 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 8: Takeda ( 2007 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 9: Vigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2016 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 10: Brandt, Dupuy & Bowler ( 2019 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Comparison of the three closest separated resolved white dwarf–brown dwarf binary systems, GD 165AB, PHL 5038AB, and SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Binary M WD (M ⊙) T eff (K) log g M BD ( M Jup ) Spectral Type Separation (au) Age (Gyr) Ref GD 165AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='02 12130 ± 450 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='035 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='58 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='57 L4 123 ± 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='5 1, 2, 3, 4, 5 PHL 5038AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='15 8000 ± 100 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='1 60 L8 69 ± 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='9–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7 6 SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='66 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 06 10926 ± 246 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='214 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='168 25–53 L4 207 + 13 −12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='2–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='4 7, This Work Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' References are 1: Giammichele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2016 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2: Filippazzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2015 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 3: Becklin & Zuckerman ( 1988 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 4: Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 1993 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 5: Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 1999 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6: Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2009 );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 7: Anguiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ( 2017 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 6 CONCLUSIONS We confirm SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB as a wide, comoving white dw arf–brown dw arf binary, which has now become resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Alongside the photometry measurements, the near-IR spectrum taken by GNIRS shows an IR excess that indicates a brown dwarf companion of spectral type L4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We determine the absolute K s -band magnitude of the brown dwarf as 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='18, which is consistent with an L4 ± 1 spectral type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We calculate the white dwarf mass as 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 66 + 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 11 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 06 M ⊙ and the total system age as 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 97 + 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 41 −0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 76 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We use the Sonora–Bobcat evolutionary models to estimate the mass of the companion as 25–53 M Jup and its radius as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='101–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='128 R ⊙, confirming that it is a brown dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The white dwarf shows no metal- line pollution that would indicate the presence of another companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' The acquisition image from the GNIRS spectrum and subsequent NIRI imaging confirm that SDSS J222551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='65 + 001637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='7AB is spa- tially resolved with an angular separation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='9498 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='0022 arcsec, 8 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' French et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' MNRAS 00, 1 (2022) which corresponds to a projected separation of 207 + 13 −12 au at the Gaia eDR3 distance of 218 + 14 −13 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We calculate UVW space velocities to demonstrate that this system is likely a member of the thick disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We estimate the minimum orbital period of this binary as P = 3560 ± 383 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Due to the wide separation, it is unlikely that the brown dwarf companion altered the primary progenitor’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This system is only the 8th confirmed wide comoving white dw arf–brown dw arf binary and constitutes the third closest separated resolved system after GD 165AB (Becklin & Zuckerman 1988 ) and PHL 5038AB (Steele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' 2009 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors would like to thank Kathleen Labrie of Noirlab for her help in reducing the data using the Dragons pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' JRF acknowledges support of a University of Leicester College of Science and Engineering PhD studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' SLC acknowledges the support of a Science and Technology Facilities Council Ernest Rutherford Fello wship (ST/R003726/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' ECM ackno wledges the support of the Heising Simons Foundation 51 Pegasi b Fellowship (#21-0684).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This work is based on observations obtained at the international Gemini Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' a program of the National Science Foundation’s National Optical-Infared Astronomy Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini Observatory partner- ship: the National Science Foundation (United States),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' National Research Council (Canada),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Agencia Nacional de Investigaci ´on y Desarrollo (Chile),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Ministerio de Ciencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Tecnolog ´ıa e Inno- vaci ´on (Argentina),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Misit ´erio da Ci ˆ encia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Tecnologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' Inova c ¸˜ oes e Comunica c ¸˜ oes (Brazil),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' and Korea Astronomy and Space Science Institute (Republic of Korea).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This w ork w as enabled by observations made from the Gemini North telescope, located within the Maunakea Science Reserve and adjacent to the summit of Maunakea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We are grateful for the privilege of observing the Universe from a place that is unique in both its astronomical quality and its cultural significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' This research has made use of data obtained from or tools provided by the portal exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content='eu of The Extrasolar Planets Encyclopaedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' We thank the anonymous re vie 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=', Reid I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=', Hunsch M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} +page_content=', 2003, ApJ, 596, 477 This paper has been typeset from a T E X/L A T E X file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9A0T4oBgHgl3EQfKP9c/content/2301.02101v1.pdf'} diff --git a/otAzT4oBgHgl3EQfqv1Z/content/tmp_files/2301.01633v1.pdf.txt b/otAzT4oBgHgl3EQfqv1Z/content/tmp_files/2301.01633v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cd14be6fa96e8bc7c243cbe7a3a37489f083bf0b --- /dev/null +++ b/otAzT4oBgHgl3EQfqv1Z/content/tmp_files/2301.01633v1.pdf.txt @@ -0,0 +1,1116 @@ + +1 +Physical properties, electronic structure, and strain-tuned monolayer of the weak topological +insulator RbTi3Bi5 with Kagome lattice + +Ying Zhou1, 2, #, Long Chen1, 2, #, Xuecong Ji1, 2, #, Chen Liu3, Ke Liao 1, 2, Zhongnan Guo4, Jia’ou +Wang3, Hongming Weng1, 2, 5, *, Gang Wang1, 2, 5, * + +1 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese +Academy of Sciences, Beijing 100190, China +2 University of Chinese Academy of Sciences, Beijing 100049, China +3 Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of +Sciences, Beijing 100049, China +4 Department of Chemistry, School of Chemistry and Biological Engineering, University of +Science and Technology Beijing, Beijing 100083, China +5 Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China +# These authors contributed equally to this work. +*Corresponding author. Email: gangwang@iphy.ac.cn; hmweng@iphy.ac.cn; + + + +Kagome metals AV3Sb5 (A = K, Rb, and Cs) with a V-Kagome lattice acting as a fertile platform to +investigate geometric frustration, electron correlation, superconductivity, and nontrivial band +topology, have attracted tremendous attention. Here we reported the structure and properties of +ATi3Bi5 (A = Rb, Cs) family with a Ti-Kagome lattice, specifically focusing on the electronic +structure and nontrivial band topology of RbTi3Bi5. ATi3Bi5 (A = Rb, Cs) is found to be non- +superconducting metal with strong quasi-two-dimensional feature, moderate electron correlation, +and small Pauli paramagnetism. Based on first principles calculations, RbTi3Bi5 is determined to be +a weak topological insulator with gapless surface states along (100) plane, and the electronic band +structure along (001) plane is in great agreement with experimentally observed one. In particular, +the electronic properties of the RbTi3Bi5 monolayer can be efficiently tuned by a biaxial strain +according to calculation, with its lower saddle points coming from Kagome lattice approaching the +Fermi level. These results highlight ATi3Bi5 (A = Rb, Cs) with Ti-Kagome lattice is a new Kagome +metal to explore nontrivial band topology and exotic phases. + +1 Introduction +Kagome metals, featuring two-dimensional (2D) Kagome sub-lattice with corner-sharing +triangles, have been serving as a fertile platform to investigate various electronic phenomena due to +the strong geometric frustration [1] and nontrivial band topology [2]. As a model system to realize +potential quantum spin liquid [3], the properties of spins on the Kagome lattice have been +extensively studied [4, 5]. Owing to the unique crystal structure, Kagome metals naturally support +electronic structures host Dirac points, flat bands, and saddle points, resulting in topologically +nontrivial states [6-8], possible fractionalization [9, 10] and Fermi surface (FS) instabilities [11], +respectively. With the combination of magnetism and topology, a 2D “Chern gap” [12] and a large +anomalous Hall response [13-15] have been found in Kagome metals with the existing long-range +magnetic order. By changing the filling of electrons in Kagome lattice, exotic phenomenon such as + + +2 +density wave orders [2, 16, 17], charge fractionalization [18, 19], and superconductivity [11, 20], +could be realized. Especially when Fermi level (EF) is near the saddle points, the spin or charge +density wave (CDW) states would happen [11, 21-23]. +Recently, the AV3Sb5 (A = K, Rb, and Cs) family with V-Kagome lattice has been discovered [24] +and attracted tremendous attention due to its CDW states, superconductivity, and nontrivial band +topology [25, 26]. All the AV3Sb5 compounds were found to be superconducting at low temperatures +(TC ~ 2.3 K for CsV3Sb5 [24, 25], TC ~ 0.75 K for RbV3Sb5 [27], and TC ~ 0.93 K for KV3Sb5 [28]), +together with an unconventional long-range CDW transition occurring at high temperatures (TCDW +~ 94 K for CsV3Sb5 [25], TCDW ~ 103 K for RbV3Sb5 [27], and TCDW ~ 78 K for KV3Sb5 [24]). +Though without long-range magnetic order, an extraordinarily large anomalous Hall effect was +observed in KV3Sb5 and CsV3Sb5 [29, 30]. By applying hydrostatic pressure or carrier doping, +multiple superconducting domes can be induced in AV3Sb5, showing the competition between CDW +and superconductivity [31-36]. According to density functional (DFT) calculations and angle- +resolved photoemission spectroscopy (ARPES) measurements, the AV3Sb5 materials are effectively +modeled as weakly correlated systems with strong quasi-2D nature [25, 37, 38]. In addition, AV3Sb5 +is found to be a Z2 topological metal with unconventional surface states [25, 39]. These researches +highlight the materials with Kagome lattice, such as AV3Sb5, being a promising platform to explore +topological superconductivity, correlated electronic states, and topological quantum computations. +In the exploration of new Kagome metals, ATi3Bi5 (A = Rb, Cs) [40, 41] stands out as another family +with possible superconductivity [41], orbital-selective nematic order [42] or electronic nematicity +[43], and nontrivial topology [40, 41]. However, most of attention has been focused on CsTi3Bi5 +rather than RbTi3Bi5, and their topological properties remain controversial and is still lack +systematical investigations. +Here we report the structure and properties of ATi3Bi5 (A = Rb, Cs) family with a Ti-Kagome +lattice, focusing on the electronic structure and nontrivial band topology of RbTi3Bi5. ATi3Bi5 (A = +Rb, Cs) shares a similar quasi-2D structure motif with that of AV3Sb5, featuring Ti-Bi layer in +Kagome lattice which is sandwiched by A+ as separator. The resistivity of ATi3Bi5 (A = Rb, Cs) +exhibits a metallic-like behavior without obvious anomaly down to 2 K, showing a quite large +anisotropy and moderate electron correlation. A small magnetic susceptibility has been observed +without any long-range magnetic order or superconducting transition in the whole measured +temperature range. The electronic band structure of RbTi3Bi5 has been characterized by ARPES and +fits well with DFT calculations with spin-orbit coupling (SOC), from which RbTi3Bi5 is determined +to be a weak topological insulator with gapless surface states. The upper saddle points coming from +Kagome lattice are proved to be far away (~ 0.8 eV) from EF, whereas the lower saddle points are +close (~ 0.3 eV) to EF, which may result interesting states like superconductivity or CDW by +changing the filling of electrons. Furthermore, due to its strong quasi-2D feature, a biaxial strain +could be used to effectively modulate the electronic properties of the RbTi3Bi5 monolayer based on +first principles calculations. + +2 Experimental +Single Crystal Growth. ATi3Bi5 (A = Rb, Cs) single crystals were grown by a high-temperature +solution method using Bi as flux. Rb/Cs chunk (99.75%, Alfa Aesar), Ti powder (99.95%, Alfa +Aesar), and Bi granules (99.999%, Sinopharm) were mixed using a molar ratio of Rb/Cs : Ti : Bi = + + +3 +2 : 4 : 12 in a fritted alumina crucible set (Canfield crucible set) [44] and sealed in a fused-silica +ampoule at vacuum. The ampoule was heated to 1073 K over 15 h, held at the temperature for 24 h, +and then slowly cooled down to 873 K at a rate of 2 K/h. At 873 K, large single crystals with size +up to 5 mm ×5 mm × 0.5 mm were separated from the remaining liquid by centrifuging the ampoule. +The obtained single crystals are shiny-silver plates and air-sensitive, so all manipulations and +specimen preparation for structure characterization and property measurements were handled in an +argon-filled glovebox. +Structure Characterization and Composition Analysis. X-ray diffraction data were obtained +using a PANalytical X’Pert PRO diffractometer (Cu Kα radiation, λ = 1.54178 Å) with a graphite +monochromator in a reflection mode (2θ = 5–80, step size = 0.017) operated at 40 kV voltage +and 40 mA current. Indexing and Rietveld refinement were performed using the DICVOL91 and +FULLPROF programs [45]. Single crystal X-ray diffraction (SCXRD) data were collected using a +Bruker D8 VENTURE with Mo Kα radiation (λ = 0.71073 Å) at 280 K for RbTi3Bi5 and a four- +circle diffractometer (Rigaku XtaLAB Synergy R-DW, HyPix) with multilayer mirror graphite- +monochromatized Mo Kα radiation (λ = 0.71073 Å) at 180 K for CsTi3Bi5. The structure was solved +using a direct method and refined with the Olex2 package [46]. The morphology and element +analyses were characterized using a scanning electron microscope (SEM, Phenom Prox) equipped +with an electron microprobe analyzer for semiquantitative elemental analysis in energy-dispersive +spectroscopy (EDS) mode. Five spots in different areas were measured on one crystal using EDS. +Physical Property Measurements. The resistivity, magnetic susceptibility, and heat capacity +measurements were carried out using a physical property measurement system (Quantum Design, 9 +T). The resistivity was measured using the standard four-probe configuration with the applied +current (about 2 mA) parallel to the ab plane (I // ab) or along the c axis (I // c). Magnetic +susceptibility was measured under a magnetic field (0.5 T) parallel (H // ab) and normal (H ⊥ ab) +to the ab plane using the zero-field-cooling (ZFC) and field-cooling (FC) protocols [47]. +First Principles Calculations. The first principles calculations were carried out with the projector +augmented wave method as implemented in the Vienna ab initio simulation Package [48-50]. The +generalized gradient approximation [51] of the Perdew-Burke-Ernzerhof [49] type was adopted for +the exchange-correlation function. The cutoff energy of the plane-wave basis was 500 eV and the +energy convergence standard was set to 10-6 eV. The 10 × 10 × 5 Monkhorst-Pack K-point mesh +was employed for the Brillouin zone sampling of the 1 × 1 × 1 unit cell. The experimental crystal +data were adopted to perform static calculations on RbTi3Bi5 both with and without SOC. To explore +the edge states, maximally localized Wannier functions (MLWFs) for the d orbitals of Ti and p +orbitals of Bi have been constructed [52, 53]. In addition, atomic SOC is added to the MLWFs based +Tight Binding (TB) Hamiltonian by fitting the first principles calculations. The TB model under +Wannier basis and iterative Green’s function method were used to calculate the surface state [54]. +Electronic Structure Measurement. Synchrotron ARPES and X-ray photoelectron spectroscopy +measurements with various photon energies on RbTi3Bi5 were performed at beamline 4B9B of the +Beijing Synchrotron Radiation Facility. In addition, a helium discharge lump (hv = 21.2 eV) was +used as a light source during ARPES measurement. The ARPES system was equipped with a + + +4 +ScientaR4000 electron analyzer and the base pressure is 7 × 10−11 Torr. The overall energy and +angular resolution were 17 meV and 0.3°, respectively. RbTi3Bi5 single crystals were cleaved in situ +along the (00l) (l = integer) plane. +3 Results and Discussion +Crystal structure, electrical transport, and magnetization. + +Fig. 1. Crystal structure and physical properties of ATi3Bi5. (a) Crystal structure of ATi3Bi5 (A += Rb, Cs) and the Ti-Bi layer with Ti-Kagome lattice. (b) X-ray diffraction patterns of as-grown +ATi3Bi5 (A = Rb, Cs) single crystals, showing (00l) (l = integer) reflections. The insets are the +optical photographs of ATi3Bi5 (A = Rb, Cs) single crystals. (c) Enlarged (004) diffraction peak of +ATi3Bi5 (A = Rb, Cs). (d) Temperature-dependent resistivity of ATi3Bi5 (A = Rb, Cs) with I // ab +and I // c. The insets show the corresponding measurement configurations. (e) Temperature- +dependent susceptibility of ATi3Bi5 (A = Rb, Cs) with H // ab and H ⊥ ab. +The crystal structures of ATi3Bi5 (A = Rb, Cs) are determined based on the SCXRD data and +summarized in Table SI – SIII. ATi3Bi5 (A = Rb, Cs) crystallizes in the hexagonal space group +P6/mmm (No. 191) with a = b = 5.8077(7) Å, c = 9.1297(11) Å for RbTi3Bi5 and a = b = 5.8079(2) +Å, c = 9.2400(4) Å for CsTi3Bi5, where α = β = 90° and γ = 120°. As shown in Fig. 1a, ATi3Bi5 (A += Rb, Cs) shares a similar quasi-2D structure motif with that of AV3Sb5 (A = K, Rb, and Cs), +featuring Ti-Bi layer with Ti-Kagome lattice, which is sandwiched by A+ acting as separator. As + +(a) +(b) +(c) +(004) +RbTi,Bis +CsTi,Bis +(004) +2 mm +(a.u.) +Ti +(600) +. +002) +Bi +Intensity ( +. +ORb/Cs +10.57° +20 +40 +60 +80 +39 +40 +20 (degree) +20 (degree) +(d) +Rb H // ab +Rb HI ab +CsH//ab×25 +3 +2.4 +Cs Hlab x3 +cm) +02 +ab +Q +Rb/ //ab×20 +1 +2.0 +Rb / l/ c +Cs / // ab ×50 +X +Cs I ll c +0 +1.8 +0 +100 +200 +300 +0 +100 +200 +300 +T (K) +T (K) +5 +shown in the insets of Fig. 1b, the as-grown single crystals of ATi3Bi5 (A = Rb, Cs) are plate-like +flakes with shiny metal luster, indicating a clear quasi-2D feature. The X-ray diffraction patterns of +the as-grown crystals are plotted in Fig. 1b with a preferential (00l) (l = integer) reflections. Fig. 1c +shows the enlarged (004) diffraction peaks for ATi3Bi5, where the peak of RbTi3Bi5 shifts 0.57 to +higher angle compared to that of CsTi3Bi5, indicating a shrinkage of distance between the structural +units along the c axis. The shrinkage corresponds well with the lattice parameters derived from +SCXRD due to the smaller cation radius of Rb+. The chemical composition is determined to be A : +Ti : Bi ~ 1 :3 : 5 according to the results of EDS (Fig. S1). +With current (~2 mA) being applied in the ab plane (I // ab) or along the c axis (I // c), the +resistivity of ATi3Bi5 (A = Rb, Cs) monotonically decreases with decreasing temperature (Fig. 1d), +showing a metallic-like behavior without obvious anomaly down to 2 K. Both the in-plane resistivity +of ATi3Bi5 (A = Rb, Cs) are much smaller than the out-of-plane resistivity, showing a quasi-2D +feature. The residual-resistance ratios (RRR) are calculated to be 47.6 (I // ab) and 26.2 (I // c) for +RbTi3Bi5, 8.6 (I // c) and 10.3 (I // ab) for CsTi3Bi5, hinting the good crystallinity of ATi3Bi5 (A = +Rb, Cs) single crystals. Similiar to AV3Sb5 compounds, all the resistivity below 40 K can be well +fitted using ρ = ρ0 + ATα with value of power (α) close to 2 (Fig. S2), indicating a Fermi liquid +behavior with moderate correlation between electrons [24]. Fig. 1e shows the magnetic +susceptibilities of ATi3Bi5 (A = Rb, Cs) with magnetic field (0.5 T) parallel (H // ab) and normal (H +⊥ ab) to the ab plane. Above 50 K, both the samples exhibit weak Pauli paramagnetism with the +small magnetic susceptibility (< 2.4  10-2 emu mol-1 Oe-1). The upturn at low temperature (< 50 K) +may originate from magnetic impurities. +Electronic structure of RbTi3Bi5. + + +6 + + +Fig. 2. Calculated bulk and surface band structures of RbTi3Bi5. (a) Calculated bulk band +structure of RbTi3Bi5 with SOC along high symmetry lines in the first Brillouin zone (BZ). The +green dots highlight the dispersions contributed mostly by the [Ti3Bi5]- layer with Ti-Kagome lattice, +with the upper saddle points, Dirac points, and lower saddle points donated as S1, D1, and S2, +respectively. The shaded areas (I, II, and III) denote the possible global gaps when considering SOC +and the bands separating them are labeled as b1 and b2. (b) Bulk BZ and (100) surface BZ with high +symmetry points and paths. (c) The schematic typical band structure and the chemical potential +difference between [Vb3Sb5]- and [Ti3Bi5]- layer with Kagome sub-lattice. (d) Total parity of +occupied states at eight time-reversal invariant momenta (TRIM) in bulk when considering shaded +area II as the global gap. (e) Parity of the two bands b1 and b2 at TRIM. (f) Surface band structure +of (100) plane. Zoomed-in surface band structure around E - EF = 0.8 eV of (100) plane along (g)X- +-X and (h)Z--Z. +Due to the quite similar crystal structure and physical properties, we only focus on the electronic +structure and topological properties of RbTi3Bi5 in this work. The electronic band structure with +SOC for RbTi3Bi5 is shown in Fig. 2a, where the bands along L-M and H-K are both fairly flat, +indicating a strong quasi-2D feature. There are a number of dispersions crossing EF, showing a +metallic feature that is consistent with the resistivity measurement. According to the density of states +(DOS) and fat band (Fig. S3), the bands near EF are mainly contributed by Ti and Bi atoms in the +Kagome lattice. The typical dispersions contributed by the Ti-Kagome lattice in [Ti3Bi5]- layer can +be clearly observed in the calculated band structures both with and without SOC, showing saddle + +(a) +(b) +d +With soC +M +M+ +E- E (eV) +M+ +M+ +(e) +b +b2 +[V3Sbs] +S1 +●D1 +M +S2 +[Ti,Bis] +L +A +M +K +A +L +H +A/L +MH +K +(f) +(h) +0.8 +E (eV) +(eV) +山 +山 +山0.6 +0.6 +0.4 +X +M +x +X +7 +7 +7 +points above EF at M (S1 around 0.8 eV and S2 around 0.3 eV) and Dirac points at K (D1 around +0.5 eV). Compared with RbV3Sb5 [24, 27], the number of valence electrons of RbTi3Bi5 decreases +by three in each unit cell, resulting in the large downward shift of chemical potential (Fig. 2c). The +upper saddle points (S1) and Dirac points coming from Kagome lattice in RbTi3Bi5 stay highly +above EF, making the FS instability being hard to happen, which should be the reason for the lack +of CDW or superconductivity in RbTi3Bi5. By continuous substitution of [Vb3Sb5]- with [Ti3Bi5]- +sub-lattice, EF would go through the upper saddle point S1, Dirac point D1, and lower saddle point +S2 consequently with emerging interesting phases. +Nevertheless, due to the dominated influence of Kagome lattice, RbTi3Bi5 is anticipated to have +nontrivial topological properties. Compared with the electronic band structure without SOC (Fig. +S3), a number of band crossings are gapped and a global gap (shaded area I) forms far above EF. In +particular, some electron pockets are formed by moving down the electron band at  point, which +gaps the band crossing along -M and defines another global gap (shaded area II) near EF. The +Dirac point at K is also gapped and seems to form a third global gap (shaded area III). According to +the location of EF, the parity of high symmetry points when taking shaded area II as the global gap +is calculated. As shown in Fig. 2d, RbTi3Bi5 is found to be a weak topological insulator having +nontrivial topological invariants (Z2; Z2, Z2, Z2) = (0; 0, 0, 1) with the coexisting inversion symmetry +and time reversal symmetry [39]. The topological properties of the other global gap (shaded area I) +far above EF can be easily derived from the parity of the band separating them (Fig. 2e). By +increasing the number of occupation states, like intercalation or electrical gating, RbTi3Bi5 can be +tuned into a trivial insulator when shaded area I is considered as the global gap. According to the +nontrivial topological invariants (Z2; Z2, Z2, Z2) = (0; 0, 0, 1), topological surface states are +anticipated to exist in the planes parallel to [001] direction. In order to gain a deeper insight of the +topological properties, we calculated the surface states along (100) plane of RbTi3Bi5. As shown in +Fig. 2f, a plenty of floating surface states can be clearly observed due to the breaking of translation +symmetry and reconstruction at the surface. The dispersions of surface states also show large +anisotropy, with complicated dispersions along X--X but simple ones along Z--Z. In the +global gap (~50 meV) around high symmetry point , two brunches of surface states link the bulk +valence states and conductance states (Fig. 2g, h). In particular, these two brunches of surface states +form a crossing at , which should be a surface Dirac cone. These results demonstrate RbTi3Bi5 is +a weak topological insulator with both nontrivial topological invariants and gapless surface states. + + + +8 + +Fig. 3. Experimentally observed band structure for RbTi3Bi5 and its comparison with DFT +calculations. (a) Calculated FS in -M-K plane for RbTi3Bi5 at E - EF = 0 eV, with high-symmetry +points Γ, M, and K labeled. (b) FS intensity plot in -M-K plane of RbTi3Bi5 recorded at hν = 21.2 +eV, obtained by integrating the spectral weight within ±10 meV with respect to EF. i (i = 1, 2, 3), +, and  donate the multiple pockets around high-symmetry points Γ, M, and K, respectively. (c) +ARPES intensity maps of -M-K plane at E - EF = -0.2 eV. (d) Photoemission intensity plot along +K--K with hν = 21.2 eV and (e) its second derivative intensity plot. (f) Photoemission intensity +plot along M--M with hν = 21.2 eV and (g) its second derivative intensity plot. The arrows indicate +the crossing points at EF and labeled as in (b). The red lines are the calculated band structures with +SOC. Intensity plots of the ARPES data along -M at (h) E − EF = 0 eV and (i) E - EF = -0.2 eV +collected in a range of photon energy from 20 eV to 55 eV. The triangles indicate the corresponding +crossing points at EF as in (b) and (f). +Fig. 3a shows the calculated FS in -M-K plane of RbTi3Bi5 with SOC, which is well consistent +with the measured FS of RbTi3Bi5 (Fig. 3b). On the in situ cleaved surface of (001) plane (Fig. S4), +the electron pocket 1 around  point can be clearly observed, indicating the existence of large SOC +interactions. Two larger hexagonal-flower-shaped electron pockets around  point (2 and 3) can + +(a) +(b) +kx (A-1) +(c) +kx (A-1) +kx +kz = 0 +E-E=0eV +E-E=-0.2eV +0.6 +0.6 +ky +K +K ++M +M +M +0 +ky +-0.6 +-0.6 +-0.6 +0 +0.6 +-0.6 +0 +0.6 +(d) +(f) +(h) +0.0 +0ev +α2α3 +50- +40 +hv (eV) +E -1.0 - +30 +hv = 21.2 eV +hv = 21.2 eV +-1.5 +20 +(e) +g +(i) +0.0 +-0.2ev +50- +hv (eV), +40 +E-1.0 +30 +-1.5 +20 +K +K +M +M +M +9 +be further resolved, with a diamond-shaped hole pocket at M point () and a triangular-shaped +electron pocket at K point (). Fig. 3c shows the ARPES intensity maps of -M-K plane at E - EF = +-0.2 eV, showing the shrinkage of electron pockets around  or K points and the enlargement of +hole pocket at M point with the increasing binding energy. Fig. 3d is the experimentally observed +bands along K--K at hν = 21.2 eV, showing four crossing points at EF in a sequence of (1, 2, 3, +). As for the experimentally observed bands along M--M at hν = 21.2 eV (Fig. 3f), another four +crossing points in a sequence of (1, 2, 3, ) can be observed, where the intensity of 2 and 3 +cannot be well resolved because they are spatially close in momentum space. By noise reduction +using a machine learning process [55], a small intensity of 2 shows up (Fig. S5). The corresponding +second derivative intensity plots (Fig. 3e and g) show remarkable agreement with the calculated +electronic structure for RbTi3Bi5 with SOC. The experimentally observed bands can be fully +recovered by calculation from E - EF = -1.0 eV up to EF, with the calculated EF only move down +0.05 eV. The band structure along -M has been measured by varying the photon energy from 20 +eV to 55 eV. As shown in Fig. 3h and i, only the change in intensity is observed at different photon +energies for both E − EF = 0 and -0.2 eV, indicating a strong quasi-2D feature in RbTi3Bi5. +Both the electron pocket 1 around  and its well consistency with calculation indicate the +existence of a strong SOC interaction, which would induce a great splitting and form a relatively +large gap at M point. Thus, RbTi3Bi5 should be an experimentally observable weak topological +insulator with surface states showing up at the global gap (shaded area II in Fig. 2a). Unlike the Z2 +topological insulator AV3Sb5, the bulk gap of RbTi3Bi5 is far above EF and the gapless surface states +reside on the (100) plane. To experimentally observe the surface states, cleaving along the (100) +plane and pumping the electrons up would be necessary. In most previous study of Kagome lattice, +whether theoretically or experimentally, the upper saddle points were the focus due to the position +of EF and the existence of surface states around [11, 23, 56-58]. By changing the filling of electrons +and getting the upper saddle points close to EF, the possible superconductivity, CDW or chiral spin +order would happen. However, the lower saddle point has received little attention. Due to the large +downward shift of chemical potential, the lower saddle point of RbTi3Bi5 is much more close to EF. +It would be of great interest to explore the possible new phases related to the lower saddle points by +lifting EF using electron doping or strain engineering. + + + +(a) +(b) +5% +=5% +E- (eV) +- +E +M +K +M +K +M +K +10 +Fig. 4. Strain engineering in RbTi3Bi5 monolayer. (a) Schematic pictures of RbTi3Bi5 monolayer. +Electronic structure of RbTi3Bi5 monolayer with SOC under (b) no strain ( = 0), (c) -5% +compressive strain ( = 0.95) and (d) 5% tensile strain ( = 1.05). The typical dispersions contributed +by the [Ti3Bi5]- layer with Ti-Kagome lattice is highlighted with green dots, and the lower saddle +points is marked by a red dot. +Owing to its strong quasi-2D feature, RbTi3Bi5 single crystal can be mechanically exfoliated +into thin flakes with thickness down to 10 nm using the scotch-tape method (Fig. S6) [59], which +contains several unit cells. This shows a potential for RbTi3Bi5 to reach the monolayer limit, making +it an ideal candidate for strain engineering or electrical gating, which would be efficient in regulating +its properties. Here the in-plane biaxial strain is theoretically tried to explore the effectiveness of +strain engineering for modulating the electronic properties of RbTi3Bi5 monolayer. As shown in Fig. +4a, the RbTi3Bi5 monolayer is a hexagonal lattice with [Ti3Bi5]- layer sandwiched by Rb atoms, +which is indeed stoichiometric Rb2Ti3Bi5 and would result in an electron doping. The in-plane +biaxial strain is defined as ε = (a - a0)/a0 × 100%, where a and a0 is the in-plane lattice constant of +the strained and unstrained monolayer, respectively. +In RbTi3Bi5 monolayer, the typical dispersions contributed by the [Ti3Bi5]- layer with Ti- +Kagome lattice can also be resolved (Fig. S7 and S8). Compared with the electronic structure of +bulk RbTi3Bi5, the chemical potential of unstrained RbTi3Bi5 monolayer shifts upward for about +0.15 eV, making the lower saddle points at M (S2) closer to EF (~ 0.05 eV) (Fig. 4b). The band +structure of RbTi3Bi5 monolayer was changed clearly with biaxial strain from −5% to 5%. With the +compressive strain, the location of S2 remains above EF and goes up further (Fig. 4c), reaching 0.3 +eV up to -5% compressive strain. Under the tensile strain, the location of S2 goes down and crosses +EF with 5% tensile strain (Fig. 4d). By the in-plane biaxial strain, the position of the lower saddle +points of RbTi3Bi5 monolayer can be continually tuned, which may induce interesting phases, +including possible superconductivity, CDW or chiral spin order. +4 Conclusion + +In summary, a new ATi3Bi5 (A = Rb, Cs) family with Ti-Kagome lattice is synthesized. ATi3Bi5 +(A = Rb, Cs) is found to own strong quasi-2D feature, moderate electron correlation, and small Pauli +paramagnetism. Based on first principles calculations considering SOC, RbTi3Bi5 is predicted to be +a weak topological insulator with gapless surface states in its global bulk gap along the (100) plane. +The experimentally observed band structure along (001) plane by ARPES fits well with the +calculation and indicates the existence of strong SOC. The in-plane biaxial strain can effectively +tune the position of lower saddle point coming from the Ti-Kagome lattice in its monolayer. These +results show ATi3Bi5 (A = Rb, Cs) is a new weak topological insulator candidate and good platform +to explore new phases of Kagome metals. +Supporting information +Tables SI-SIII show the crystal structure of ATi3Bi5 (A = Rb, Cs) single crystals. Figure S1 shows +the SEM images and chemical composition of ATi3Bi5 (A = Rb, Cs) single crystals. Figure S2 shows + + +11 +the power law fittings of the resistivity below 40 K. Figure S3 shows the calculated electronic +structure without SOC, DOS with SOC, and fat bands both with and without SOC. Figure S4 shows +the low energy electron diffraction pattern of (001) plane after in situ cleavage. Figure S5 shows the +noise reduction of the photoemission intensity plot along M--M using a machine learning process. +Figure S6 shows thin flakes of RbTi3Bi5 by mechanical exfoliation. Figure S7 shows the strain- +tunable electronic structure of RbTi3Bi5 monolayer without SOC. Figure S8 shows the strain-tunable +fat band of RbTi3Bi5 monolayer with SOC. +. +Author information +Notes +The authors declare no competing financial interest. + +Note added: During the preparation of this manuscript, we recognized some unpublished preprints +about CsTi3Bi5 on its superconductivity, topology, and nematicity (https://arxiv.org/abs/2209.03840, +https://arxiv.org/abs/2211.12264v1, and https://arxiv.org/abs/2211.16477). + +Acknowledgements +Y. Zhou, L. Chen, and X. C. Ji contributed equally to this work. Y. Zhou, L. Chen, and G. Wang +would like to thank Prof. X. L. Chen and Prof. J. P. Hu of the Institute of Physics, Chinese Academy +of Sciences for helpful discussions. This work was partially supported by the National Key Research +and Development Program of China (Grant Nos. 2018YFE0202600 and 2022YFA1403900) and the +National Natural Science Foundation of China (Grant No. 51832010, 11888101, Grants No. +11925408, No. 11921004, and No. 12188101), the Ministry of Science and Technology of China +(Grants No. 2018YFA0305700 and No. 2022YFA1403800), the Chinese Academy of Sciences +(Grant No. XDB33000000) and the Informatization Plan of the Chinese Academy of Sciences +(Grant No. CAS WX2021SF-0102), and the Center for Materials Genome. + +References +[1] J. Strečka, L. Čanová, M. Jaščur, M. Hagiwara, Exact solution of the geometrically frustrated spin- +1/2 Ising-Heisenberg model on the triangulated kagome (triangles-in-triangles) lattice, Phys. Rev. B, 78 +(2008) 024427. +[2] H.M. Guo, M. Franz, Topological insulator on the kagome lattice, Phys. Rev. B, 80 (2009) 113102. +[3] S. Yan, D.A. Huse, S.R. 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Van Noorden, Production beyond sticky tape, Nature, 483 (2012) S32-S33. + + + + + + +16 +Supporting information for +Physical properties, electronic structure, and strain-tuned monolayer of the weak topological +insulator RbTi3Bi5 with Kagome lattice + +Ying Zhou1, 2, #, Long Chen1, 2, #, Xuecong Ji1, 2, #, Chen Liu3, Ke Liao 1, 2, Zhongnan Guo4, Jia-Ou +Wang3, Hongming Weng1, 2, 5, *, Gang Wang1, 2, 5, * + +1 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese +Academy of Sciences, Beijing 100190, China +2 University of Chinese Academy of Sciences, Beijing 100049, China +3 Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of +Sciences, Beijing 100049, China +4 Department of Chemistry, School of Chemistry and Biological Engineering, University of +Science and Technology Beijing, Beijing 100083, China +5 Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China +# These authors contributed equally to this work. +*Corresponding author. Email: gangwang@iphy.ac.cn; hmweng@iphy.ac.cn. + + + + +17 +1 Crystal structure and composition of ATi3Bi5 (A = Rb, Cs). +Table SI. Crystallographic data and structure refinement of ATi3Bi5 (A = Rb, Cs). +Empirical formula +RbTi3Bi5 +CsTi3Bi5 +f.u. weight (g/mol) +1278.27 +1321.51 +S.G. / Z +P6/mmm (No.191) / 1 +Unit cell +parameter + +a (Å) +5.8077(7) +5.8079(2) +b (Å) +5.8077(7) +5.8079(2) +c (Å) +9.1297(11) +9.2400(4) +α, β () +90 +90 +γ () +120 +120 +V (Å3) +266.68(6) +269.92(2) +dcal (g/cm3) +7.928 +8.130 +Refl. Collectd / unique + 2679 / 119 +1652 / 151 +Rint +0.0739 +0.0925 +Goodness-of-fit +1.537 +1.073 +R1 / wR2 (I > 2σ(I)) +0.0227 / 0.0489 +0.0251 / 0.0651 +R1 / wR2 (all) +0.0238 / 0.0495 +0.0258 / 0.0655 + +Table SII. Atomic coordinates and equivalent isotropic displacement parameters for RbTi3Bi5. +Atom +Wyck. +Sym. +x/a +y/b +z/c +Occ. +U(eq)(Å2) +Rb +1b +6/mmm +1.0 +1.0 +0 +1.0 +0.0338(19) +Ti1 +3g +mmm +0.5 +0.5 +0.5 +1.0 +0.0083(13) +Bi1 +1a +6/mmm +1.0 +1.0 +0.5 +1.0 +0.0109(6) +Bi2 +4h +3m +0.6667 +0.3333 +0.23386(12) +1.0 +0.0126(4) + +Table SIII. Atomic coordinates and equivalent isotropic displacement parameters for CsTi3Bi5. +Atom +Wyck. +Sym. +x/a +y/b +z/c +Occ. +U(eq)(Å2) +Cs +1b +6/mmm +1.0 +1.0 +0 +1.0 +0.0144(6) +Ti1 +3g +mmm +0.5 +0.5 +0.5 +1.0 +0.0040(8) +Bi1 +1a +6/mmm +1.0 +1.0 +0.5 +1.0 +0.0056(4) +Bi2 +4h +3m +0.6667 +0.3333 +0.23839(7) +1.0 +0.0066(4) + + + + +18 + +Figure S1. SEM images of as-grown (a) RbTi3Bi5 and (c) CsTi3Bi5 single crystals and +corresponding elemental mapping. Typical EDSs and elemental composition collected on the flat +clean surface of (b) RbTi3Bi5 and (d) CsTi3Bi5 single crystals. + +Figure S2. Power law fittings of the in-plane resistivity (ab) of (a) RbTi3Bi5 and (b) CsTi3Bi5 single +crystals below 40 K. + +2 Electronic structure of ATi3Bi5 (A = Rb, Cs). + + +(a) +Rb +(b) +Rb:9.56% +Ti: 33.94% +Bi:56.60% +Bi +30μm +0 +2 +4 +6 +8 +keV +(c) +CS +(d) +Cs: 11.44% +Ti: 33.44% +Bi:55.12% +Bi +30um +- +6 +8 keV(a) +6 +(b) 10 +0—Pab - RbTi,Bis +0 Pab - CsTigBis +p=P+AT +p=P+ATα +Po = 3.8(5) × 10-6 m2 cm +P = 5.2(1) × 10-6 mΩ2 cm +8 +A = 6.41(1) × 10-9 m2 cm K-1.93 +A = 1.81(2) × 10-9 m2 cm k-2.04 +α = 1.93(3) +α= 2.04(3) +6 +0 +4 +0 +10 +20 +30 +40 +50 +0 +10 +20 +30 +40 +50 +T (K) +T (K) +19 + +Figure S3. (a) Calculated band structure of RbTi3Bi5 without SOC along high symmetry lines in the +first Brillouin zone. The green dots highlight the dispersions contributed by the [Ti3Bi5]- layer with +Ti-Kagome lattice, and the orange circles highlight the Dirac points and band crossings. (b) Total +and partial density of states near the Fermi level. Fat band of RbTi3Bi5 (c) without SOC and (d) with +SOC. + + +Figure S4. Low energy electron diffraction pattern of RbTi3Bi5 (001) plane after cleavage at hv = +98 eV, showing a clear hexagonal shape. + +(a) 2 +(b) +2 +Without soc +Rb +Bi +Total +E- E (eV) +E- E (eV) +EF +-1 +2 +.2 +M +K +A +L +H +A/L M|H +K +c +(d) 2 +Rb +Without soC +Rb +Withsoc +Bi +(eV) +(eV) +出 +M +K +厂 +A +L +H +A|L M|H K +M +K +A +L +H +A/L MH K +20 + + +Figure S5. (a) Photoemission intensity plot along M--M measured with hν = 30 eV and its +momentum distribution curve (MDC) at E - EF = - 0.08 eV (red line). (b) Photoemission intensity +plot after noise reduction of (a) using a machine learning process and its MDC at E - EF = - 0.08 eV. +The white dashed line is the Fermi level and i (i = 1, 2, 3),  are the multiple pockets along M-- +M. (c) MDC plot of (b). The black line highlights the Fermi level and the triangles outline the shape +of the dispersions. +Since the count signal collected on the angle-resolved photoemission spectroscopy detector +introduces white Gaussian noise, we use the semi-supervised Noise2Noise algorithm [1], which has +good ability to remove white Gaussian noise in depth learning, to suppress white Gaussian noise +from raw data. Figure S5a shows the simple sum of 10 independent data acquisition of +photoemission intensity plot along M--M with hν = 30 eV. And the results of white Gaussian noise +suppression using 10 independently collected image plots as training data are shown in Figure S5b. + + +Figure S6. (a) Atomic force microscopy image of thin flakes of RbTi3Bi5 by exfoliation and (b) +corresponding height profile. +Thin flakes of RbTi3Bi5 crystal were exfoliated using the scotch-tape method [2] and transferred +onto a Si/SiO2 substrate. Atomic force microscopy of thin flakes was performed using an atomic force +microscope (Multimode 8.0, Bruker, USA) in a ScanAsyst mode. + + + +(a) +(b) +C +0 +0 +(eV) +-0.4 +山 +ntensity (a.u.) +-0.8 +-0.8 +a +a +Intensity (a.u.) +Intensity (a.u.) +0.8 +0.4 +0 +-0.4 +0.8 +0.4 +0 +-0.4 +0.8 +0.4 +0 +kx (A-1) +kx (A-1) +kx (A-1)(a) +(nm) +(b) +20 +50 +二L2 +40 +.10 +30 +0 +30nm +20 +20nm +MM +10 +-10 +10nm +um +0 +-20 +0 +1 +2 +3 +4 +5 +Distance (um) +21 + +Figure S7. (a) Electronic structure of RbTi3Bi5 monolayer without SOC under (a) no strain (ε = +0), (b) -5% compressive strain (ε = 0.95) and (c) 5% tensile strain (ε = 1.05). The typical dispersions +mostly contributed by the [Ti3Bi5]- layer with Ti-Kagome lattice is highlighted with green dots, and +the lower saddle points is marked by a red dot. + + +Figure S8. Fat band of RbTi3Bi5 monolayer with SOC under (a) no strain (ε = 0), (b) -5% +compressive strain (ε = 0.95) and (c) 5% tensile strain (ε = 1.05). + + +[1] Noise2Noise: Learning image restoration without clean data, arXiv:1803.04189, DOI. +[2] R. Van Noorden, Production beyond sticky tape, Nature, 483 (2012) S32-S33. + + +a) +(b +C +=0 +=0.95 += 1.05 +E (eV) +- +E +-1 +-2 +M +K +M +K +M +Ka +b +C +Rb +Rb +=0.95 +Rb +8=0 += 1.05 +B +Ti +1 +Bi +E- E- (eV) +(eV) +(eV) +C +-2 +2 +M +K +M +K +M +K \ No newline at end of file diff --git a/otAzT4oBgHgl3EQfqv1Z/content/tmp_files/load_file.txt b/otAzT4oBgHgl3EQfqv1Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b3c8953d606822291e6836d3032ebda65f6cc810 --- /dev/null +++ b/otAzT4oBgHgl3EQfqv1Z/content/tmp_files/load_file.txt @@ -0,0 +1,1302 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf,len=1301 +page_content='1 Physical properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' electronic structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' and strain-tuned monolayer of the weak topological insulator RbTi3Bi5 with Kagome lattice Ying Zhou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Long Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Xuecong Ji1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chen Liu3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Ke Liao 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Zhongnan Guo4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Jia’ou Wang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Hongming Weng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Gang Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' * 1 Beijing National Laboratory for Condensed Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 2 University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 3 Beijing Synchrotron Radiation Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 4 Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' School of Chemistry and Biological Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 5 Songshan Lake Materials Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Dongguan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Guangdong 523808,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China # These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Email: gangwang@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' hmweng@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Kagome metals AV3Sb5 (A = K, Rb, and Cs) with a V-Kagome lattice acting as a fertile platform to investigate geometric frustration, electron correlation, superconductivity, and nontrivial band topology, have attracted tremendous attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Here we reported the structure and properties of ATi3Bi5 (A = Rb, Cs) family with a Ti-Kagome lattice, specifically focusing on the electronic structure and nontrivial band topology of RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' ATi3Bi5 (A = Rb, Cs) is found to be non- superconducting metal with strong quasi-two-dimensional feature, moderate electron correlation, and small Pauli paramagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Based on first principles calculations, RbTi3Bi5 is determined to be a weak topological insulator with gapless surface states along (100) plane, and the electronic band structure along (001) plane is in great agreement with experimentally observed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In particular, the electronic properties of the RbTi3Bi5 monolayer can be efficiently tuned by a biaxial strain according to calculation, with its lower saddle points coming from Kagome lattice approaching the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' These results highlight ATi3Bi5 (A = Rb, Cs) with Ti-Kagome lattice is a new Kagome metal to explore nontrivial band topology and exotic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1 Introduction Kagome metals, featuring two-dimensional (2D) Kagome sub-lattice with corner-sharing triangles, have been serving as a fertile platform to investigate various electronic phenomena due to the strong geometric frustration [1] and nontrivial band topology [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As a model system to realize potential quantum spin liquid [3], the properties of spins on the Kagome lattice have been extensively studied [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Owing to the unique crystal structure, Kagome metals naturally support electronic structures host Dirac points, flat bands, and saddle points, resulting in topologically nontrivial states [6-8], possible fractionalization [9, 10] and Fermi surface (FS) instabilities [11], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' With the combination of magnetism and topology, a 2D “Chern gap” [12] and a large anomalous Hall response [13-15] have been found in Kagome metals with the existing long-range magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By changing the filling of electrons in Kagome lattice, exotic phenomenon such as 2 density wave orders [2, 16, 17], charge fractionalization [18, 19], and superconductivity [11, 20], could be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Especially when Fermi level (EF) is near the saddle points, the spin or charge density wave (CDW) states would happen [11, 21-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Recently, the AV3Sb5 (A = K, Rb, and Cs) family with V-Kagome lattice has been discovered [24] and attracted tremendous attention due to its CDW states, superconductivity, and nontrivial band topology [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' All the AV3Sb5 compounds were found to be superconducting at low temperatures (TC ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3 K for CsV3Sb5 [24, 25], TC ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='75 K for RbV3Sb5 [27], and TC ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='93 K for KV3Sb5 [28]), together with an unconventional long-range CDW transition occurring at high temperatures (TCDW ~ 94 K for CsV3Sb5 [25], TCDW ~ 103 K for RbV3Sb5 [27], and TCDW ~ 78 K for KV3Sb5 [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Though without long-range magnetic order, an extraordinarily large anomalous Hall effect was observed in KV3Sb5 and CsV3Sb5 [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By applying hydrostatic pressure or carrier doping, multiple superconducting domes can be induced in AV3Sb5, showing the competition between CDW and superconductivity [31-36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' According to density functional (DFT) calculations and angle- resolved photoemission spectroscopy (ARPES) measurements, the AV3Sb5 materials are effectively modeled as weakly correlated systems with strong quasi-2D nature [25, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In addition, AV3Sb5 is found to be a Z2 topological metal with unconventional surface states [25, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' These researches highlight the materials with Kagome lattice, such as AV3Sb5, being a promising platform to explore topological superconductivity, correlated electronic states, and topological quantum computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In the exploration of new Kagome metals, ATi3Bi5 (A = Rb, Cs) [40, 41] stands out as another family with possible superconductivity [41], orbital-selective nematic order [42] or electronic nematicity [43], and nontrivial topology [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' However, most of attention has been focused on CsTi3Bi5 rather than RbTi3Bi5, and their topological properties remain controversial and is still lack systematical investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Here we report the structure and properties of ATi3Bi5 (A = Rb, Cs) family with a Ti-Kagome lattice, focusing on the electronic structure and nontrivial band topology of RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' ATi3Bi5 (A = Rb, Cs) shares a similar quasi-2D structure motif with that of AV3Sb5, featuring Ti-Bi layer in Kagome lattice which is sandwiched by A+ as separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The resistivity of ATi3Bi5 (A = Rb, Cs) exhibits a metallic-like behavior without obvious anomaly down to 2 K, showing a quite large anisotropy and moderate electron correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' A small magnetic susceptibility has been observed without any long-range magnetic order or superconducting transition in the whole measured temperature range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The electronic band structure of RbTi3Bi5 has been characterized by ARPES and fits well with DFT calculations with spin-orbit coupling (SOC), from which RbTi3Bi5 is determined to be a weak topological insulator with gapless surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The upper saddle points coming from Kagome lattice are proved to be far away (~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 eV) from EF, whereas the lower saddle points are close (~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3 eV) to EF, which may result interesting states like superconductivity or CDW by changing the filling of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Furthermore, due to its strong quasi-2D feature, a biaxial strain could be used to effectively modulate the electronic properties of the RbTi3Bi5 monolayer based on first principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2 Experimental Single Crystal Growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' ATi3Bi5 (A = Rb, Cs) single crystals were grown by a high-temperature solution method using Bi as flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Rb/Cs chunk (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='75%, Alfa Aesar), Ti powder (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='95%, Alfa Aesar), and Bi granules (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='999%, Sinopharm) were mixed using a molar ratio of Rb/Cs : Ti : Bi = 3 2 : 4 : 12 in a fritted alumina crucible set (Canfield crucible set) [44] and sealed in a fused-silica ampoule at vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The ampoule was heated to 1073 K over 15 h, held at the temperature for 24 h, and then slowly cooled down to 873 K at a rate of 2 K/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' At 873 K, large single crystals with size up to 5 mm ×5 mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 mm were separated from the remaining liquid by centrifuging the ampoule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The obtained single crystals are shiny-silver plates and air-sensitive, so all manipulations and specimen preparation for structure characterization and property measurements were handled in an argon-filled glovebox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Structure Characterization and Composition Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' X-ray diffraction data were obtained using a PANalytical X’Pert PRO diffractometer (Cu Kα radiation, λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='54178 Å) with a graphite monochromator in a reflection mode (2θ = 5\uf0b0–80\uf0b0, step size = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='017\uf0b0) operated at 40 kV voltage and 40 mA current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Indexing and Rietveld refinement were performed using the DICVOL91 and FULLPROF programs [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Single crystal X-ray diffraction (SCXRD) data were collected using a Bruker D8 VENTURE with Mo Kα radiation (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='71073 Å) at 280 K for RbTi3Bi5 and a four- circle diffractometer (Rigaku XtaLAB Synergy R-DW, HyPix) with multilayer mirror graphite- monochromatized Mo Kα radiation (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='71073 Å) at 180 K for CsTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The structure was solved using a direct method and refined with the Olex2 package [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The morphology and element analyses were characterized using a scanning electron microscope (SEM, Phenom Prox) equipped with an electron microprobe analyzer for semiquantitative elemental analysis in energy-dispersive spectroscopy (EDS) mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Five spots in different areas were measured on one crystal using EDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Physical Property Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The resistivity, magnetic susceptibility, and heat capacity measurements were carried out using a physical property measurement system (Quantum Design, 9 T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The resistivity was measured using the standard four-probe configuration with the applied current (about 2 mA) parallel to the ab plane (I // ab) or along the c axis (I // c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Magnetic susceptibility was measured under a magnetic field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 T) parallel (H // ab) and normal (H ⊥ ab) to the ab plane using the zero-field-cooling (ZFC) and field-cooling (FC) protocols [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' First Principles Calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The first principles calculations were carried out with the projector augmented wave method as implemented in the Vienna ab initio simulation Package [48-50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The generalized gradient approximation [51] of the Perdew-Burke-Ernzerhof [49] type was adopted for the exchange-correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The cutoff energy of the plane-wave basis was 500 eV and the energy convergence standard was set to 10-6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The 10 × 10 × 5 Monkhorst-Pack K-point mesh was employed for the Brillouin zone sampling of the 1 × 1 × 1 unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The experimental crystal data were adopted to perform static calculations on RbTi3Bi5 both with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' To explore the edge states, maximally localized Wannier functions (MLWFs) for the d orbitals of Ti and p orbitals of Bi have been constructed [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In addition, atomic SOC is added to the MLWFs based Tight Binding (TB) Hamiltonian by fitting the first principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The TB model under Wannier basis and iterative Green’s function method were used to calculate the surface state [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Electronic Structure Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Synchrotron ARPES and X-ray photoelectron spectroscopy measurements with various photon energies on RbTi3Bi5 were performed at beamline 4B9B of the Beijing Synchrotron Radiation Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In addition, a helium discharge lump (hv = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV) was used as a light source during ARPES measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The ARPES system was equipped with a 4 ScientaR4000 electron analyzer and the base pressure is 7 × 10−11 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The overall energy and angular resolution were 17 meV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3°, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' RbTi3Bi5 single crystals were cleaved in situ along the (00l) (l = integer) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3 Results and Discussion Crystal structure, electrical transport, and magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Crystal structure and physical properties of ATi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Crystal structure of ATi3Bi5 (A = Rb, Cs) and the Ti-Bi layer with Ti-Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (b) X-ray diffraction patterns of as-grown ATi3Bi5 (A = Rb, Cs) single crystals, showing (00l) (l = integer) reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The insets are the optical photographs of ATi3Bi5 (A = Rb, Cs) single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (c) Enlarged (004) diffraction peak of ATi3Bi5 (A = Rb, Cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (d) Temperature-dependent resistivity of ATi3Bi5 (A = Rb, Cs) with I // ab and I // c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The insets show the corresponding measurement configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (e) Temperature- dependent susceptibility of ATi3Bi5 (A = Rb, Cs) with H // ab and H ⊥ ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The crystal structures of ATi3Bi5 (A = Rb, Cs) are determined based on the SCXRD data and summarized in Table SI – SIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' ATi3Bi5 (A = Rb, Cs) crystallizes in the hexagonal space group P6/mmm (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 191) with a = b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8077(7) Å, c = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='1297(11) Å for RbTi3Bi5 and a = b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8079(2) Å, c = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2400(4) Å for CsTi3Bi5, where α = β = 90° and γ = 120°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1a, ATi3Bi5 (A = Rb, Cs) shares a similar quasi-2D structure motif with that of AV3Sb5 (A = K, Rb, and Cs), featuring Ti-Bi layer with Ti-Kagome lattice, which is sandwiched by A+ acting as separator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As (a) (b) (c) (004) RbTi,Bis CsTi,Bis (004) 2 mm (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=') Ti (600) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 002) Bi Intensity ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' ORb/Cs 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='57° 20 40 60 80 39 40 20 (degree) 20 (degree) (d) Rb H // ab Rb HI ab CsH//ab×25 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 Cs Hlab x3 cm) 02 ab Q Rb/ //ab×20 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 Rb / l/ c Cs / // ab ×50 X Cs I ll c 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 0 100 200 300 0 100 200 300 T (K) T (K) 5 shown in the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1b, the as-grown single crystals of ATi3Bi5 (A = Rb, Cs) are plate-like flakes with shiny metal luster, indicating a clear quasi-2D feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The X-ray diffraction patterns of the as-grown crystals are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1b with a preferential (00l) (l = integer) reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1c shows the enlarged (004) diffraction peaks for ATi3Bi5, where the peak of RbTi3Bi5 shifts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='57\uf0b0 to higher angle compared to that of CsTi3Bi5, indicating a shrinkage of distance between the structural units along the c axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The shrinkage corresponds well with the lattice parameters derived from SCXRD due to the smaller cation radius of Rb+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The chemical composition is determined to be A : Ti : Bi ~ 1 :3 : 5 according to the results of EDS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' With current (~2 mA) being applied in the ab plane (I // ab) or along the c axis (I // c), the resistivity of ATi3Bi5 (A = Rb, Cs) monotonically decreases with decreasing temperature (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1d), showing a metallic-like behavior without obvious anomaly down to 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Both the in-plane resistivity of ATi3Bi5 (A = Rb, Cs) are much smaller than the out-of-plane resistivity, showing a quasi-2D feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The residual-resistance ratios (RRR) are calculated to be 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 (I // ab) and 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 (I // c) for RbTi3Bi5, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 (I // c) and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3 (I // ab) for CsTi3Bi5, hinting the good crystallinity of ATi3Bi5 (A = Rb, Cs) single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Similiar to AV3Sb5 compounds, all the resistivity below 40 K can be well fitted using ρ = ρ0 + ATα with value of power (α) close to 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S2), indicating a Fermi liquid behavior with moderate correlation between electrons [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 1e shows the magnetic susceptibilities of ATi3Bi5 (A = Rb, Cs) with magnetic field (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 T) parallel (H // ab) and normal (H ⊥ ab) to the ab plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Above 50 K, both the samples exhibit weak Pauli paramagnetism with the small magnetic susceptibility (< 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 \uf0b4 10-2 emu mol-1 Oe-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The upturn at low temperature (< 50 K) may originate from magnetic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Electronic structure of RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Calculated bulk and surface band structures of RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Calculated bulk band structure of RbTi3Bi5 with SOC along high symmetry lines in the first Brillouin zone (BZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The green dots highlight the dispersions contributed mostly by the [Ti3Bi5]- layer with Ti-Kagome lattice, with the upper saddle points, Dirac points, and lower saddle points donated as S1, D1, and S2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The shaded areas (I, II, and III) denote the possible global gaps when considering SOC and the bands separating them are labeled as b1 and b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (b) Bulk BZ and (100) surface BZ with high symmetry points and paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (c) The schematic typical band structure and the chemical potential difference between [Vb3Sb5]- and [Ti3Bi5]- layer with Kagome sub-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (d) Total parity of occupied states at eight time-reversal invariant momenta (TRIM) in bulk when considering shaded area II as the global gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (e) Parity of the two bands b1 and b2 at TRIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (f) Surface band structure of (100) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Zoomed-in surface band structure around E - EF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 eV of (100) plane along (g)\uf060X- \uf060\uf047-\uf060X and (h)\uf060Z-\uf060\uf047-\uf060Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Due to the quite similar crystal structure and physical properties, we only focus on the electronic structure and topological properties of RbTi3Bi5 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The electronic band structure with SOC for RbTi3Bi5 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2a, where the bands along L-M and H-K are both fairly flat, indicating a strong quasi-2D feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' There are a number of dispersions crossing EF, showing a metallic feature that is consistent with the resistivity measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' According to the density of states (DOS) and fat band (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S3), the bands near EF are mainly contributed by Ti and Bi atoms in the Kagome lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The typical dispersions contributed by the Ti-Kagome lattice in [Ti3Bi5]- layer can be clearly observed in the calculated band structures both with and without SOC, showing saddle (a) (b) d With soC M M+ E- E (eV) M+ M+ (e) b b2 [V3Sbs] S1 D1 M S2 [Ti,Bis] L A M K A L H A/L MH K (f) (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 E (eV) (eV) 山 山 山0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 X M x X 7 7 7 points above EF at M (S1 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 eV and S2 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3 eV) and Dirac points at K (D1 around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Compared with RbV3Sb5 [24, 27], the number of valence electrons of RbTi3Bi5 decreases by three in each unit cell, resulting in the large downward shift of chemical potential (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The upper saddle points (S1) and Dirac points coming from Kagome lattice in RbTi3Bi5 stay highly above EF, making the FS instability being hard to happen, which should be the reason for the lack of CDW or superconductivity in RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By continuous substitution of [Vb3Sb5]- with [Ti3Bi5]- sub-lattice, EF would go through the upper saddle point S1, Dirac point D1, and lower saddle point S2 consequently with emerging interesting phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Nevertheless, due to the dominated influence of Kagome lattice, RbTi3Bi5 is anticipated to have nontrivial topological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Compared with the electronic band structure without SOC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S3), a number of band crossings are gapped and a global gap (shaded area I) forms far above EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In particular, some electron pockets are formed by moving down the electron band at \uf047 point, which gaps the band crossing along \uf047-M and defines another global gap (shaded area II) near EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The Dirac point at K is also gapped and seems to form a third global gap (shaded area III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' According to the location of EF, the parity of high symmetry points when taking shaded area II as the global gap is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2d, RbTi3Bi5 is found to be a weak topological insulator having nontrivial topological invariants (Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Z2, Z2, Z2) = (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 0, 0, 1) with the coexisting inversion symmetry and time reversal symmetry [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The topological properties of the other global gap (shaded area I) far above EF can be easily derived from the parity of the band separating them (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By increasing the number of occupation states, like intercalation or electrical gating, RbTi3Bi5 can be tuned into a trivial insulator when shaded area I is considered as the global gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' According to the nontrivial topological invariants (Z2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Z2, Z2, Z2) = (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 0, 0, 1), topological surface states are anticipated to exist in the planes parallel to [001] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In order to gain a deeper insight of the topological properties, we calculated the surface states along (100) plane of RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2f, a plenty of floating surface states can be clearly observed due to the breaking of translation symmetry and reconstruction at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The dispersions of surface states also show large anisotropy, with complicated dispersions along \uf060X-\uf060\uf047-\uf060X but simple ones along \uf060Z-\uf060\uf047-\uf060Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In the global gap (~50 meV) around high symmetry point \uf060\uf047, two brunches of surface states link the bulk valence states and conductance states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In particular, these two brunches of surface states form a crossing at \uf060\uf047, which should be a surface Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' These results demonstrate RbTi3Bi5 is a weak topological insulator with both nontrivial topological invariants and gapless surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Experimentally observed band structure for RbTi3Bi5 and its comparison with DFT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Calculated FS in \uf047-M-K plane for RbTi3Bi5 at E - EF = 0 eV, with high-symmetry points Γ, M, and K labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (b) FS intensity plot in \uf047-M-K plane of RbTi3Bi5 recorded at hν = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV, obtained by integrating the spectral weight within ±10 meV with respect to EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' \uf061i (i = 1, 2, 3), \uf062, and \uf067 donate the multiple pockets around high-symmetry points Γ, M, and K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (c) ARPES intensity maps of \uf047-M-K plane at E - EF = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (d) Photoemission intensity plot along K-\uf047-K with hν = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV and (e) its second derivative intensity plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (f) Photoemission intensity plot along M-\uf047-M with hν = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV and (g) its second derivative intensity plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The arrows indicate the crossing points at EF and labeled as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The red lines are the calculated band structures with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Intensity plots of the ARPES data along \uf047-M at (h) E − EF = 0 eV and (i) E - EF = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV collected in a range of photon energy from 20 eV to 55 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The triangles indicate the corresponding crossing points at EF as in (b) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3a shows the calculated FS in \uf047-M-K plane of RbTi3Bi5 with SOC, which is well consistent with the measured FS of RbTi3Bi5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' On the in situ cleaved surface of (001) plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S4), the electron pocket \uf0611 around \uf047 point can be clearly observed, indicating the existence of large SOC interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Two larger hexagonal-flower-shaped electron pockets around \uf047 point (\uf0612 and \uf0613) can (a) (b) kx (A-1) (c) kx (A-1) kx kz = 0 E-E=0eV E-E=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 ky K K +M M M 0 ky 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6 (d) (f) (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0ev α2α3 50- 40 hv (eV) E -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 - 30 hv = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV hv = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 20 (e) g (i) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2ev 50- hv (eV), 40 E-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 20 K K M M M 9 be further resolved, with a diamond-shaped hole pocket at M point (\uf062) and a triangular-shaped electron pocket at K point (\uf067).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3c shows the ARPES intensity maps of \uf047-M-K plane at E - EF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV, showing the shrinkage of electron pockets around \uf047 or K points and the enlargement of hole pocket at M point with the increasing binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3d is the experimentally observed bands along K-\uf047-K at hν = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV, showing four crossing points at EF in a sequence of (\uf0611, \uf0612, \uf0613, \uf067).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As for the experimentally observed bands along M-\uf047-M at hν = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3f), another four crossing points in a sequence of (\uf0611, \uf0612, \uf0613, \uf062) can be observed, where the intensity of \uf0612 and \uf0613 cannot be well resolved because they are spatially close in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By noise reduction using a machine learning process [55], a small intensity of \uf0612 shows up (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The corresponding second derivative intensity plots (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3e and g) show remarkable agreement with the calculated electronic structure for RbTi3Bi5 with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The experimentally observed bands can be fully recovered by calculation from E - EF = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 eV up to EF, with the calculated EF only move down 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='05 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The band structure along \uf047-M has been measured by varying the photon energy from 20 eV to 55 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 3h and i, only the change in intensity is observed at different photon energies for both E − EF = 0 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 eV, indicating a strong quasi-2D feature in RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Both the electron pocket \uf0611 around \uf047 and its well consistency with calculation indicate the existence of a strong SOC interaction, which would induce a great splitting and form a relatively large gap at M point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Thus, RbTi3Bi5 should be an experimentally observable weak topological insulator with surface states showing up at the global gap (shaded area II in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Unlike the Z2 topological insulator AV3Sb5, the bulk gap of RbTi3Bi5 is far above EF and the gapless surface states reside on the (100) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' To experimentally observe the surface states, cleaving along the (100) plane and pumping the electrons up would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In most previous study of Kagome lattice, whether theoretically or experimentally, the upper saddle points were the focus due to the position of EF and the existence of surface states around [11, 23, 56-58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By changing the filling of electrons and getting the upper saddle points close to EF, the possible superconductivity, CDW or chiral spin order would happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' However, the lower saddle point has received little attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Due to the large downward shift of chemical potential, the lower saddle point of RbTi3Bi5 is much more close to EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' It would be of great interest to explore the possible new phases related to the lower saddle points by lifting EF using electron doping or strain engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) (b) 5% =5% E- (eV) E M K M K M K 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Strain engineering in RbTi3Bi5 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Schematic pictures of RbTi3Bi5 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Electronic structure of RbTi3Bi5 monolayer with SOC under (b) no strain (\uf065 = 0), (c) -5% compressive strain (\uf065 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='95) and (d) 5% tensile strain (\uf065 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The typical dispersions contributed by the [Ti3Bi5]- layer with Ti-Kagome lattice is highlighted with green dots, and the lower saddle points is marked by a red dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Owing to its strong quasi-2D feature, RbTi3Bi5 single crystal can be mechanically exfoliated into thin flakes with thickness down to 10 nm using the scotch-tape method (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S6) [59], which contains several unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' This shows a potential for RbTi3Bi5 to reach the monolayer limit, making it an ideal candidate for strain engineering or electrical gating, which would be efficient in regulating its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Here the in-plane biaxial strain is theoretically tried to explore the effectiveness of strain engineering for modulating the electronic properties of RbTi3Bi5 monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 4a, the RbTi3Bi5 monolayer is a hexagonal lattice with [Ti3Bi5]- layer sandwiched by Rb atoms, which is indeed stoichiometric Rb2Ti3Bi5 and would result in an electron doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The in-plane biaxial strain is defined as ε = (a - a0)/a0 × 100%, where a and a0 is the in-plane lattice constant of the strained and unstrained monolayer, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' In RbTi3Bi5 monolayer, the typical dispersions contributed by the [Ti3Bi5]- layer with Ti- Kagome lattice can also be resolved (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' S7 and S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Compared with the electronic structure of bulk RbTi3Bi5, the chemical potential of unstrained RbTi3Bi5 monolayer shifts upward for about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='15 eV, making the lower saddle points at M (S2) closer to EF (~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='05 eV) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The band structure of RbTi3Bi5 monolayer was changed clearly with biaxial strain from −5% to 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' With the compressive strain, the location of S2 remains above EF and goes up further (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 4c), reaching 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3 eV up to -5% compressive strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Under the tensile strain, the location of S2 goes down and crosses EF with 5% tensile strain (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 4d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' By the in-plane biaxial strain, the position of the lower saddle points of RbTi3Bi5 monolayer can be continually tuned, which may induce interesting phases, including possible superconductivity, CDW or chiral spin order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 4 Conclusion In summary, a new ATi3Bi5 (A = Rb, Cs) family with Ti-Kagome lattice is synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' ATi3Bi5 (A = Rb, Cs) is found to own strong quasi-2D feature, moderate electron correlation, and small Pauli paramagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Based on first principles calculations considering SOC, RbTi3Bi5 is predicted to be a weak topological insulator with gapless surface states in its global bulk gap along the (100) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The experimentally observed band structure along (001) plane by ARPES fits well with the calculation and indicates the existence of strong SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The in-plane biaxial strain can effectively tune the position of lower saddle point coming from the Ti-Kagome lattice in its monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' These results show ATi3Bi5 (A = Rb, Cs) is a new weak topological insulator candidate and good platform to explore new phases of Kagome metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Supporting information Tables SI-SIII show the crystal structure of ATi3Bi5 (A = Rb, Cs) single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S1 shows the SEM images and chemical composition of ATi3Bi5 (A = Rb, Cs) single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S2 shows 11 the power law fittings of the resistivity below 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S3 shows the calculated electronic structure without SOC, DOS with SOC, and fat bands both with and without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S4 shows the low energy electron diffraction pattern of (001) plane after in situ cleavage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S5 shows the noise reduction of the photoemission intensity plot along M-\uf047-M using a machine learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S6 shows thin flakes of RbTi3Bi5 by mechanical exfoliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S7 shows the strain- tunable electronic structure of RbTi3Bi5 monolayer without SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S8 shows the strain-tunable fat band of RbTi3Bi5 monolayer with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Author information Notes The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Note added: During the preparation of this manuscript, we recognized some unpublished preprints about CsTi3Bi5 on its superconductivity, topology, and nematicity (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='org/abs/2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='03840, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='12264v1, and https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='16477).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Acknowledgements Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chen, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Ji contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chen, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Wang would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chen and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Hu of the Institute of Physics, Chinese Academy of Sciences for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' This work was partially supported by the National Key Research and Development Program of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2018YFE0202600 and 2022YFA1403900) and the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 51832010, 11888101, Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 11925408, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 11921004, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 12188101), the Ministry of Science and Technology of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2018YFA0305700 and No.' metadata={'source': 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properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' electronic structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' and strain-tuned monolayer of the weak topological insulator RbTi3Bi5 with Kagome lattice Ying Zhou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Long Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Xuecong Ji1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' #,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chen Liu3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Ke Liao 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Zhongnan Guo4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Jia-Ou Wang3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Hongming Weng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' *,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Gang Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' * 1 Beijing National Laboratory for Condensed Matter Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 2 University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 3 Beijing Synchrotron Radiation Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 4 Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' School of Chemistry and Biological Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' University of Science and Technology Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Beijing 100083,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China 5 Songshan Lake Materials Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Dongguan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Guangdong 523808,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' China # These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Email: gangwang@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' hmweng@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 17 1 Crystal structure and composition of ATi3Bi5 (A = Rb, Cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Table SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Crystallographic data and structure refinement of ATi3Bi5 (A = Rb, Cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Empirical formula RbTi3Bi5 CsTi3Bi5 f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' weight (g/mol) 1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='27 1321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='51 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' / Z P6/mmm (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='191) / 1 Unit cell parameter a (Å) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8077(7) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8079(2) b (Å) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8077(7) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8079(2) c (Å) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='1297(11) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2400(4) α, β (\uf0b0) 90 90 γ (\uf0b0) 120 120 V (Å3) 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='68(6) 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='92(2) dcal (g/cm3) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='928 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='130 Refl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Collectd / unique 2679 / 119 1652 / 151 Rint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0925 Goodness-of-fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='537 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='073 R1 / wR2 (I > 2σ(I)) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0227 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0251 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0651 R1 / wR2 (all) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0238 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0258 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0655 Table SII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Atomic coordinates and equivalent isotropic displacement parameters for RbTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Atom Wyck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' x/a y/b z/c Occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' U(eq)(Å2) Rb 1b 6/mmm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0338(19) Ti1 3g mmm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0083(13) Bi1 1a 6/mmm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0109(6) Bi2 4h 3m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='23386(12) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0126(4) Table SIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Atomic coordinates and equivalent isotropic displacement parameters for CsTi3Bi5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Atom Wyck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Sym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' x/a y/b z/c Occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' U(eq)(Å2) Cs 1b 6/mmm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0144(6) Ti1 3g mmm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0040(8) Bi1 1a 6/mmm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0056(4) Bi2 4h 3m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='6667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='3333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='23839(7) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0066(4) 18 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' SEM images of as-grown (a) RbTi3Bi5 and (c) CsTi3Bi5 single crystals and corresponding elemental mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Typical EDSs and elemental composition collected on the flat clean surface of (b) RbTi3Bi5 and (d) CsTi3Bi5 single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Power law fittings of the in-plane resistivity (\uf072ab) of (a) RbTi3Bi5 and (b) CsTi3Bi5 single crystals below 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' 2 Electronic structure of ATi3Bi5 (A = Rb, Cs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Rb (b) Rb:9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='56% Ti: 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='94% Bi:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='60% Bi 30μm 0 2 4 6 8 keV (c) CS (d) Cs: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='44% Ti: 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='44% Bi:55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='12% Bi 30um 6 8 keV(a) 6 (b) 10 0—Pab - RbTi,Bis 0 Pab - CsTigBis p=P+AT p=P+ATα Po = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8(5) × 10-6 m2 cm P = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2(1) × 10-6 mΩ2 cm 8 A = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='41(1) × 10-9 m2 cm K-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='93 A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='81(2) × 10-9 m2 cm k-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='04 α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='93(3) α= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='04(3) 6 0 4 0 10 20 30 40 50 0 10 20 30 40 50 T (K) T (K) 19 Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Calculated band structure of RbTi3Bi5 without SOC along high symmetry lines in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The green dots highlight the dispersions contributed by the [Ti3Bi5]- layer with Ti-Kagome lattice, and the orange circles highlight the Dirac points and band crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (b) Total and partial density of states near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fat band of RbTi3Bi5 (c) without SOC and (d) with SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Low energy electron diffraction pattern of RbTi3Bi5 (001) plane after cleavage at hv = 98 eV, showing a clear hexagonal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) 2 (b) 2 Without soc Rb Bi Total E- E (eV) E- E (eV) EF 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='2 M K A L H A/L M|H K c (d) 2 Rb Without soC Rb Withsoc Bi (eV) (eV) 出 M K 厂 A L H A|L M|H K M K A L H A/L MH K 20 Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Photoemission intensity plot along M-\uf047-M measured with hν = 30 eV and its momentum distribution curve (MDC) at E - EF = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='08 eV (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (b) Photoemission intensity plot after noise reduction of (a) using a machine learning process and its MDC at E - EF = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='08 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The white dashed line is the Fermi level and \uf061i (i = 1, 2, 3), \uf062 are the multiple pockets along M-\uf047- M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (c) MDC plot of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The black line highlights the Fermi level and the triangles outline the shape of the dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Since the count signal collected on the angle-resolved photoemission spectroscopy detector introduces white Gaussian noise, we use the semi-supervised Noise2Noise algorithm [1], which has good ability to remove white Gaussian noise in depth learning, to suppress white Gaussian noise from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S5a shows the simple sum of 10 independent data acquisition of photoemission intensity plot along M-\uf047-M with hν = 30 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' And the results of white Gaussian noise suppression using 10 independently collected image plots as training data are shown in Figure S5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Atomic force microscopy image of thin flakes of RbTi3Bi5 by exfoliation and (b) corresponding height profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Thin flakes of RbTi3Bi5 crystal were exfoliated using the scotch-tape method [2] and transferred onto a Si/SiO2 substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Atomic force microscopy of thin flakes was performed using an atomic force microscope (Multimode 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='0, Bruker, USA) in a ScanAsyst mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) (b) C 0 0 (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 山 ntensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 a a Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=') Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='4 0 kx (A-1) kx (A-1) kx (A-1)(a) (nm) (b) 20 50 二L2 40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='10 30 0 30nm 20 20nm MM 10 10 10nm um 0 20 0 1 2 3 4 5 Distance (um) 21 Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' (a) Electronic structure of RbTi3Bi5 monolayer without SOC under (a) no strain (ε = 0), (b) -5% compressive strain (ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='95) and (c) 5% tensile strain (ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' The typical dispersions mostly contributed by the [Ti3Bi5]- layer with Ti-Kagome lattice is highlighted with green dots, and the lower saddle points is marked by a red dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Fat band of RbTi3Bi5 monolayer with SOC under (a) no strain (ε = 0), (b) -5% compressive strain (ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='95) and (c) 5% tensile strain (ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' [1] Noise2Noise: Learning image restoration without clean data, arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content='04189, DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otAzT4oBgHgl3EQfqv1Z/content/2301.01633v1.pdf'} +page_content=' Van Noorden, Production beyond sticky tape, Nature, 483 (2012) S32-S33.' 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”harmony with diversity” state within a attraction-repulsion model +framework +Peng-Bi Cui (崔鹏碧)1, ∗ +1International Academic Center of Complex Systems, Beijing Normal University, Zhuhai, 519087, China +Opinion evolution is generally subject to either global neutral consensus under influence mechanism, or po- +larization into two opposing camps in face of distant views. However, as a third novel state, the formation +of ”harmony with diversity”, where individuals freely express various viewpoints to a certain extent to sustain +integration of diversity and at the same time shared values ensure social coherence to avoid ideological split, +still remains unclear, as well as its unique dynamic features. Since a general model framework to generate the +desired state together with global consensus and polarization is still lacked. To address this issue, we develop +an attraction-repulsion model based on the general simple assumption that individuals tend to either reach an +agreement with shared opinions or to amplify difference from others with distant opinions, which allows us +to take account into the three core parameters: interaction strength, individuals’ susceptibility and tolerance to +others’ opinions. We are concerned with the effect of not only time-varying topology but also fixed interactions +imposed by static social network, where the tasks of heterogeneous individuals’ attributes are also performed. +Remarkably, the simple model rules successfully generate the three phases, along with three different transitions +and the triple point, regardless of whether the interactions are time-varying or fixed. Especially, state of ”har- +mony with diversity” emerges for sufficient susceptibility, intermediate interaction strength and high tolerance. +Whereas the population instead gets polarized for strong interaction or low tolerance. Fixed interactions can in- +troduce cluster-level self-reinforced mechanism which can unexpectedly suppress the emergence of ”harmony +with diversity”, and instead promote polarization due to negative influence. Moreover, heterogeneous attributes +such as susceptibility turns out to be a facilitating factor for achieving the desired state, while heterogeneous +tolerance can make things worse, which should be avoided. In any case, the simulations validate our method to +identify the boundaries between different phases. Overall, this study gains a number of crucial insights that can +be subject to further empirical analysis. +I. +INTRODUCTION +Opinion evolution is about crucial issues which may gener- +ally rises three distinct states: (i) Global consensus (GC state) +in which people’s view evolves towards one common opin- +ion, showing a narrow unimodal distribution centering on the +neutral consensus [1]. (ii) Polarization (P state) into two op- +posing clusters or camps which is characterized by a bimodal +distribution around the neutral consensus [2–5], further mak- +ing compromise impossible and giving rise to social oppo- +sition [4, 5]. (iii) Fragmentation (F state) is the state with +more than two distinct clusters in the spectrum [6, 7]. Es- +pecially, polarization is here conceived as a process of grow- +ing constraint in people’s viewpoints, and further emergence +of alignments along multiple or even opposite lines of poten- +tial disagreement with respect to political views, immigration, +biotechnology applications, LGBT rights, mask wearing and +viral nucleic acid detection in the COVID-19 [1, 4, 5, 8]. +In addition, a striking state besides GC, F and P state can- +not be neglected, in which individuals’ views or positions self- +organize into only one cluster with wide spectrum and fluctu- +ations. It is definitely different from F and GC state that, with +respect to ideological issues, the emergence of one opinion +cluster with wide spectrum can be defined as a desired state +of ”Harmony with diversity” (HD state), which has also been +stressed by Confucius. Since the presence of HD state reveals +that individuals can freely express different viewpoints to a +∗ cuisir610@gmail.com +certain extent while the free internal debates result in fluctu- +ations, and at the same time shared values ensure social co- +herence and the avoidance of ideological split, or even sharp +conflicts. This robust empirical phenomenon is actually vary +crucial to healthy functioning of social systems. For example, +except the United Kingdom, member states of the European +Union (EU) over the last two decades have been straining will- +ingness to comply with EU agreements, aiming to sustain an +internal single market based on the common Motto ”United +in Diversity”, regardless of the fact they have various eco- +nomic demands [9]. Moreover, the softness of the laws or +decrees targeting COVID-19, if they are embodied in a basic +loose national agreement that this disease is harmful and in- +fectious, tends to guarantee least controversial measures such +as wearing face masks, social distancing and home quaran- +tine [10, 11]. Unlike shelter-in-place ordinances, lockdowns, +or capacity restrictions, the three measures can diminish the +spread of an infection, and simultaneously do least harm to +economic activity [10, 12–16]. However, to the best of our +knowledge, the factors contributing to the emergence of HD +state still remain unclear. +Considerable efforts have been recently put into model- +ing polarization dynamics to uncover the mechanisms behind +it [8, 17–27]. One of progresses is to measure polarization +with the variance or SD of continuous ideological positions in +a population [23, 24, 28]. However, to the best of our knowl- +edge, the emergence and persistence of HD state has received +too little attention in the growing literature. Since most com- +putational attempts are trapped in the framework of two-state +model, failing to additionally reproduce a HD state. The main +approaches of these studies are based on traditional attraction- +arXiv:2301.03042v1 [physics.soc-ph] 8 Jan 2023 + +2 +repulsion model (ARM) with a confidence bound defining a +reference that an individual compares with the opinion dis- +tance to a neighbor, and based on it, the individuals decide to +move closer or farther from its neighbor [2, 3, 23, 24, 28–33]. +According to such mechanism, the contest between attracted +and repelled forces can accordingly facilitate either GC or P +states [2, 3]. Alternatively, some modeling approaches inves- +tigate the coevolution of the network topology and opinion +dynamics as the result of reinforcement and rewiring activi- +ties driven by structural similarity [23] or homophily [21, 26]: +the two individuals sharing similar connections or opinions +are more likely to interact. Whatever the case, the HD state +does not appear to emerge. Although, some seminal works +on cultural diversity show that the emergence of clustering +phase is possible [6], GC and P states cannot be theoretically +reproduced within the same model rules. Moreover, due to +the lack of HD state, the physical pictures of opinion evolu- +tion from the attempts capturing biased assimilation [34] or +information accumulation [35] are not integrated. The prob- +lem rises two fundamental questions we want to answer in this +study: whether simple settings can stably facilitate HD state in +addition to GC and P states, and how to effectively distinguish +the three phases from each other. +Influence plays a decisive role in driving the evolution +of individuals’ opinions, which is responsible for individu- +als’ tendency to either reach agreement with shared opin- +ions (positive influence) [1, 17, 36, 37] or to amplify dif- +ference from others with distant opinions (negative influ- +ence) [2, 3, 29, 30, 32, 33]. +Consequently, the attraction- +repulsion model (ARM) can well curve opinion dynamics by +setting a threshold for tolerance of disagreement. Within the +framework, rules specify the mechanisms for interaction be- +tween individuals, they can thus provide insight about impor- +tant mechanisms and the role they play in regulating emergent +properties of system, and highlight the consequences from a +few simple assumptions. In addition, other mechanisms such +as homophily and media influence can be more easily incor- +porated into ARM to successfully map more empirical polar- +ization changes [8, 38, 39]. +Therefore, to address the above two questions, we develop +an agent-based attraction-repulsion model (ABARM) featur- +ing the emergence of GC, HD and P state from microscopic in- +teractions between individuals. Therefore, this paper focuses +narrowly on the phase analysis with respect to ideological is- +sues, rather than elucidating new mechanisms behind polar- +ization. In our paradigm, opinion evolution rules are built on +the general simple assumption that individual’s attraction to or +repulsion from others is only dominated by their opinion sim- +ilarity in one-dimensional topic space. It allows us to uncover +the exact mechanisms that yield particular outcomes, with- +out loss of consideration of three core parameters: interaction +strength, individuals’ susceptibility and tolerance to different +opinions of others. We are concerned with the effect of not +only time-varying topology but also the static social network +(part of Facebook friendships network) on opinion dynamics. +Moreover, the tasks with heterogeneous susceptibility or tol- +erance are also considered and performed to make the study +closer to the reality. +Regardless of the interaction structures and the heteroge- +neous attributes, our model successfully generates the three +states: (i) global convergence towards a neutral consensus, +(ii) a fluctuated cluster with a wide spectrum of opinion, (iii) +polarization which opinions split into two opposing camps, +which are defined as GC, HD and P states, respectively. It can +help us uncover the mechanism for facilitating HD state. In +addition to the SD of population opinion, we additionally de- +fine opinion entropy as a novel order parameter of the system. +Crucially, a novel available method which indicating the max- +imum susceptibility of opinion entropy is correspondingly de- +veloped to numerically identify the boundaries between the +phases. Note the good agreement between simulations and +the estimated boundaries, validating the phase-identification +method. We thus conclude that the three phases are distinct +in our model. Interestingly, comparison of the results from +time-varying networks and part of Facebook friendships net- +work reveals that the reinforcement from spatial opinion clus- +ters can increase the likelihood of high-level polarization, in- +stead of extreme polarization. This highlights complex roles +of group-level struggles. In addition, the present model suc- +cessfully reproduces the fluctuations of cluster without the +noise effect, which has been previously considered as the cen- +tral mechanism to facilitate the fluctuation of opinion clus- +ters [6, 40–42]. For the first time, our study proposes a basic +framework to explore the formation of HD state, and thus the +unique dynamic features of HD. +The rest of the paper is organized as follows. In Sec. II, +we introduce the model, as well as the phase-identification +method. In Sec. III A, we firstly simulate our model with ho- +mogeneous and heterogeneous individual attributes on time- +varying networks, and then on parts of empirical social net- +works with fixed interactions in Sec. III C, attempting to make +deep comparison. We conclude with a summary of the results +and an outlook for future studies in Sec. IV. +II. +MODEL AND PHASE-IDENTIFICATION METHOD +A. +Model +The developed ABARM basically assumes a threshold of +tolerance to determine whether a repulsive or attractive inter- +action happens, in line with the assumption of bounded con- +fidence which has been widely confirmed by previous stud- +ies [24, 37, 43]. The threshold defines a reference distance in +terms of opinion difference, based on which an agent would +decides whether to move closer or farther from the selected +neighbor. If opinion difference between the individual and +its neighbor is smaller than the threshold, an individual will +tolerate the position of its neighbor, and move closer to the +neighbor. Otherwise, it will move farther from the neighbor. +After defining the original model, in turn, we can proceed our +explorations by accomplishing the following. (i) Make the at- +tributes individuals in response to the others’ position hetero- +geneous. (ii) Next, endow the parts of social networks with +local structure to fix the interactions among individuals. A +systematic comparison of the three cases (including the basic + +3 +model) with regard to phase areas allows us to fully under- +stand the formation of HD. +The basic model considers a population of size N where +each individual is typified by an opinion xi(t) at time t, which +is a real number in the interval xi(t) ∈ [−10, + 10]. Inspired +by recent studies to capture polarization process on social me- +dia networks [21, 23, 26, 44], the updating of individual’s +opinion xi(t) is solely driven by the interactions among indi- +viduals and is described by the following N coupled ordinary +differential equations: +˙xi(t) = +� +A tanh(αiDji(t)) +if |Dji(t)| < Ti; +A tanh(αiσ(Dji(t))(xi(t) − |Dji(t)|)) +if |Dji(t)| ≥ Ti. +(1) +Dji(t) = xj(t)−xi(t) is the opinion difference between i and +j at time t. Ti denotes the tolerance threshold of individual i. +αi can be interpreted as the controversialness of the topic, and +thus gauge nonlinearity of interaction or susceptibility of in- +dividual. Moreover, αi positively associates with the extent to +which individual i is passionate, attentive or sensitive, as a re- +sult, susceptible to be socially influenced. It is obvious that Ti +and αi are largely related to intrinsic preferences of individual +i. Nonlinear shape of the influence function tanh(x) is con- +trolled by α. While A quantifies interaction strength, which +is actually the upper bound of opinion shift driven by each in- +teraction, indicating that the influence exerted by individuals +on others is capped, in accordance with the experimental find- +ings [45]. σ(Dji(t)) extracts the sign of Dji(t). Note that in +our model the opinion difference is responsible for the evolu- +tion of opinions, attempting to capture both the repulsive and +attractive forces, which differs from the model settings focus- +ing mainly on reinforcement and homophily [21, 23, 26, 44]. +Firstly, we define the activity of each individual by means of +activity driven (AD) model [26, 46–48], so that opinion evolu- +tion is coupled to an underlying time-varying network. More +in detail, ki(t) represents the number of interactions they can +have with others within a given time step t. It hence gives rise +to a temporal network determine by the temporal adjacency +matrix Aij(t), where Aij(t) = 1 if individual i contacts in- +dividual j, otherwise Aij(t) = 0. More specifically, individ- +ual i connects ki distinct random other individuals, such that +ki = �N +j Aij(t) is satisfied throughout the simulation. Con- +sidering the empirical statistics that activities of people are +generally heterogeneous [21, 47, 48], we assume the interac- +tions are extracted from a power-law distribution p(k) ∼ k−γ. +Intrinsic preferences toward topics are always proved to be +heterogeneous in reality [49, 50]. Therefore, we next assign +each individual a susceptibility αi or tolerance threshold Ti, +which is correspondingly selected from the following power- +law distributions, respectively. +p(α) ∼ α−η, +(2) +p(T) ∼ T −ξ. +(3) +η is susceptibility exponent, and ξ is tolerance threshold ex- +ponent. In such case, we still perform simulations on time- +varying networks. +Online social networks are increasingly used to access +opinion information, engage with COVID-19 vaccines, gun- +control, abortion and so on. These platforms can reduce bar- +riers and cost to information and, further, allow individuals to +freely voice their viewpoints, consequently improving rate of +opinion exchanges. On the other hand, the structures of social +networks combined with other mechanism such as reinforce- +ment can drive the emergence of echo chambers [23, 51, 52], +where the segregation in opinion space is reflected in inter- +actions among individuals [33]. Therefore, we finally fix the +interactions among individuals by embedding the population +into the parts of the social media networks such as Facebook. +In such cases, ki corresponds to the number of edges that in- +dividual i stretch to its neighbors, and the individual is repre- +sented by a node in the networks. As a result, the connected +neighbors of each individual keep unchanged. We perform the +above two cases on the static networks. +In numerical simulations, we set the simulation parameters +to be the following values: N = 1000 for time-varying net- +works and γ = 2.1, that will be specified when needed. The +control parameters of the present model is A, αi, Ti, η and +ξ. The final results are obtained from Nr = 100 independent +realizations, after at least 500 time steps. For each simula- +tion realization, the initial opinions attributed to each individ- +ual is independently and randomly sampled from the interval +[−1.0, 1.0]. Then at each time step t, opinion evolves as fol- +lowing: (i) In random order, each individual (i) is selected +from the population. (ii) Within the framework of AD, the +initial temporal adjacency matrix Aij(t) is the zero matrix, +and i can randomly choose ki new neighbors out of all indi- +viduals. Nevertheless, in the cases considering a static social +media network, the neighbors of i keep unchanged. (iii) Then, +one by one, i compares its opinion with that of the neighbors, +attempting to update its opinion according to Eq. 1. +We operationalize the degree of polarization through the +standard deviation (SD) in opinions SD(x0, ..., xN), and +measure the opinion diversity by calculating the opinion en- +tropy (S) of the population: S = �xmax +xmin xρx. ρx = Nx +N is +the density of individual owning opinion x; Nx denotes the +population of opinion x; xmin = −10 and xmax = 10. +B. +Phase-Identification Method +In our model, SD gets larger due to increasing degree of +global polarization with a bimodal distribution, and the mini- +mum of the polarization SD = 0 suggests a narrow unimodal +distribution and thus the existence of GC state. While HD + +4 +state corresponds to that a opinion cluster with wide spectrum. +The changes of opinion entropy are therefore sensitive to the +transitions between different state. Inspired by this fact, we +employ the susceptibility of S to numerically determine the +thresholds between different sates. +χ(S) = +� +⟨S2⟩ − ⟨S⟩2 +⟨S⟩ +, +(4) +where ⟨S⟩ is the ensemble average of S, which can be ob- +tained by averaging S from Nr independent realizations. ⟨S2⟩ +is the secondary moment of the ensemble distribution. Based +on the principle that χ(S) exhibits a maximum value at the +threshold, one can further identify the boundaries between +GC, HD and P states. +III. +RESULTS +A. +The results from time-varying networks +Three different cases occur in order with increasing inter- +action strength A, as shown in Fig. 1. The evolution of a +system following Eq. 1 is reported in Fig. 2. The population +firstly converge to the neutral consensus within a short time +(see Fig. 2(a)) when individuals are more or less independent +due to weak interaction. Then a striking phenomenon occurs +at intermediate levels, which produces a mediate polarization +that begins to increase slowly after a transition whose thresh- +old is denoted by Ac1 (see the secondary peak illustrated in +the subplots of Figs. 1(a1) and (b1)) and then takes hold, cor- +responding rightly to a highland of entropy (see Figs. 1(a2) +and (b2)). In such case, as shown in Figs. 2(b) and (c), a fluc- +tuated integral opinion cluster in ideological space persists for +the entire simulation. Specifically, an inflation point of opin- +ion entropy can also be indicated by the position of the first +main peak susceptibility of opinion entropy. While spectrum +width of the cluster on both sides of the peak was claimed +to be rather different, and the spectrum width tend to extend +sharply as A > A +′ +c (compare Figs. 2(b) and (c)). Moreover, +it is important to note that the occurrence of an inflation point +imply the existence of a transition. In other words, the infla- +tion point is dependent on whether the state of opinion clus- +ter with wide spectrum exists or not. Though, the interest- +ing phenomenon is qualitatively different from the dependent +transition class uncovered by microcanonical statistical anal- +ysis [53]. The system governed by the intermediate range of +interaction strength regime Ac1 < A < Ac2 provides suffi- +cient evidence for the existence of HD state. +Yet, the polarization starts to rise uncontrollably at the sec- +ond threshold Ac2, at which the transition involves an asym- +metric temporal trajectory (see Figs. 2(d) and (e)), and after +which the population goes to the extreme i.e., P state due to +considerable negative influence [2, 3, 28]. The asymmetric +temporal trajectory at Ac2 suggests an asymmetric polariza- +tion which has also been the focus of recent studies [44, 54]. +In P state, opinion symmetrically splits into two opposite ex- +treme camps, as shown in Fig. 2(f). Obviously, the three dif- +ferent cases correspond to GC, HD and P state, respectively. +Moreover, as shown in Figs. 1 (c1) and (c2), individual’s sus- +ceptibility has similar effects on opinion dynamics. +To achieve a better understand the most basic effects of in- +teraction strength and individual’s susceptibility on opinion +dynamics, we then extend the previous observations to a wide +range of (A, α) in Fig. 3, where the color encodes the values +of SD (top panels) and S (bottom panels). We observe two +types of transitions: from GC to HD phase when α is small, +and from HD to P phase for large α. The transition from HD to +P is largely invariant with respect to A if individuals are sen- +sitive enough to the dissimilarity in opinions (α is not small). +While the regions of stable neutral consensus are character- +ized by small values of A and α, and regions of HD can be +obtained by for increasing A and α. Moreover, a comparison +among the top (bottom) panels of Fig. 3 reveals that increasing +tolerance threshold T can expand HD regions being robust to +change of susceptibility, by simultaneously occupying the re- +gions of GC and P. The new interesting phenomenon is largely +in accordance with the empirical results [7, 55] and the nature +of the social moral rules requiring a high level of tolerance, +and thus one of novel findings of our present study. We can +see in Figs. 3(b1)-(b4) that highlands of opinion entropy per- +fectly indicate the regions of HD. The identified boundaries +afford a precise division of the regions of different phases, +validating our phase-identification method. +Figs. 4(a1)-(a4) show the phase diagrams in terms of SD in +(A, T) space. We find not only transition from GC to P phase +if T is small but larger than a certain value, but also transitions +from GC to HD phase, and from HD to P phase as T getting +larger. Therefore, there definitely exists a triple point denoted +by a vanishing entropy highland (see the entropy highlands +shown in Figs. S1(a1)-(a4)), above which the emergence of +GC is completely dominated by A, and independent of the +change of T. If the tolerance threshold is large enough, weak +interaction becomes a fully dominant factor for a global con- +vergence, and intermediate A can generate HD state whose +region tends to expand linearly with increasing T. For all +large susceptibility promotes HD to erode the regions of GC, +and in turn P occupy the regions of HD, leading to a shrink- +age of GC regions until its disappearance when α = 10.0 +(combine with the entropy highlands shown in Figs. S1(a1)- +(a4)). Figs. 4(b1)-(b4) offers a comprehensive view of the +effects of individual’s susceptibility on opinion dynamics for +different levels of tolerance. We find that, if α becomes large, +polarization is likely to emerge, along with extinction of GC +and decreasing likelihood of HD. Intermediate levels of tol- +erance guarantees an intermediate range of susceptibility for +HD state when A is not small, as shown in Figs. 1(c1)-(c2) +and Figs. S2(b1)-(b4). However, high tolerance would result +in saturation effect of large α, and role of susceptibility is lim- +ited. In line with the results illustrated in Figs. 1 and 3, strong +interaction can make things worse, and it is hard to achieve +HD state. Still, the system can produce a triple point, and +there occur three different transitions under certain parameter +conditions: from GC to P, GC to HD, and HD to P. +Up to now, the results from the basic model on time-varying +networks reveals that HD state can be easily facilitated by +sufficient susceptibility, intermediate interaction strength and + +5 +0.5 +1 +1.5 +2 +2.5 +3 +Interaction strength, A +0 +5 +10 +Variables +T=7.0, +=1.0 +(a1) +Ac1 +Ac2 +SD +(S) +0.5 +1 +1.5 +2 +2.5 +3 +Interaction strength, A +0 +2 +4 +6 +Variables +(a2) +Ac1 A' +c +Ac2 +S +(S) +0.5 +1 +1.5 +2 +2.5 +3 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Variables +=1.0, T=6.5 +Ac1 +Ac2 +(b1) +SD +(S) +0.5 +1 +1.5 +2 +2.5 +3 +Interaction strength, A +0 +2 +4 +6 +Variables +Ac1 A' +c +Ac2 +(b2) +S +(S) +0.5 +1 +1.5 +2 +Susceptibility, +0 +2 +4 +6 +8 +10 +Variables +A=1.5, T=4.0 +c1 +c2 +(c1) +SD +(S) +0.5 +1 +1.5 +2 +Susceptibility, +0 +2 +4 +6 +Variables +c1 +' +c +c2 +(c2) +S +(S) +0.9 +1.1 +1.3 +0 +1 +0.9 +1.1 +1.3 +0 +1 +0.9 +1.1 +1.3 +0 +1 +0.9 +1.1 +1.3 +0 +1 +0.6 +0.9 +0 +1 +0.6 +0.9 +0 +1 +FIG. 1. The dependence of SD (red circles), S (blue squares), χ(S) (brown curves and blue cures in the insets) on interaction strength A in +time-varying networks. Pink vertical line labels the position of Ac1 at which HD state begins to appear, while light blue vertical line indicates +the position of Ac2 at which HD state vanishes. In particular, A +′ +c denotes an inflection point in S followed by a sharp growth of S. The values +of T and α are correspondingly listed in titles of (a1), (b1) and (c1). +FIG. 2. Temporal evolution of the individuals’ opinions for six different values of A. The six cases are also indicated by the dots in Fig. 3(a3) +for a more explicit presentation. The values of the parameters are listed in the titles of the subplots. +high tolerance are responsible. In contrast, strong interaction +or low tolerance can make things worse, and the population +can easily be polarized and extreme. Moreover, as another +novel finding, the system formulated by our model can gener- +ally generate a triple point in different parameter spaces. + +(a) T=7.0, α=1.0, A=0.9 +(b) T=7.0, Qα=1.0, A=1.095 +10 +10 +5 +5 +0 +0 +X +X +-5 +-5 +10(c) T=7.0, α=1.0, A=2.1 +10 +5 +0 +-5 +10100 +105 +100 +Time +Time +(d) T=7.0, α=1.0, A=2.375 +(e) T=7.0, α=1.0, A=2.38 +10 +10 +5 +5 +t +0 +0 +X +-5 +-5 +-10 +-10 +100 +105 +100 +Time +Time100 +105 +Time +(f) T=7.0, α=1.0, A=3.0 +10 +5 +0 +-5 +-10 +100 +105 +Time6 +(a1) Polarization, SD +T=1.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +(a2) Polarization, SD +T=2.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +(a3) Polarization, SD +T=7.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +(a4) Polarization, SD +T=10.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(b1) Entropy, S +T=1.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +(b2) Entropy, S +T=2.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +(b3) Entropy, S +T=7.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +(b4) Entropy, S +T=10.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +FIG. 3. Phase diagrams in (A, α) space for four different values of T. We run simulations with polarization degree SD in (a1)-(a3), and with +opinion entropy S in (b1)-(b3). The lines consisting of pink triangles depict the boundaries from the first threshold Ac1, separating the regions +of GC and HD phase; while the lines consisting of yellow circles depict the boundaries from the second threshold Ac2, separating the regions +of HD and P phase, as given by our method. The regions belonging to different states are labeled in the subplots. The six dots correspond to +the subplots of six parameter combinations in Fig. 2. +(a1) Polarization, SD +=0.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(a2) Polarization, SD +=1.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(a3) Polarization, SD +=2.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(a4) Polarization, SD +=10.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(b1) Polarization, SD +A=0.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +(b2) Polarization, SD +A=1.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +(b3) Polarization, SD +A=2.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +(b4) Polarization, SD +A=2.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +FIG. 4. (a1)-(a4) Phase diagrams in (A, T) space for four different values of α. (b1)-(b4) Phase diagrams in (α, T) space for four different +values of A. We run simulations with polarization degree SD in all subplots. The lines consisting of pink triangles depict the boundaries from +the first threshold Ac1, separating the regions of GC and HD phase; while the lines consisting of yellow circles depict the boundaries from the +second threshold Ac2, separating the regions of HD (GC) and P phase, as given by our method. The regions belonging to different states are +labeled in the subplots. The light blue pentagrams indicate the triple points which, however, do not present in (A, α) space. + +7 +B. +The results from heterogeneous attributes +Individuals are always heterogeneous in their roles, their +power and their capacity to influence others, which is mostly +rooted in their heterogeneous intrinsic attributes. Within our +model framework, individuals differ in terms of their suscep- +tibility or tolerance. In this subsection, we focus on the effects +of heterogeneous susceptibility or tolerance, which follows +power law distributions given by Eqs. (2) and (3). This lends +a different and distinct perspective to our studies of HD state +through comparing the results with the homogeneous case in +Subsec. III B. +Fig. 5 depicts phase diagrams in terms of SD in parame- +ter spaces of (A, T), (A, η) and (T, η). We firstly note that, +in Figs. 5(a1)-(a3), the layouts of phase diagrams are similar +to those illustrated in Figs. 4(a1)-(a4). The difference is that +GC emerges only if the susceptibility exponent increases from +”scale-free” regime (η < 3) to ”small-wold” regime (η > 3) +where the value of SD can still identifies three types of tran- +sitions as identified in figures from Figs. 4(a1)-(a4) to Fig. 5. +Therefore, there is still a triple point. However, in (A, η) +space, things are different. In this case, we find at most two +different transitions for high-level tolerance: from GC to HD +phase, and from HD to P phase (see Figs. 5(b1)-(b3)), let alone +the existence of the triple point. In line with what we have ob- +served in Figs. 5(a1)-(a3), we can obtain GC state for η >∼ 3. +It is evident that ηc ≈ 3.0 can thus be considered as a thresh- +old for GC state. In addition, intermediate A can generate +HD state whose regions expand with T, slightly shrink with +η. This means that heterogeneity of susceptibility prevents the +population from converging to a global neutral consensus. As +expected, increasing T can effectively promote the population +into HD phase after a transition. It is consistent with the ho- +mogeneous case. Although it becomes harder to achieve the +desired state for large A which though broaden the spectrum +of opinion cluster (see Figs. S2(c1)-(c3)). Moreover, HD re- +gions gradually shrink with η along with a expansion of P re- +gions, suggesting a positive role of susceptibility heterogene- +ity. In comparison with the parameters A and T, the effect +from susceptibility heterogeneity is less pronounced. How- +ever, to obtain a triple point or GC state, η must be firstly in +”small-world” range. +Correspondingly, Fig. 6 further approves that the dynamic +behaviors do not qualitatively change by exhibiting three +states of opinion evolution. Especially, the middle subplots, +as well as Fig. 6(c3), show the temporal behaviors of individ- +uals’ opinions in HD state, where the opinion cluster is fluc- +tuating irregularly with time, but integral even when the clus- +ter touches the boundary of ideological space. This further +supports our statement about the system going into HD state +where patterns of opinion cluster keep robust against time. +We next focus on the effect of heterogeneous individuals’ +tolerance on opinion dynamics. Remarkably, the population +with heterogeneous tolerance leads to a qualitative change of +the results with respect to both phase diagram and temporal +evolution. In Fig. 7, in any case, we find that heterogeneity of +tolerance largely facilitate the emergence of P state, and the +population will be finally polarized when A or α is beyond a +small value. This is definitely in contrast to the convergence +properties of bounded confidence models [56]. In such case, +most individuals are closed-minded, i.e., has a smaller toler- +ance threshold, and therefore have a stronger tendency to am- +plify differences from others with slightly dissimilar opinions. +Moreover, saying that the blue regions between GC and P +phases (see Figs. 7(a1)-(a3) and (b1)) correspond to HD phase +is not appropriate. It is instead qualitatively different from the +aforementioned HD state. Since Figs. S4(b1) and (b2) show +that the population is eventually polarized as the opinion clus- +ter touches the boundaries of the ideological space, and gets +absorbed. In this region, we have checked that it is rather hard +for the system to reach a steady state within a long time. From +these results, we find that the population with heterogeneous +tolerance cannot generate a HD state, as well as that the prob- +ability of converging to a global neutral state is extremely low +unless both A and α are rather small. Interaction strength A +largely determines the population’s outcome. Tolerance het- +erogeneity can largely increase the tendency of the population +getting trapped in P state, as shown in Fig. 7(b1). + +8 +(a1) Polarization, SD +=2.1 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(a2) Polarization, SD +=3.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(a3) Polarization, SD +=6.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(b1) Polarization, SD +T=1.0 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(b2) Polarization, SD +T=2.0 +GC +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(b3) Polarization, SD +T=10.0 +GC +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(c1) Polarization, SD +A=0.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(c2) Polarization, SD +A=1.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(c3) Polarization, SD +A=2.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +FIG. 5. Phase diagrams for the population with heterogeneous susceptibility. In more detail, (a1)-(a3) phase diagrams in (A, T) space for +three different values of η. (b1)-(b3) phase diagrams in (α, η) space for three different values of T. (c1)-(c3) phase diagrams in (T, η) +for three different values of A. We run simulations with polarization degree SD in all subplots. The lines consisting of different markers +denote the boundaries between different phases, which are the same as those presented in Fig. 4. The regions belonging to the three phases +are correspondingly labeled in the subplots. The light blue pentagram presented in (a3) indicates the triple point in (A, T) space. The dots +presented in (a3), (b3) and (c1) correspond to the subplots (a1)-(a3), (b1)-(b3) and (c1)-(c3) in Fig. 6, respectively. + +9 +FIG. 6. Temporal evolution of the individuals’ opinions for the population with heterogeneous susceptibility. The values of parameters are +correspondingly listed in the titles of the subplots. + +(a1) n=6.0, T=6.0, A=0.4 +(a2) n=6.0, T=6.0, A=1.7 +10 +10 +T +0 +0 +-10 +-10 +100 +105 +100 +Time +Time(a3) m=6.0, T=6.0, A=7.0 +10 +0 +-10 +100 +105 +Time10 +10 +T +0 +X +-10 +-10 +100 +105 +100 +Time +Time +(c1)A=0.5, n=3.0,T=0.5 +(c2) A=0.5, n=2.1, T=6.0 +10 +10 +TL +0 +X +-10 +-10 +100 +105 +100 +Time +Time10 +1 = 10.0,=0.0, H=0.0 +0 +X +-10 +100 +105 +Time +(c3) A=0.5, n=5.9, T=6.0 +10 +0 +-10 +100 +105 +Time10 +(a1) Polarization, SD +=2.1 +GC +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(a2) Polarization, SD +=2.5 +GC +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(a3) Polarization, SD +=3.0 +GC +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(b1) Polarization, SD +A=0.5 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(b2) Polarization, SD +A=1.0 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(b3) Polarization, SD +A=1.5 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +(c1) Polarization, SD +=0.5 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(c2) Polarization, SD +=1.0 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(c3) Polarization, SD +=2.0 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +FIG. 7. Phase diagram for the population with heterogeneous tolerance. In more detail, (a1)-(a3) phase diagrams in (A, α) space for three +different values of ξ, (b1)-(b3) phase diagrams in (α, ξ) space for three different values of A, (c1)-(c3) phase diagrams (A, ξ) for three +different values of α. The values of parameters are listed in corresponding subplots. We run simulations with SD in all subplots. The lines +consisting of either pink triangles or yellow circles depict the boundaries separating the regions of GC and P state, as given by our method. +The regions of different phases are correspondingly labeled in the subplots. + +11 +C. +The results from static networks +We subsequently embed our model into the social networks +of size N = 43952, where individuals interact through fixed +connections, so as to explore the joint effects of the network +topology and influence mechanism on the formation of HD. +The phase diagrams are presented in Fig. 8 in three differ- +ent parameter spaces: (A, T), (A, η) and (T, η). Our results +more or less remain robust with respect to the engagement +of fixed interactions. It is evident that the phase layouts are +largely dominated by model rules rather than fixed interac- +tions, and thus similar to those illustrated in Figs. 3 and 4. +In Fig. 9, we see that the behaviors of SD, S and χ(S) are +qualitatively similar to the case of time-varying network (see +Fig. 1). Whereas it is important to note that the introduction +of fixed interactions can remarkably give rise to much sharper +dependence of SD, S and χ(S) (see Fig. S5), and much larger +regions of P phase where extreme polarization (SD = 10) co- +exists with high-level polarization with large SD (see Fig. 8 +for the light red regions in P phases). Also notice that there +are less triple points in comparison with the results reported +in Figs. 3 and 3. +Figs. 9(a1)-(a6) further illustrate hot maps of the density +of users in the (x, ⟨x⟩NN) plane, for six values of A, where +the results reveals the correlation between the opinion of an +individual i, xi, and the average opinions of their nearest +neighbors. This can measure an individual’s preference to +connect to peers sharing similar opinions [21], which fosters +information about the effects of cluster-level self-reinforced +mechanism. In our study, cluster-level self-reinforced mech- +anism is defined that the like-minded members tend to clus- +ter together and persistently support those on the boundaries +to repel distant ones. Attributing to the symmetrical nature +of our model between positive and negative range, the con- +figurations of positive clusters i.e. +the clusters of positive +opinions, are similar to those of negative clusters with re- +spect to size and number. We thus calculate the statistics of +positive clusters by means of statistical variables reported in +Figs. 9(b) and (c), which provide an overall picture of cluster- +level self-reinforced mechanism. +This naturally gives rise +to the question whether abundance or size of these spatial +opinion clusters guarantee the presence of cluster-level self- +reinforced mechanism. +As we can see in Figs. 9(b1)-(b6), plotting the distributions +of positive clusters for six different values of A shows that +there are always giant positive clusters indicated by isolated +peaks, as well as power-law cluster-size distributions; which +is independent of the values of A. Hence the observed phe- +nomenon is not symbol of distributions transitions, instead, it +is rooted in the model rules and initial assignment of individ- +uals’ opinions. +(i) In the first parameter regime of HD state i.e., Ac1 < A < +A +′ +c, Figs. 9(a3) shows bright spindle-shaped diagonal speck- +les that stretch the first and third quadrants. This identifies a +high density of users hold moderate opinions of their own as +clusters of size 1, while most individuals are within the largest +clusters and unbiased, as we can see the first three markers in +Figs. 9(c2)-(c4). +(ii) In the range A +′ +c < A < Ac2, increasing interaction +strength makes more individuals split off from the largest clus- +ter to form more clusters of size 1 (see corresponding markers +in Fig. 9(c2)) and more smaller modest clusters (see corre- +sponding markers in Figs. 9(c3)-(c6)) which are not aligned. +There are longer boundaries between positive and negative +opinions, and this results in more coherent intra-group and +intenser group-level struggles among clusters of the opposite +sides i.e., stronger cluster-level self-reinforced mechanism. +We can see that the bright speckles become long streaks, and +several short streaks start to appear with the same direction in +the second and forth quadrants (see Figs. 9(a4) and (a5)). +(iii) When the population gets polarized for A > Ac2, +in addition to persistent modest clusters (see Fig. 9(c4)) +and the shrinking largest clusters of opposing camps (see +Fig. 9(c2)), smallest clusters of size 1 are more abundant +(see Figs. 9(c3)). They guarantee the strongest self-reinforced +mechanism, leading to longer speckles through the origin of +coordinates (see Figs. 9(a5) and (a6)). +With stronger sup- +port from like-mind group members, those streaks starts to +stretch out to the first and third quadrants (see Figs. 9(a5) +and (a6)). Whatever, unbiased individuals themselves have a +stronger preference to cluster together within the largest clus- +ters and therefore the patterns of the longest speckles are al- +ways brighter. This is definitely different from those previ- +ously uncovered echo chambers determined by large opposed +clusters [21], but in agreement with the empirical findings that +unbiased individuals themselves can form clusters [57]. +The above results firstly indicate that the abundance of clus- +ters of size 1 and modest clusters may actually be more im- +portant than their sizes in determining the effects of cluster- +level self-reinforced mechanism. Secondly, one can avoid ex- +treme polarization for increasing T and α, but intermediate +A. Under such parameter condition, people who are reluc- +tant to hearing different opinions can create many different +relatively modest ideological islands that people will be stuck +on. This may leads to low exposure which may prevent the +emergence of two opposing extreme opinions [24]. Conse- +quently, extreme polarization is replaced by high-level polar- +ization (see the light red regions in Fig. 9). On the other hand, +like-minded group members persistently instead support those +on the boundaries to repel distant ones, although the innermost +members are relatively unbiased. Polarized views can more +easily arise from the interaction between individuals at the +boundaries of clusters, making polarization more likely (see +larger red regions in Fig. 9), which is associated with lower +likelihood of a triple point. +The same is the case for heterogeneous susceptibility, still +the cluster-level self-reinforced mechanism is positively re- +lated to A. We can thus observe slight shrinkage of the regions +of GC and HD phase when A <∼ 2.0. In contrast, regions of +P phase instead compress the other two phases with increas- +ing A (see Fig. S6). We can confirm that the mechanism still +works, and to achieve a HD state is still impossible. + +12 +FIG. 8. Phase diagram for the population embed on part of Facebook network, where interactions among individuals are fixed. (a1)-(a3) Phase +diagrams in (A, α) space for four different values of T. (b1)-(b4) Phase diagram in (A, T) space for four different values of α. (c1)-(c4) +Phase diagram in (α, T) space for four different values of A. We run simulations with SD in all subplots. The lines consisting of different +markers denote the boundaries between different phases, which are the same as those presented in Fig. 4. The regions of the three phases are +correspondingly labeled in the subplots. The light blue pentagrams indicate the triple points which are less frequent. The dots presented in +(a3),(b2) and (c1) correspond to the subplots (a1)-(a3), (b1)-(b3) and (c1)-(c3) in Fig. S4, respectively. + +(a1) Polarization, SD +(a2) Polarization, SD +(a3) Polariz +10 +10 +10 +10 +10 +T=1.0 +T=2.0 +★HD +Susceptibility,α +Susceptibility,α +Susceptibility ,α +HD +8 +8 +8 +8 +8 +６ +６ +６ +6 +6 +P +4 +4 +4 +4 +4 +2 +¥2 +2 +2 +2 +GC +GC +. +0 +0 +0 +0 +0 +4 +6 +8 +¥10 +0 +2 +4 +6 +8 +10 +0 +2 +2 +4 +Interaction strength, A +Interaction strength, A +Interaction +G +h3)Polazation, SD +(a4) Polarization, SD +10 +10 +10 +T=7.0 +T=10.0 +Susceptibility,α +8 +8 +HD +8 +６ +6 +6 +P +P +4 +4 +4 +GC +2 +2 +2 +0 +0 +0 +6 +810 +0 +2 +6810 +strength, A +Interaction strength, A +sn10 +10 +10 +10 +10 +Q=0.5 +HD +8 +8 +8 +HD +8 +8 +HD +Tolerance, +¥６ +６ +6 +6 +P +P +４ +4 +4 +4 +GC +¥2 +2 +2 +2 +2 +GC +Q=1.0 +0 +0 +0 +0 +0 +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +0 +2 +4 +Interaction strength, A +Interaction strength, A +Interaction +(c1) Polarization, SD +(c2) Polarization, SD +(c3) Polariz +10 +10 +10 +10 +10 +A=0.5 +GC +GC +HD +8 +8 +8 +8 +8 +T +T +T +Tolerance, +Tolerance, +Tolerance, +HD +９ +6 +6 +6 +6 +HD +4 +4 +4 +P +4 +4 +2 +2 +2 +2 +A=1.5 +P +0 +0 +0 +0 +0 +0 +2 +4 +8 +10 +0 +2 +4 +6 +8 +10 +0 +2 +4 +Susceptibility,α +Susceptibility,α +Suscept10 +10 +10 +α=2.0 +HD +Q=10.0 +8 +8 +8 +T +Tolerance, +¥6 +6 +6 +P +P +4 +4 +4 +¥2 +2 +2 +0 +0 +0 +6 +¥810 +0 +2 +468 +¥10 +strength, A +Interaction strength, A +zation, SD +(c4) Polarization, SD +10 +10 +10 +８６ +8 +HD +8 +Tolerance, +6 +6 +P +P +4 +4 +¥2 +2 +2 +A=2.0 +A=2.5 +0 +0 +0 +6 +8 +10 +0 +2 +4 +6 +8 +10 +ibility,α +Susceptibility,α13 +FIG. 9. (a1)-(a6) Hot maps for the average opinion of the nearest neighbors ⟨x⟩NN against an individual’s opinion x, for Nr = 500 simulations +of independent realizations. (b1)-(b6) The distributions of positive clusters for six different values of A, where positive clusters refer to those +consisting of individuals owning positive opinions. (c1) The size of the population with positive opinions NP against A; (c2) The size of +largest positive cluster SP GC against A; (c3) The number of positive clusters nP C against A; (c4) The number of modest positive clusters +nMP C against A, which exclude positive clusters of size 1 and the largest positive clusters; (c5) The mean size of positive clusters SP against +A; (c6) The mean size of the modest positive clusters SMP C against A. The six markers presented in (c1)-(c6) rightly correspond the six +values of A given in (a1)-(a6) and (b1)-(b6). The values of other parameters are T = 7.0 and α = 1.0. + +(a2) A,=1.24 +(a3) A=1.43 +(a1) A=1 +(a4) A=2 +10 +10 +C1 +10 +10 +５ +5 +5 +5 +NN +0 +0 +0 + +-5 +-5 +-5 +-5 +-10 +-10 +-10 +-10 +-10 +0 +10 +-10 +0 +10 +-10 +0 +10 +-10 +0 +1 +X +X +X +X(a5) A.=2.41 +(a6) A=3 +10 +C2 +10 +4 +5 +5 +3 +0 +0 +2 +-5 +-5 +1 +-10 +-10 +0 +0 +-10 +0 +10 +-10 +0 +10 +X +Xb2 +(b1) A=1 +(b4) A=2 +10°℃ +C1 +C +10°g +0 +口 +S +d +5 +8 +0 +0 +0 +100 +105 +100 +105 +100 +100 +1 +S +s +S +S +(c1) +(c2) +(c3) +(c4) +X10 +2.5 ,X104 +2斤 +6000 +1000 +s +GN +2.4 +PGC +PC +P +5000 +1.8 +2.3 +4000 +500 +1.6 +3000 +2.1 +AA +V +MPC +2000 +2 +1.4 +0 +2 +3 +2 +1 +3 +2 +3 +2 +1 +A +A +A +A(b5) Ac2=2.41 +(b6) A=3 +c2 +109 +10° +0 +0 +100 +105 +100 +105 +s +s +(c5) +(c6) +12 +31 +-× SMPC +10 +2.8 +8 +2.6 +6 +XX! +4 +2.4 +3 +1 +2 +2 +3 +3 +A +A14 +IV. +DISCUSSIONS AND CONCLUSIONS +In this paper, we have proposed a simple model based on +one core assumption that individuals tend to amplify differ- +ence from others with dissimilar opinions according to nega- +tive influence, and to attract toward others with similar opin- +ions due to positive influence. We correspondingly consider +three key realistic ingredients to regulate the opinion dynam- +ics: interaction strength, individuals’ susceptibility to opinion +dissimilarity between it and its neighbor, and individuals’ tol- +erance to different views determines whether the result of an +interaction is attractive or repulsive. +For time-varying networks, within our model HD state +emerges for sufficient susceptibility, intermediate interaction +strength and high tolerance. Whereas strong interaction or +low tolerance makes this desired state not accessible, and +the population instead gets polarized. Remarkably, the sim- +ple model rules successfully generate the three phases, along +with three different transitions. Especially, as another novel +finding, there exists a triple point in the planes of (A, T) or +(α, T). To the best of our knowledge, these two important and +interesting phenomena have never been uncovered and men- +tioned in previous studies, especially those consider influence +mechanism. +In the second part of the paper, we have extended our study +to the heterogeneous cases where individuals’ susceptibility +or tolerance follows a power-law distribution whose exponent +are η and ξ, respectively. Through a comparison with the re- +sults in Subsec. III A, we find that, although the effect of sus- +ceptibility heterogeneity is less pronounced, it can facilitate +HD to a certain extent, and instead prevent the population +from achieving a global neutral consensus. +Unfortunately, +heterogeneous tolerance can largely facilitate the emergence +of P state such that HD is not achievable. To achieve a global +neutral consensus is nearly impossible unless both A and α +are rather small. Whatever, interaction strength A largely de- +termines the population’s outcome in such cases. Therefore, +it is evident that the effects of heterogeneous individuals’ in- +trinsic attributes such as susceptibility or tolerance cannot be +ignored. +In the last case, we have applied our model on empirical +static networks where the interactions among individuals are +fixed throughout. Although the final overall phenomenology +is similar to the time-varying case with respect to phase lay- +outs, the population has larger regions of P phase where ex- +treme polarization coexist with high-level polarization. We +have uncovered that cluster-level self-reinforced mechanism +is responsible for these phenomena, which is dependent on +whether the spatial opinion clusters of opinions are abundant. +In presence of such mechanism, the struggles among clusters +formed by like-minded individuals protects inner ones from +the influence of the majority of the population, allowing them +to persistently support the group members on the boundaries +in face of distant individuals of other clusters. It is also the +cause of the other two results: the coexistence of extreme po- +larization and high-level polarization, and the triple point is +less likely to obtain. Moreover, we have confirmed that, for +heterogeneous susceptibility or tolerance, this mechanism still +works, and to achieve a HD state thus becomes harder or even +impossible in presence of fixed interactions. +For the first time, this paper proposes a basic theory frame- +work to understand the formation of HD, and we can con- +clude the four following remarks from this paper: (i) Most +importantly, through simplest possible setting, our proposed +model can generally generate three phases: GC, HD and P, +along with a triple point, regardless of that the interactions +are time-varying or fixed. Which has never been stressed or +mentioned by previous opinion models. (ii) We correspond- +ingly propose an effective method to numerically identify the +boundaries between these phases though calculating the sus- +ceptibility of opinion entropy, and the simulations validate this +method. (iii) Heterogeneous attributes such as susceptibility +turns out to be a facilitating factor for achieving a HD state. +However, heterogeneous tolerance which makes things worse, +should be avoided. (iv) Fixed interactions create a negative +impact on the emergence of HD. They can generate cluster- +level self-reinforced mechanism through abundant clusters of +size 1 and modest opinion clusters, which can unexpectedly +promote polarization. +However, it still remains a challenge to theoretically locate +the position of the triple points, and uncover the nature of the +transitions on both side of the triple point. Moreover, it is +worth noticing that our model is based on a minimal number +of assumptions. It does not take into account some empiri- +cal features of networks or individuals which might generate +different scenarios, such as heterogeneous duration time of in- +teractions, and or different social positions of individuals. In +light of this fact, it is essential to further extend our study in +the future. Also, this study opens one interesting issues for fu- +ture research: whether HD state becomes more easily acces- +sible with the involvement of some optimization strategies? +And if the answer is yes, how much it will do. +ACKNOWLEDGMENTS +This work was supported by the Key Program of +the National Natural Science Foundation of China (Grant +No. 71731002), and by Guangdong Basic and Applied Basic +Research Foundation (Grant No. 2021A1515011975). P.-B. +C. thanks Kai Qi for helpful discussions. +APPENDIX +Appendix A + +15 +(a1) Entropy, S +=0.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(a2) Entropy, S +=1.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(a3) Entropy, S +=2.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(a4) Entropy, S +=10.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Tolerance, T +(b1) Entropy, S +A=0.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +(b2) Entropy, S +A=1.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +(b3) Entropy, S +A=2.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +(b4) Entropy, S +A=2.5 +GC +HD +P +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +1 +2 +3 +4 +5 +6 +FIG. S1. (a1)-(a4) The dependence of opinion entropy S on A and T for four different values of α. (b1)-(b4) The dependence S on α and +T for four different values of A. Different regions of the three states are correspondingly labeled. The lines consisting of different markers +denote the boundaries between different phases, which are the same as those presented in Fig. 3. The light blue pentagrams indicate the triple +points. +APPENDIX B + +16 +(a1) Entropy, S +=2.1 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +(a2) Entropy, S +=3.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +(a3) Entropy, S +=6.0 +GC +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +(b1) Entropy, S +T=1.0 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +(b2) Entropy, S +T=2.0 +GC +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +(b3) Entropy, S +T=10.0 +GC +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +(c1) Entropy, S +A=0.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +(c2) Entropy, S +A=1.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +(c3) Entropy, S +A=2.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +FIG. S2. (a1)-(a3) The dependence of S on A and T for three different values of η. (b1)-(b3) The dependence of S on A and η for three +different values of T. (c1)-(c3) The dependence of S on T and η for three different values of A. Different regions of the three phases are +correspondingly labeled. The lines consisting of different markers denote the boundaries between different phases, which are the same as those +presented in Fig. 3. There is only one triple point indicated by light blue pentagram in (a3). + +17 +(a1) Entropy, S +=2.1 +GC +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +1 +2 +3 +4 +(a2) Entropy, S +=2.5 +GC +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +1 +2 +3 +4 +(a3) Entropy, S +=3.0 +GC +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +1 +2 +3 +4 +(b1) Entropy, S +A=0.5 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +1 +2 +3 +4 +(b2) Entropy, S +A=1.0 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +0.5 +1 +1.5 +2 +(b3) Entropy, S +A=1.5 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Susceptibility, +0 +0.5 +1 +1.5 +2 +(c1) Entropy, S +=0.5 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +1 +2 +3 +4 +(c2) Entropy, S +=1.0 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +1 +2 +3 +4 +(c3) Entropy, S +=2.0 +GC +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +1 +2 +3 +4 +FIG. S3. (a1)-(a3) The dependence of S on A and α for three different values of ξ. (b1)-(b3) The dependence of S on α and ξ for three +different values of A. (c1)-(c3) The dependence of S on A and ξ for three different values of α. Different regions of the two states GC and P +phase are correspondingly labeled. +FIG. S4. The evolution of the entropy of opinion distribution for three different values of susceptibility α. The values of other parameters such +as interaction strength A and tolerance threshold exponent ξ are listed in the titles of subplots. + +(a) A=0.5, =2.2, α=0.1 +10 +10 +5 +5 +0 +0 +X +X +-5 +-5(b1) A=0.5, =2.2, α=0.5101 +102 +103 +104 +105 +100 +100 +Time +(c) A=0.5, =2.2, α=4.0 +10 +10 +5 +5 +0 +0 +X +X +-5 +-5 +-10 +-10 +101 +102 +103 +104 +100 +105 +100 +Time101 +102 +103 +104 +105 +Time +(b2) A=0.5, =2.2, α=0.5 +101 +102 +103 +104 +105 +Time18 +APPENDIX C +[1] R. Axelrod, J. Confl. Resolut. 41, 203 (1997). +[2] N. P. Mark, Am. Sociol. Rev. , 319 (2003). +[3] D. 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X 10, 041042 (2020). + +19 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Variables +T=7.0, +=1.0 +(a1) +Ac1 +Ac2 +SD +(S) +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Interaction strength, A +0 +2 +4 +6 +Variables +(a2) +Ac1 A' +c +Ac2 +S +(S) +0.5 +1 +1.5 +2 +2.5 +3 +Interaction strength, A +0 +2 +4 +6 +8 +10 +Variables +=1.0, T=6.5 +Ac1 +Ac2 +(b1) +SD +(S) +0.5 +1 +1.5 +2 +2.5 +3 +Interaction strength, A +0 +2 +4 +6 +Variables +Ac1A' +c +Ac2 +(b2) +S +(S) +0.5 +1 +1.5 +2 +Susceptibility, +0 +2 +4 +6 +8 +10 +Variables +A=1.5, T=4.0 +c1 +c2 +(c1) +SD +(S) +0.5 +1 +1.5 +2 +Susceptibility, +0 +2 +4 +6 +Variables +c1 +' +c +c2 +(c2) +S +(S) +1.1 +1.3 +1.5 +0 +1 +1.1 1.3 1.5 +0 +1 +0.9 +1.1 +1.3 +0 +1 +0.9 +1.1 +1.3 +0 +1 +0.6 +0.9 +0 +1 +0.6 +0.9 +0 +1 +FIG. S5. The dependence of SD (red circles), S (blue squares), χ(S) (brown curves and blue cures in the insets) on A in part of Facebook +network. Pink vertical line labels the position of Ac1 at which HD state begins to appear, while light blue vertical line indicates the position of +Ac2 at which HD state vanishes. In particular, A +′ +c denotes an inflection point in S followed by a sharp growth of S. The values of T and α are +correspondingly listed in titles of (a1), (b1) and (c1). + +20 +(a1) Polarization, SD +=2.1 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(a2) Polarization, SD +=3.0 +HD +P +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(a3) Polarization, SD +=6.0 +HD +P +GC +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +5 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(b1) Polarization, SD +T=1.0 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(b2) Polarization, SD +T=2.0 +GC +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(b3) Polarization, SD +T=10.0 +GC +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Interaction strength, A +0 +2 +4 +6 +8 +10 +(c1) Polarization, SD +A=0.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(c2) Polarization, SD +A=1.5 +HD +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +(c3) Polarization, SD +A=2.5 +P +2 +3 +4 +5 +6 +Heterogeneity, +0 +2 +4 +6 +8 +10 +Tolerance, T +0 +2 +4 +6 +8 +10 +FIG. S6. Phase diagrams for the population embed in part of Facebook network, where interaction among individuals are fixed. In more detail, +(a1)-(a3) phase diagrams in (A, T) space for three different values of η. (b1)-(b3) phase diagrams in (α, η) space for three different values +of T. (c1)-(c3) phase diagrams in (T, η) for three different values of A. We run simulations with polarization degree SD in all subplots. The +lines consisting of different markers denote the boundaries between different phases, which are the same as those presented in Fig. 3. The +regions belonging to the three phases are correspondingly labeled in the subplots. The light blue pentagram presented in (a3) indicates the +triple point in (A, T) space. + diff --git a/uNE1T4oBgHgl3EQfQgPj/content/tmp_files/load_file.txt b/uNE1T4oBgHgl3EQfQgPj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7631e18b98006a5ba23532f98bad29259dee7a4 --- /dev/null +++ b/uNE1T4oBgHgl3EQfQgPj/content/tmp_files/load_file.txt @@ -0,0 +1,1246 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf,len=1245 +page_content='Exploring the formation of ”harmony with diversity” state within a attraction-repulsion model framework Peng-Bi Cui (崔鹏碧)1, ∗ 1International Academic Center of Complex Systems, Beijing Normal University, Zhuhai, 519087, China Opinion evolution is generally subject to either global neutral consensus under influence mechanism, or po- larization into two opposing camps in face of distant views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, as a third novel state, the formation of ”harmony with diversity”, where individuals freely express various viewpoints to a certain extent to sustain integration of diversity and at the same time shared values ensure social coherence to avoid ideological split, still remains unclear, as well as its unique dynamic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Since a general model framework to generate the desired state together with global consensus and polarization is still lacked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' To address this issue, we develop an attraction-repulsion model based on the general simple assumption that individuals tend to either reach an agreement with shared opinions or to amplify difference from others with distant opinions, which allows us to take account into the three core parameters: interaction strength, individuals’ susceptibility and tolerance to others’ opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We are concerned with the effect of not only time-varying topology but also fixed interactions imposed by static social network, where the tasks of heterogeneous individuals’ attributes are also performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Remarkably, the simple model rules successfully generate the three phases, along with three different transitions and the triple point, regardless of whether the interactions are time-varying or fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Especially, state of ”har- mony with diversity” emerges for sufficient susceptibility, intermediate interaction strength and high tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Whereas the population instead gets polarized for strong interaction or low tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Fixed interactions can in- troduce cluster-level self-reinforced mechanism which can unexpectedly suppress the emergence of ”harmony with diversity”, and instead promote polarization due to negative influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, heterogeneous attributes such as susceptibility turns out to be a facilitating factor for achieving the desired state, while heterogeneous tolerance can make things worse, which should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In any case, the simulations validate our method to identify the boundaries between different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Overall, this study gains a number of crucial insights that can be subject to further empirical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' INTRODUCTION Opinion evolution is about crucial issues which may gener- ally rises three distinct states: (i) Global consensus (GC state) in which people’s view evolves towards one common opin- ion, showing a narrow unimodal distribution centering on the neutral consensus [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (ii) Polarization (P state) into two op- posing clusters or camps which is characterized by a bimodal distribution around the neutral consensus [2–5], further mak- ing compromise impossible and giving rise to social oppo- sition [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (iii) Fragmentation (F state) is the state with more than two distinct clusters in the spectrum [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Es- pecially, polarization is here conceived as a process of grow- ing constraint in people’s viewpoints, and further emergence of alignments along multiple or even opposite lines of poten- tial disagreement with respect to political views, immigration, biotechnology applications, LGBT rights, mask wearing and viral nucleic acid detection in the COVID-19 [1, 4, 5, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In addition, a striking state besides GC, F and P state can- not be neglected, in which individuals’ views or positions self- organize into only one cluster with wide spectrum and fluctu- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is definitely different from F and GC state that, with respect to ideological issues, the emergence of one opinion cluster with wide spectrum can be defined as a desired state of ”Harmony with diversity” (HD state), which has also been stressed by Confucius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Since the presence of HD state reveals that individuals can freely express different viewpoints to a ∗ cuisir610@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='com certain extent while the free internal debates result in fluctu- ations, and at the same time shared values ensure social co- herence and the avoidance of ideological split, or even sharp conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This robust empirical phenomenon is actually vary crucial to healthy functioning of social systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' For example, except the United Kingdom, member states of the European Union (EU) over the last two decades have been straining will- ingness to comply with EU agreements, aiming to sustain an internal single market based on the common Motto ”United in Diversity”, regardless of the fact they have various eco- nomic demands [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, the softness of the laws or decrees targeting COVID-19, if they are embodied in a basic loose national agreement that this disease is harmful and in- fectious, tends to guarantee least controversial measures such as wearing face masks, social distancing and home quaran- tine [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Unlike shelter-in-place ordinances, lockdowns, or capacity restrictions, the three measures can diminish the spread of an infection, and simultaneously do least harm to economic activity [10, 12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, to the best of our knowledge, the factors contributing to the emergence of HD state still remain unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Considerable efforts have been recently put into model- ing polarization dynamics to uncover the mechanisms behind it [8, 17–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' One of progresses is to measure polarization with the variance or SD of continuous ideological positions in a population [23, 24, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, to the best of our knowl- edge, the emergence and persistence of HD state has received too little attention in the growing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Since most com- putational attempts are trapped in the framework of two-state model, failing to additionally reproduce a HD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The main approaches of these studies are based on traditional attraction- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='03042v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='soc-ph] 8 Jan 2023 2 repulsion model (ARM) with a confidence bound defining a reference that an individual compares with the opinion dis- tance to a neighbor, and based on it, the individuals decide to move closer or farther from its neighbor [2, 3, 23, 24, 28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' According to such mechanism, the contest between attracted and repelled forces can accordingly facilitate either GC or P states [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Alternatively, some modeling approaches inves- tigate the coevolution of the network topology and opinion dynamics as the result of reinforcement and rewiring activi- ties driven by structural similarity [23] or homophily [21, 26]: the two individuals sharing similar connections or opinions are more likely to interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Whatever the case, the HD state does not appear to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Although, some seminal works on cultural diversity show that the emergence of clustering phase is possible [6], GC and P states cannot be theoretically reproduced within the same model rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, due to the lack of HD state, the physical pictures of opinion evolu- tion from the attempts capturing biased assimilation [34] or information accumulation [35] are not integrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The prob- lem rises two fundamental questions we want to answer in this study: whether simple settings can stably facilitate HD state in addition to GC and P states, and how to effectively distinguish the three phases from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Influence plays a decisive role in driving the evolution of individuals’ opinions, which is responsible for individu- als’ tendency to either reach agreement with shared opin- ions (positive influence) [1, 17, 36, 37] or to amplify dif- ference from others with distant opinions (negative influ- ence) [2, 3, 29, 30, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Consequently, the attraction- repulsion model (ARM) can well curve opinion dynamics by setting a threshold for tolerance of disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Within the framework, rules specify the mechanisms for interaction be- tween individuals, they can thus provide insight about impor- tant mechanisms and the role they play in regulating emergent properties of system, and highlight the consequences from a few simple assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In addition, other mechanisms such as homophily and media influence can be more easily incor- porated into ARM to successfully map more empirical polar- ization changes [8, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, to address the above two questions, we develop an agent-based attraction-repulsion model (ABARM) featur- ing the emergence of GC, HD and P state from microscopic in- teractions between individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, this paper focuses narrowly on the phase analysis with respect to ideological is- sues, rather than elucidating new mechanisms behind polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In our paradigm, opinion evolution rules are built on the general simple assumption that individual’s attraction to or repulsion from others is only dominated by their opinion sim- ilarity in one-dimensional topic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It allows us to uncover the exact mechanisms that yield particular outcomes, with- out loss of consideration of three core parameters: interaction strength, individuals’ susceptibility and tolerance to different opinions of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We are concerned with the effect of not only time-varying topology but also the static social network (part of Facebook friendships network) on opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, the tasks with heterogeneous susceptibility or tol- erance are also considered and performed to make the study closer to the reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Regardless of the interaction structures and the heteroge- neous attributes, our model successfully generates the three states: (i) global convergence towards a neutral consensus, (ii) a fluctuated cluster with a wide spectrum of opinion, (iii) polarization which opinions split into two opposing camps, which are defined as GC, HD and P states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It can help us uncover the mechanism for facilitating HD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In addition to the SD of population opinion, we additionally de- fine opinion entropy as a novel order parameter of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Crucially, a novel available method which indicating the max- imum susceptibility of opinion entropy is correspondingly de- veloped to numerically identify the boundaries between the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Note the good agreement between simulations and the estimated boundaries, validating the phase-identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We thus conclude that the three phases are distinct in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Interestingly, comparison of the results from time-varying networks and part of Facebook friendships net- work reveals that the reinforcement from spatial opinion clus- ters can increase the likelihood of high-level polarization, in- stead of extreme polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This highlights complex roles of group-level struggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In addition, the present model suc- cessfully reproduces the fluctuations of cluster without the noise effect, which has been previously considered as the cen- tral mechanism to facilitate the fluctuation of opinion clus- ters [6, 40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' For the first time, our study proposes a basic framework to explore the formation of HD state, and thus the unique dynamic features of HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' II, we introduce the model, as well as the phase-identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' III A, we firstly simulate our model with ho- mogeneous and heterogeneous individual attributes on time- varying networks, and then on parts of empirical social net- works with fixed interactions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' III C, attempting to make deep comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We conclude with a summary of the results and an outlook for future studies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' MODEL AND PHASE-IDENTIFICATION METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Model The developed ABARM basically assumes a threshold of tolerance to determine whether a repulsive or attractive inter- action happens, in line with the assumption of bounded con- fidence which has been widely confirmed by previous stud- ies [24, 37, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The threshold defines a reference distance in terms of opinion difference, based on which an agent would decides whether to move closer or farther from the selected neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' If opinion difference between the individual and its neighbor is smaller than the threshold, an individual will tolerate the position of its neighbor, and move closer to the neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Otherwise, it will move farther from the neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' After defining the original model, in turn, we can proceed our explorations by accomplishing the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (i) Make the at- tributes individuals in response to the others’ position hetero- geneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (ii) Next, endow the parts of social networks with local structure to fix the interactions among individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' A systematic comparison of the three cases (including the basic 3 model) with regard to phase areas allows us to fully under- stand the formation of HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The basic model considers a population of size N where each individual is typified by an opinion xi(t) at time t, which is a real number in the interval xi(t) ∈ [−10, + 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Inspired by recent studies to capture polarization process on social me- dia networks [21, 23, 26, 44], the updating of individual’s opinion xi(t) is solely driven by the interactions among indi- viduals and is described by the following N coupled ordinary differential equations: ˙xi(t) = � A tanh(αiDji(t)) if |Dji(t)| < Ti;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' A tanh(αiσ(Dji(t))(xi(t) − |Dji(t)|)) if |Dji(t)| ≥ Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (1) Dji(t) = xj(t)−xi(t) is the opinion difference between i and j at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Ti denotes the tolerance threshold of individual i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' αi can be interpreted as the controversialness of the topic, and thus gauge nonlinearity of interaction or susceptibility of in- dividual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, αi positively associates with the extent to which individual i is passionate, attentive or sensitive, as a re- sult, susceptible to be socially influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is obvious that Ti and αi are largely related to intrinsic preferences of individual i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Nonlinear shape of the influence function tanh(x) is con- trolled by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' While A quantifies interaction strength, which is actually the upper bound of opinion shift driven by each in- teraction, indicating that the influence exerted by individuals on others is capped, in accordance with the experimental find- ings [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' σ(Dji(t)) extracts the sign of Dji(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Note that in our model the opinion difference is responsible for the evolu- tion of opinions, attempting to capture both the repulsive and attractive forces, which differs from the model settings focus- ing mainly on reinforcement and homophily [21, 23, 26, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Firstly, we define the activity of each individual by means of activity driven (AD) model [26, 46–48], so that opinion evolu- tion is coupled to an underlying time-varying network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' More in detail, ki(t) represents the number of interactions they can have with others within a given time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It hence gives rise to a temporal network determine by the temporal adjacency matrix Aij(t), where Aij(t) = 1 if individual i contacts in- dividual j, otherwise Aij(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' More specifically, individ- ual i connects ki distinct random other individuals, such that ki = �N j Aij(t) is satisfied throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Con- sidering the empirical statistics that activities of people are generally heterogeneous [21, 47, 48], we assume the interac- tions are extracted from a power-law distribution p(k) ∼ k−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Intrinsic preferences toward topics are always proved to be heterogeneous in reality [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, we next assign each individual a susceptibility αi or tolerance threshold Ti, which is correspondingly selected from the following power- law distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' p(α) ∼ α−η, (2) p(T) ∼ T −ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (3) η is susceptibility exponent, and ξ is tolerance threshold ex- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In such case, we still perform simulations on time- varying networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Online social networks are increasingly used to access opinion information, engage with COVID-19 vaccines, gun- control, abortion and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' These platforms can reduce bar- riers and cost to information and, further, allow individuals to freely voice their viewpoints, consequently improving rate of opinion exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' On the other hand, the structures of social networks combined with other mechanism such as reinforce- ment can drive the emergence of echo chambers [23, 51, 52], where the segregation in opinion space is reflected in inter- actions among individuals [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, we finally fix the interactions among individuals by embedding the population into the parts of the social media networks such as Facebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In such cases, ki corresponds to the number of edges that in- dividual i stretch to its neighbors, and the individual is repre- sented by a node in the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' As a result, the connected neighbors of each individual keep unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We perform the above two cases on the static networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In numerical simulations, we set the simulation parameters to be the following values: N = 1000 for time-varying net- works and γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1, that will be specified when needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The control parameters of the present model is A, αi, Ti, η and ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The final results are obtained from Nr = 100 independent realizations, after at least 500 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' For each simula- tion realization, the initial opinions attributed to each individ- ual is independently and randomly sampled from the interval [−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Then at each time step t, opinion evolves as fol- lowing: (i) In random order, each individual (i) is selected from the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (ii) Within the framework of AD, the initial temporal adjacency matrix Aij(t) is the zero matrix, and i can randomly choose ki new neighbors out of all indi- viduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Nevertheless, in the cases considering a static social media network, the neighbors of i keep unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (iii) Then, one by one, i compares its opinion with that of the neighbors, attempting to update its opinion according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We operationalize the degree of polarization through the standard deviation (SD) in opinions SD(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=', xN), and measure the opinion diversity by calculating the opinion en- tropy (S) of the population: S = �xmax xmin xρx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' ρx = Nx N is the density of individual owning opinion x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Nx denotes the population of opinion x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' xmin = −10 and xmax = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phase-Identification Method In our model, SD gets larger due to increasing degree of global polarization with a bimodal distribution, and the mini- mum of the polarization SD = 0 suggests a narrow unimodal distribution and thus the existence of GC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' While HD 4 state corresponds to that a opinion cluster with wide spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The changes of opinion entropy are therefore sensitive to the transitions between different state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Inspired by this fact, we employ the susceptibility of S to numerically determine the thresholds between different sates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' χ(S) = � ⟨S2⟩ − ⟨S⟩2 ⟨S⟩ , (4) where ⟨S⟩ is the ensemble average of S, which can be ob- tained by averaging S from Nr independent realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' ⟨S2⟩ is the secondary moment of the ensemble distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Based on the principle that χ(S) exhibits a maximum value at the threshold, one can further identify the boundaries between GC, HD and P states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The results from time-varying networks Three different cases occur in order with increasing inter- action strength A, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The evolution of a system following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1 is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The population firstly converge to the neutral consensus within a short time (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2(a)) when individuals are more or less independent due to weak interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Then a striking phenomenon occurs at intermediate levels, which produces a mediate polarization that begins to increase slowly after a transition whose thresh- old is denoted by Ac1 (see the secondary peak illustrated in the subplots of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1(a1) and (b1)) and then takes hold, cor- responding rightly to a highland of entropy (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1(a2) and (b2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In such case, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2(b) and (c), a fluc- tuated integral opinion cluster in ideological space persists for the entire simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Specifically, an inflation point of opin- ion entropy can also be indicated by the position of the first main peak susceptibility of opinion entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' While spectrum width of the cluster on both sides of the peak was claimed to be rather different, and the spectrum width tend to extend sharply as A > A ′ c (compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2(b) and (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, it is important to note that the occurrence of an inflation point imply the existence of a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In other words, the infla- tion point is dependent on whether the state of opinion clus- ter with wide spectrum exists or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Though, the interest- ing phenomenon is qualitatively different from the dependent transition class uncovered by microcanonical statistical anal- ysis [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The system governed by the intermediate range of interaction strength regime Ac1 < A < Ac2 provides suffi- cient evidence for the existence of HD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Yet, the polarization starts to rise uncontrollably at the sec- ond threshold Ac2, at which the transition involves an asym- metric temporal trajectory (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2(d) and (e)), and after which the population goes to the extreme i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=', P state due to considerable negative influence [2, 3, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The asymmetric temporal trajectory at Ac2 suggests an asymmetric polariza- tion which has also been the focus of recent studies [44, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In P state, opinion symmetrically splits into two opposite ex- treme camps, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Obviously, the three dif- ferent cases correspond to GC, HD and P state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1 (c1) and (c2), individual’s sus- ceptibility has similar effects on opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' To achieve a better understand the most basic effects of in- teraction strength and individual’s susceptibility on opinion dynamics, we then extend the previous observations to a wide range of (A, α) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3, where the color encodes the values of SD (top panels) and S (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We observe two types of transitions: from GC to HD phase when α is small, and from HD to P phase for large α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The transition from HD to P is largely invariant with respect to A if individuals are sen- sitive enough to the dissimilarity in opinions (α is not small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' While the regions of stable neutral consensus are character- ized by small values of A and α, and regions of HD can be obtained by for increasing A and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, a comparison among the top (bottom) panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3 reveals that increasing tolerance threshold T can expand HD regions being robust to change of susceptibility, by simultaneously occupying the re- gions of GC and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The new interesting phenomenon is largely in accordance with the empirical results [7, 55] and the nature of the social moral rules requiring a high level of tolerance, and thus one of novel findings of our present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We can see in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3(b1)-(b4) that highlands of opinion entropy per- fectly indicate the regions of HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The identified boundaries afford a precise division of the regions of different phases, validating our phase-identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4(a1)-(a4) show the phase diagrams in terms of SD in (A, T) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We find not only transition from GC to P phase if T is small but larger than a certain value, but also transitions from GC to HD phase, and from HD to P phase as T getting larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, there definitely exists a triple point denoted by a vanishing entropy highland (see the entropy highlands shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S1(a1)-(a4)), above which the emergence of GC is completely dominated by A, and independent of the change of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' If the tolerance threshold is large enough, weak interaction becomes a fully dominant factor for a global con- vergence, and intermediate A can generate HD state whose region tends to expand linearly with increasing T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' For all large susceptibility promotes HD to erode the regions of GC, and in turn P occupy the regions of HD, leading to a shrink- age of GC regions until its disappearance when α = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 (combine with the entropy highlands shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S1(a1)- (a4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4(b1)-(b4) offers a comprehensive view of the effects of individual’s susceptibility on opinion dynamics for different levels of tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We find that, if α becomes large, polarization is likely to emerge, along with extinction of GC and decreasing likelihood of HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Intermediate levels of tol- erance guarantees an intermediate range of susceptibility for HD state when A is not small, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1(c1)-(c2) and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S2(b1)-(b4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, high tolerance would result in saturation effect of large α, and role of susceptibility is lim- ited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In line with the results illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1 and 3, strong interaction can make things worse, and it is hard to achieve HD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Still, the system can produce a triple point, and there occur three different transitions under certain parameter conditions: from GC to P, GC to HD, and HD to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Up to now, the results from the basic model on time-varying networks reveals that HD state can be easily facilitated by sufficient susceptibility, intermediate interaction strength and 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 3 Interaction strength, A 0 5 10 Variables T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 (a1) Ac1 Ac2 SD (S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content="5 3 Interaction strength, A 0 2 4 6 Variables (a2) Ac1 A' c Ac2 S (S) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 3 Interaction strength, A 0 2 4 6 8 10 Variables =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 Ac1 Ac2 (b1) SD (S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content="5 3 Interaction strength, A 0 2 4 6 Variables Ac1 A' c Ac2 (b2) S (S) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 Susceptibility, 0 2 4 6 8 10 Variables A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, T=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 c1 c2 (c1) SD (S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content="5 2 Susceptibility, 0 2 4 6 Variables c1 ' c c2 (c2) S (S) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The dependence of SD (red circles), S (blue squares), χ(S) (brown curves and blue cures in the insets) on interaction strength A in time-varying networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Pink vertical line labels the position of Ac1 at which HD state begins to appear, while light blue vertical line indicates the position of Ac2 at which HD state vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In particular, A ′ c denotes an inflection point in S followed by a sharp growth of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of T and α are correspondingly listed in titles of (a1), (b1) and (c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Temporal evolution of the individuals’ opinions for six different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The six cases are also indicated by the dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3(a3) for a more explicit presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of the parameters are listed in the titles of the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' high tolerance are responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In contrast, strong interaction or low tolerance can make things worse, and the population can easily be polarized and extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, as another novel finding, the system formulated by our model can gener- ally generate a triple point in different parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a) T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 (b) T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, Qα=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='095 10 10 5 5 0 0 X X 5 5 10(c) T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 10 5 0 5 10100 105 100 Time Time (d) T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='375 (e) T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='38 10 10 5 5 t 0 0 X 5 5 10 10 100 105 100 Time Time100 105 Time (f) T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, α=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 10 5 0 5 10 100 105 Time6 (a1) Polarization, SD T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, (a2) Polarization, SD T=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, (a3) Polarization, SD T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, (a4) Polarization, SD T=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (b1) Entropy, S T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, (b2) Entropy, S T=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, (b3) Entropy, S T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, (b4) Entropy, S T=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 2 4 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phase diagrams in (A, α) space for four different values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We run simulations with polarization degree SD in (a1)-(a3), and with opinion entropy S in (b1)-(b3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of pink triangles depict the boundaries from the first threshold Ac1, separating the regions of GC and HD phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' while the lines consisting of yellow circles depict the boundaries from the second threshold Ac2, separating the regions of HD and P phase, as given by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The regions belonging to different states are labeled in the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The six dots correspond to the subplots of six parameter combinations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1) Polarization, SD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (a2) Polarization, SD =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (a3) Polarization, SD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (a4) Polarization, SD =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (b1) Polarization, SD A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T (b2) Polarization, SD A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T (b3) Polarization, SD A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T (b4) Polarization, SD A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T 0 1 2 3 4 5 6 7 8 9 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1)-(a4) Phase diagrams in (A, T) space for four different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b4) Phase diagrams in (α, T) space for four different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We run simulations with polarization degree SD in all subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of pink triangles depict the boundaries from the first threshold Ac1, separating the regions of GC and HD phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' while the lines consisting of yellow circles depict the boundaries from the second threshold Ac2, separating the regions of HD (GC) and P phase, as given by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The regions belonging to different states are labeled in the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The light blue pentagrams indicate the triple points which, however, do not present in (A, α) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The results from heterogeneous attributes Individuals are always heterogeneous in their roles, their power and their capacity to influence others, which is mostly rooted in their heterogeneous intrinsic attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Within our model framework, individuals differ in terms of their suscep- tibility or tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In this subsection, we focus on the effects of heterogeneous susceptibility or tolerance, which follows power law distributions given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This lends a different and distinct perspective to our studies of HD state through comparing the results with the homogeneous case in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 5 depicts phase diagrams in terms of SD in parame- ter spaces of (A, T), (A, η) and (T, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We firstly note that, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 5(a1)-(a3), the layouts of phase diagrams are similar to those illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4(a1)-(a4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The difference is that GC emerges only if the susceptibility exponent increases from ”scale-free” regime (η < 3) to ”small-wold” regime (η > 3) where the value of SD can still identifies three types of tran- sitions as identified in figures from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4(a1)-(a4) to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, there is still a triple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, in (A, η) space, things are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In this case, we find at most two different transitions for high-level tolerance: from GC to HD phase, and from HD to P phase (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 5(b1)-(b3)), let alone the existence of the triple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In line with what we have ob- served in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 5(a1)-(a3), we can obtain GC state for η >∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is evident that ηc ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 can thus be considered as a thresh- old for GC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In addition, intermediate A can generate HD state whose regions expand with T, slightly shrink with η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This means that heterogeneity of susceptibility prevents the population from converging to a global neutral consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' As expected, increasing T can effectively promote the population into HD phase after a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is consistent with the ho- mogeneous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Although it becomes harder to achieve the desired state for large A which though broaden the spectrum of opinion cluster (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S2(c1)-(c3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, HD re- gions gradually shrink with η along with a expansion of P re- gions, suggesting a positive role of susceptibility heterogene- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In comparison with the parameters A and T, the effect from susceptibility heterogeneity is less pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' How- ever, to obtain a triple point or GC state, η must be firstly in ”small-world” range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Correspondingly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 6 further approves that the dynamic behaviors do not qualitatively change by exhibiting three states of opinion evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Especially, the middle subplots, as well as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 6(c3), show the temporal behaviors of individ- uals’ opinions in HD state, where the opinion cluster is fluc- tuating irregularly with time, but integral even when the clus- ter touches the boundary of ideological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This further supports our statement about the system going into HD state where patterns of opinion cluster keep robust against time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We next focus on the effect of heterogeneous individuals’ tolerance on opinion dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Remarkably, the population with heterogeneous tolerance leads to a qualitative change of the results with respect to both phase diagram and temporal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 7, in any case, we find that heterogeneity of tolerance largely facilitate the emergence of P state, and the population will be finally polarized when A or α is beyond a small value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This is definitely in contrast to the convergence properties of bounded confidence models [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In such case, most individuals are closed-minded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=', has a smaller toler- ance threshold, and therefore have a stronger tendency to am- plify differences from others with slightly dissimilar opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, saying that the blue regions between GC and P phases (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 7(a1)-(a3) and (b1)) correspond to HD phase is not appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is instead qualitatively different from the aforementioned HD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Since Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S4(b1) and (b2) show that the population is eventually polarized as the opinion clus- ter touches the boundaries of the ideological space, and gets absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In this region, we have checked that it is rather hard for the system to reach a steady state within a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' From these results, we find that the population with heterogeneous tolerance cannot generate a HD state, as well as that the prob- ability of converging to a global neutral state is extremely low unless both A and α are rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Interaction strength A largely determines the population’s outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Tolerance het- erogeneity can largely increase the tendency of the population getting trapped in P state, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 7(b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 8 (a1) Polarization, SD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 8 10 (a2) Polarization, SD =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 8 10 (a3) Polarization, SD =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 8 10 (b1) Polarization, SD T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (b2) Polarization, SD T=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (b3) Polarization, SD T=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (c1) Polarization, SD A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 8 10 (c2) Polarization, SD A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 8 10 (c3) Polarization, SD A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 8 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phase diagrams for the population with heterogeneous susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In more detail, (a1)-(a3) phase diagrams in (A, T) space for three different values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b3) phase diagrams in (α, η) space for three different values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c1)-(c3) phase diagrams in (T, η) for three different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We run simulations with polarization degree SD in all subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of different markers denote the boundaries between different phases, which are the same as those presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The regions belonging to the three phases are correspondingly labeled in the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The light blue pentagram presented in (a3) indicates the triple point in (A, T) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The dots presented in (a3), (b3) and (c1) correspond to the subplots (a1)-(a3), (b1)-(b3) and (c1)-(c3) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Temporal evolution of the individuals’ opinions for the population with heterogeneous susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of parameters are correspondingly listed in the titles of the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1) n=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='4 (a2) n=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='7 10 10 T 0 0 10 10 100 105 100 Time Time(a3) m=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, A=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 10 0 10 100 105 Time10 10 T 0 X 10 10 100 105 100 Time Time (c1)A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, n=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0,T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 (c2) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, n=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 10 10 TL 0 X 10 10 100 105 100 Time Time10 1 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, H=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 0 X 10 100 105 Time (c3) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, n=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 10 0 10 100 105 Time10 (a1) Polarization, SD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 GC P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (a2) Polarization, SD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (a3) Polarization, SD =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (b1) Polarization, SD A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (b2) Polarization, SD A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (b3) Polarization, SD A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 (c1) Polarization, SD =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (c2) Polarization, SD =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (c3) Polarization, SD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phase diagram for the population with heterogeneous tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In more detail, (a1)-(a3) phase diagrams in (A, α) space for three different values of ξ, (b1)-(b3) phase diagrams in (α, ξ) space for three different values of A, (c1)-(c3) phase diagrams (A, ξ) for three different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of parameters are listed in corresponding subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We run simulations with SD in all subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of either pink triangles or yellow circles depict the boundaries separating the regions of GC and P state, as given by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The regions of different phases are correspondingly labeled in the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 11 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The results from static networks We subsequently embed our model into the social networks of size N = 43952, where individuals interact through fixed connections, so as to explore the joint effects of the network topology and influence mechanism on the formation of HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The phase diagrams are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 8 in three differ- ent parameter spaces: (A, T), (A, η) and (T, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Our results more or less remain robust with respect to the engagement of fixed interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is evident that the phase layouts are largely dominated by model rules rather than fixed interac- tions, and thus similar to those illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9, we see that the behaviors of SD, S and χ(S) are qualitatively similar to the case of time-varying network (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Whereas it is important to note that the introduction of fixed interactions can remarkably give rise to much sharper dependence of SD, S and χ(S) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S5), and much larger regions of P phase where extreme polarization (SD = 10) co- exists with high-level polarization with large SD (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 8 for the light red regions in P phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Also notice that there are less triple points in comparison with the results reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(a1)-(a6) further illustrate hot maps of the density of users in the (x, ⟨x⟩NN) plane, for six values of A, where the results reveals the correlation between the opinion of an individual i, xi, and the average opinions of their nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This can measure an individual’s preference to connect to peers sharing similar opinions [21], which fosters information about the effects of cluster-level self-reinforced mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In our study, cluster-level self-reinforced mech- anism is defined that the like-minded members tend to clus- ter together and persistently support those on the boundaries to repel distant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Attributing to the symmetrical nature of our model between positive and negative range, the con- figurations of positive clusters i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' the clusters of positive opinions, are similar to those of negative clusters with re- spect to size and number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We thus calculate the statistics of positive clusters by means of statistical variables reported in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(b) and (c), which provide an overall picture of cluster- level self-reinforced mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This naturally gives rise to the question whether abundance or size of these spatial opinion clusters guarantee the presence of cluster-level self- reinforced mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' As we can see in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(b1)-(b6), plotting the distributions of positive clusters for six different values of A shows that there are always giant positive clusters indicated by isolated peaks, as well as power-law cluster-size distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' which is independent of the values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Hence the observed phe- nomenon is not symbol of distributions transitions, instead, it is rooted in the model rules and initial assignment of individ- uals’ opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (i) In the first parameter regime of HD state i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=', Ac1 < A < A ′ c, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(a3) shows bright spindle-shaped diagonal speck- les that stretch the first and third quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This identifies a high density of users hold moderate opinions of their own as clusters of size 1, while most individuals are within the largest clusters and unbiased, as we can see the first three markers in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(c2)-(c4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (ii) In the range A ′ c < A < Ac2, increasing interaction strength makes more individuals split off from the largest clus- ter to form more clusters of size 1 (see corresponding markers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(c2)) and more smaller modest clusters (see corre- sponding markers in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(c3)-(c6)) which are not aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' There are longer boundaries between positive and negative opinions, and this results in more coherent intra-group and intenser group-level struggles among clusters of the opposite sides i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=', stronger cluster-level self-reinforced mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We can see that the bright speckles become long streaks, and several short streaks start to appear with the same direction in the second and forth quadrants (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(a4) and (a5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (iii) When the population gets polarized for A > Ac2, in addition to persistent modest clusters (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(c4)) and the shrinking largest clusters of opposing camps (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(c2)), smallest clusters of size 1 are more abundant (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(c3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' They guarantee the strongest self-reinforced mechanism, leading to longer speckles through the origin of coordinates (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(a5) and (a6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' With stronger sup- port from like-mind group members, those streaks starts to stretch out to the first and third quadrants (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9(a5) and (a6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Whatever, unbiased individuals themselves have a stronger preference to cluster together within the largest clus- ters and therefore the patterns of the longest speckles are al- ways brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This is definitely different from those previ- ously uncovered echo chambers determined by large opposed clusters [21], but in agreement with the empirical findings that unbiased individuals themselves can form clusters [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The above results firstly indicate that the abundance of clus- ters of size 1 and modest clusters may actually be more im- portant than their sizes in determining the effects of cluster- level self-reinforced mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Secondly, one can avoid ex- treme polarization for increasing T and α, but intermediate A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Under such parameter condition, people who are reluc- tant to hearing different opinions can create many different relatively modest ideological islands that people will be stuck on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' This may leads to low exposure which may prevent the emergence of two opposing extreme opinions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Conse- quently, extreme polarization is replaced by high-level polar- ization (see the light red regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' On the other hand, like-minded group members persistently instead support those on the boundaries to repel distant ones, although the innermost members are relatively unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Polarized views can more easily arise from the interaction between individuals at the boundaries of clusters, making polarization more likely (see larger red regions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9), which is associated with lower likelihood of a triple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The same is the case for heterogeneous susceptibility, still the cluster-level self-reinforced mechanism is positively re- lated to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We can thus observe slight shrinkage of the regions of GC and HD phase when A <∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In contrast, regions of P phase instead compress the other two phases with increas- ing A (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We can confirm that the mechanism still works, and to achieve a HD state is still impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phase diagram for the population embed on part of Facebook network, where interactions among individuals are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1)-(a3) Phase diagrams in (A, α) space for four different values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b4) Phase diagram in (A, T) space for four different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c1)-(c4) Phase diagram in (α, T) space for four different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We run simulations with SD in all subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of different markers denote the boundaries between different phases, which are the same as those presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The regions of the three phases are correspondingly labeled in the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The light blue pentagrams indicate the triple points which are less frequent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The dots presented in (a3),(b2) and (c1) correspond to the subplots (a1)-(a3), (b1)-(b3) and (c1)-(c3) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1) Polarization, SD (a2) Polarization, SD (a3) Polariz 10 10 10 10 10 T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 T=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 ★HD Susceptibility,α Susceptibility,α Susceptibility ,α HD 8 8 8 8 8 ６ ６ ６ 6 6 P 4 4 4 4 4 2 ¥2 2 2 2 GC GC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 0 0 0 0 0 4 6 8 ¥10 0 2 4 6 8 10 0 2 2 4 Interaction strength, A Interaction strength, A Interaction G h3)Polazation, SD (a4) Polarization, SD 10 10 10 T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 T=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 Susceptibility,α 8 8 HD 8 ６ 6 6 P P 4 4 4 GC 2 2 2 0 0 0 6 810 0 2 6810 strength, A Interaction strength, A sn10 10 10 10 10 Q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD 8 8 8 HD 8 8 HD Tolerance, ¥６ ６ 6 6 P P ４ 4 4 4 GC ¥2 2 2 2 2 GC Q=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 0 0 0 0 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 Interaction strength, A Interaction strength, A Interaction (c1) Polarization, SD (c2) Polarization, SD (c3) Polariz 10 10 10 10 10 A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC GC HD 8 8 8 8 8 T T T Tolerance, Tolerance, Tolerance, HD ９ 6 6 6 6 HD 4 4 4 P 4 4 2 2 2 2 A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 P 0 0 0 0 0 0 2 4 8 10 0 2 4 6 8 10 0 2 4 Susceptibility,α Susceptibility,α Suscept10 10 10 α=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD Q=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 8 8 8 T Tolerance, ¥6 6 6 P P 4 4 4 ¥2 2 2 0 0 0 6 ¥810 0 2 468 ¥10 strength, A Interaction strength, A zation, SD (c4) Polarization, SD 10 10 10 ８６ 8 HD 8 Tolerance, 6 6 P P 4 4 ¥2 2 2 A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 0 0 0 6 8 10 0 2 4 6 8 10 ibility,α Susceptibility,α13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1)-(a6) Hot maps for the average opinion of the nearest neighbors ⟨x⟩NN against an individual’s opinion x, for Nr = 500 simulations of independent realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b6) The distributions of positive clusters for six different values of A, where positive clusters refer to those consisting of individuals owning positive opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c1) The size of the population with positive opinions NP against A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c2) The size of largest positive cluster SP GC against A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c3) The number of positive clusters nP C against A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c4) The number of modest positive clusters nMP C against A, which exclude positive clusters of size 1 and the largest positive clusters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c5) The mean size of positive clusters SP against A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c6) The mean size of the modest positive clusters SMP C against A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The six markers presented in (c1)-(c6) rightly correspond the six values of A given in (a1)-(a6) and (b1)-(b6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of other parameters are T = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 and α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a2) A,=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='24 (a3) A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='43 (a1) A=1 (a4) A=2 10 10 C1 10 10 ５ 5 5 5 NN 0 0 0 5 5 5 5 10 10 10 10 10 0 10 10 0 10 10 0 10 10 0 1 X X X X(a5) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='41 (a6) A=3 10 C2 10 4 5 5 3 0 0 2 5 5 1 10 10 0 0 10 0 10 10 0 10 X Xb2 (b1) A=1 (b4) A=2 10°℃ C1 C 10°g 0 口 S d 5 8 0 0 0 100 105 100 105 100 100 1 S s S S (c1) (c2) (c3) (c4) X10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 ,X104 2斤 6000 1000 s GN 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='4 PGC PC P 5000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 4000 500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='6 3000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 AA V MPC 2000 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='4 0 2 3 2 1 3 2 3 2 1 A A A A(b5) Ac2=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='41 (b6) A=3 c2 109 10° 0 0 100 105 100 105 s s (c5) (c6) 12 31 × SMPC 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='8 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='6 6 XX!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='4 3 1 2 2 3 3 A A14 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' DISCUSSIONS AND CONCLUSIONS In this paper, we have proposed a simple model based on one core assumption that individuals tend to amplify differ- ence from others with dissimilar opinions according to nega- tive influence, and to attract toward others with similar opin- ions due to positive influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We correspondingly consider three key realistic ingredients to regulate the opinion dynam- ics: interaction strength, individuals’ susceptibility to opinion dissimilarity between it and its neighbor, and individuals’ tol- erance to different views determines whether the result of an interaction is attractive or repulsive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' For time-varying networks, within our model HD state emerges for sufficient susceptibility, intermediate interaction strength and high tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Whereas strong interaction or low tolerance makes this desired state not accessible, and the population instead gets polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Remarkably, the sim- ple model rules successfully generate the three phases, along with three different transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Especially, as another novel finding, there exists a triple point in the planes of (A, T) or (α, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' To the best of our knowledge, these two important and interesting phenomena have never been uncovered and men- tioned in previous studies, especially those consider influence mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In the second part of the paper, we have extended our study to the heterogeneous cases where individuals’ susceptibility or tolerance follows a power-law distribution whose exponent are η and ξ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Through a comparison with the re- sults in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' III A, we find that, although the effect of sus- ceptibility heterogeneity is less pronounced, it can facilitate HD to a certain extent, and instead prevent the population from achieving a global neutral consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Unfortunately, heterogeneous tolerance can largely facilitate the emergence of P state such that HD is not achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' To achieve a global neutral consensus is nearly impossible unless both A and α are rather small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Whatever, interaction strength A largely de- termines the population’s outcome in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Therefore, it is evident that the effects of heterogeneous individuals’ in- trinsic attributes such as susceptibility or tolerance cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In the last case, we have applied our model on empirical static networks where the interactions among individuals are fixed throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Although the final overall phenomenology is similar to the time-varying case with respect to phase lay- outs, the population has larger regions of P phase where ex- treme polarization coexist with high-level polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We have uncovered that cluster-level self-reinforced mechanism is responsible for these phenomena, which is dependent on whether the spatial opinion clusters of opinions are abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In presence of such mechanism, the struggles among clusters formed by like-minded individuals protects inner ones from the influence of the majority of the population, allowing them to persistently support the group members on the boundaries in face of distant individuals of other clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It is also the cause of the other two results: the coexistence of extreme po- larization and high-level polarization, and the triple point is less likely to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, we have confirmed that, for heterogeneous susceptibility or tolerance, this mechanism still works, and to achieve a HD state thus becomes harder or even impossible in presence of fixed interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' For the first time, this paper proposes a basic theory frame- work to understand the formation of HD, and we can con- clude the four following remarks from this paper: (i) Most importantly, through simplest possible setting, our proposed model can generally generate three phases: GC, HD and P, along with a triple point, regardless of that the interactions are time-varying or fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Which has never been stressed or mentioned by previous opinion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (ii) We correspond- ingly propose an effective method to numerically identify the boundaries between these phases though calculating the sus- ceptibility of opinion entropy, and the simulations validate this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (iii) Heterogeneous attributes such as susceptibility turns out to be a facilitating factor for achieving a HD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, heterogeneous tolerance which makes things worse, should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (iv) Fixed interactions create a negative impact on the emergence of HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' They can generate cluster- level self-reinforced mechanism through abundant clusters of size 1 and modest opinion clusters, which can unexpectedly promote polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' However, it still remains a challenge to theoretically locate the position of the triple points, and uncover the nature of the transitions on both side of the triple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Moreover, it is worth noticing that our model is based on a minimal number of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' It does not take into account some empiri- cal features of networks or individuals which might generate different scenarios, such as heterogeneous duration time of in- teractions, and or different social positions of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In light of this fact, it is essential to further extend our study in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Also, this study opens one interesting issues for fu- ture research: whether HD state becomes more easily acces- sible with the involvement of some optimization strategies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' And if the answer is yes, how much it will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Key Program of the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 71731002), and by Guangdong Basic and Applied Basic Research Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 2021A1515011975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' thanks Kai Qi for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' APPENDIX Appendix A 15 (a1) Entropy, S =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (a2) Entropy, S =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (a3) Entropy, S =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (a4) Entropy, S =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Tolerance, T (b1) Entropy, S A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T (b2) Entropy, S A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T (b3) Entropy, S A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T (b4) Entropy, S A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC HD P 0 2 4 6 8 10 Susceptibility, 0 2 4 6 8 10 Tolerance, T 0 1 2 3 4 5 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1)-(a4) The dependence of opinion entropy S on A and T for four different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b4) The dependence S on α and T for four different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Different regions of the three states are correspondingly labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of different markers denote the boundaries between different phases, which are the same as those presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The light blue pentagrams indicate the triple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' APPENDIX B 16 (a1) Entropy, S =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 (a2) Entropy, S =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 (a3) Entropy, S =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 (b1) Entropy, S T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 (b2) Entropy, S T=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 (b3) Entropy, S T=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 (c1) Entropy, S A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 (c2) Entropy, S A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 (c3) Entropy, S A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1)-(a3) The dependence of S on A and T for three different values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b3) The dependence of S on A and η for three different values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c1)-(c3) The dependence of S on T and η for three different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Different regions of the three phases are correspondingly labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of different markers denote the boundaries between different phases, which are the same as those presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' There is only one triple point indicated by light blue pentagram in (a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 17 (a1) Entropy, S =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 GC P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 1 2 3 4 (a2) Entropy, S =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 1 2 3 4 (a3) Entropy, S =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC P 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 Susceptibility, 0 1 2 3 4 (b1) Entropy, S A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Susceptibility, 0 1 2 3 4 (b2) Entropy, S A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Susceptibility, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 (b3) Entropy, S A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Susceptibility, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 (c1) Entropy, S =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 1 2 3 4 (c2) Entropy, S =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 1 2 3 4 (c3) Entropy, S =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 1 2 3 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a1)-(a3) The dependence of S on A and α for three different values of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b3) The dependence of S on α and ξ for three different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c1)-(c3) The dependence of S on A and ξ for three different values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Different regions of the two states GC and P phase are correspondingly labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The evolution of the entropy of opinion distribution for three different values of susceptibility α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of other parameters such as interaction strength A and tolerance threshold exponent ξ are listed in the titles of subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (a) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='2, α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 10 10 5 5 0 0 X X 5 5(b1) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='2, α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5101 102 103 104 105 100 100 Time (c) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='2, α=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 10 10 5 5 0 0 X X 5 5 10 10 101 102 103 104 100 105 100 Time101 102 103 104 105 Time (b2) A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='2, α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 101 102 103 104 105 Time18 APPENDIX C [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Axelrod, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Confl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Resolut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 41, 203 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Mark, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Sociol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' , 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Guilbeault, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Becker, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Centola, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 115, 9714 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Allcott, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Boxell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Conway, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Gentzkow, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Thaler, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Public Econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 191, 104254 (2020).' metadata={'source': 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H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Noorazar, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Plus 135, 1 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Bramson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Grim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Singer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Berger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Sack, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Fisher, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Flocken, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Holman, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 84, 115 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Schimmelfennig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Leuffen, and B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Acemoglu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Ozdaglar, Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Games Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 1, 3 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Flache and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} 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Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' X 10, 041042 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 Interaction strength, A 0 2 4 6 8 10 Variables T=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 (a1) Ac1 Ac2 SD (S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content="5 Interaction strength, A 0 2 4 6 Variables (a2) Ac1 A' c Ac2 S (S) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 3 Interaction strength, A 0 2 4 6 8 10 Variables =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0, T=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 Ac1 Ac2 (b1) SD (S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content="5 3 Interaction strength, A 0 2 4 6 Variables Ac1A' c Ac2 (b2) S (S) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 2 Susceptibility, 0 2 4 6 8 10 Variables A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5, T=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 c1 c2 (c1) SD (S) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content="5 2 Susceptibility, 0 2 4 6 Variables c1 ' c c2 (c2) S (S) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 0 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='3 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='9 0 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The dependence of SD (red circles), S (blue squares), χ(S) (brown curves and blue cures in the insets) on A in part of Facebook network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Pink vertical line labels the position of Ac1 at which HD state begins to appear, while light blue vertical line indicates the position of Ac2 at which HD state vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In particular, A ′ c denotes an inflection point in S followed by a sharp growth of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The values of T and α are correspondingly listed in titles of (a1), (b1) and (c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 20 (a1) Polarization, SD =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='1 HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 8 10 (a2) Polarization, SD =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 8 10 (a3) Polarization, SD =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 HD P GC 0 2 4 6 8 10 Interaction strength, A 0 5 10 Tolerance, T 0 2 4 6 8 10 (b1) Polarization, SD T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (b2) Polarization, SD T=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (b3) Polarization, SD T=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='0 GC HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Interaction strength, A 0 2 4 6 8 10 (c1) Polarization, SD A=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 8 10 (c2) Polarization, SD A=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 HD P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 8 10 (c3) Polarization, SD A=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content='5 P 2 3 4 5 6 Heterogeneity, 0 2 4 6 8 10 Tolerance, T 0 2 4 6 8 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' Phase diagrams for the population embed in part of Facebook network, where interaction among individuals are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' In more detail, (a1)-(a3) phase diagrams in (A, T) space for three different values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (b1)-(b3) phase diagrams in (α, η) space for three different values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' (c1)-(c3) phase diagrams in (T, η) for three different values of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' We run simulations with polarization degree SD in all subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The lines consisting of different markers denote the boundaries between different phases, which are the same as those presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The regions belonging to the three phases are correspondingly labeled in the subplots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} +page_content=' The light blue pentagram presented in (a3) indicates the triple point in (A, T) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE1T4oBgHgl3EQfQgPj/content/2301.03042v1.pdf'} diff --git a/uNE3T4oBgHgl3EQf-AtL/content/tmp_files/2301.04821v1.pdf.txt b/uNE3T4oBgHgl3EQf-AtL/content/tmp_files/2301.04821v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa7fff6b96aedcbce588dad65da23b24a661b36f --- /dev/null +++ b/uNE3T4oBgHgl3EQf-AtL/content/tmp_files/2301.04821v1.pdf.txt @@ -0,0 +1,1977 @@ +arXiv:2301.04821v1 [cond-mat.stat-mech] 12 Jan 2023 +January 13, 2023 +On the Application of Non-Gaussian Noise in Stochastic Langevin Simulations +Niels Grønbech-Jensen∗ +Department of Mechanical & Aerospace Engineering +Department of Mathematics +University of California, Davis, CA 95616, U.S.A. +In light of recent advances in time step independent stochastic integrators for Langevin equations, +we revisit the considerations for using non-Gaussian distributions for the thermal noise term in +discrete-time thermostats. We find that the desirable time step invariance of the modern methods is +rooted in the Gaussian noise, and that deviations from this distribution will distort the Boltzmann +statistics arising from the fluctuation-dissipation balance of the integrators. We use the GJ methods +as the focus of our investigation since these methods are the ones that contain the most accurate +thermodynamic measures of existing methods. Within this set of methods we find that any distri- +bution of applied noise which satisfies the two first moments given by the fluctuation-dissipation +theorem, will result in correct, time step independent results that are generated by the first two +moments of the system coordinates. However, if non-Gaussian noise is applied, undesired deviations +in higher moments of the system coordinates will appear to the detriment of several important +thermodynamic measures that depend especially on the fourth moments. The deviations, induced +by non-Gaussian noise, become significant with the one time step attenuation, thereby inhibiting +the benefits of the methods. Thus, we conclude that the application of Gaussian noise is necessary +for reliable thermodynamic results when using modern stochastic thermostats with large time steps. +I. +INTRODUCTION +Over the past several decades, discrete-time Langevin +and Brownian simulations in computational statistical +mechanics have been a centerpiece in many specific sim- +ulation applications to molecular dynamics, e.g., materi- +als science, soft matter, and biomolecular modeling [1–5]. +Inherent to simulations is the fundamental complication +of time discretization, which on one hand is necessary for +advancing time, but on the other hand inevitably cor- +rupts the desired, continuous-time system features of in- +terest. An integral part of a simulation task is therefore +to explore and optimize the balance between the two con- +flicting core objectives; namely simulation efficiency by +increasing the time step, and simulation accuracy by de- +creasing it. +Since discrete-time algorithms for time evolution pro- +duce only approximate solutions to a continuous-time so- +lution, and since the errors must vanish in the limit of +infinitesimal time step, most considerations for a useful +simulation environment have been considered in the limit +of small time steps, where the simulated system variables +change only little in each step. In that limit, the discrete- +time system behaves much like the continuous-time sys- +tem, and algorithm development has therefore mostly +been conducted by formulating objectives and measures +in continuous-time whereafter time has been discretized +and executed with relatively small steps in order to retain +accuracy. +∗ ngjensen@math.ucdavis.edu +For stochastic differential equations, notably Langevin +equations, an added complication is that it is not a +specific trajectory which is of interest, but instead the +statistics of trajectories that determines if the desired +continuous-time behavior has been adequately repre- +sented. The traditional approach to these problems has +therefore been to acquire accurate statistics from accu- +rate trajectories obtained from small time steps, where +continuous-time behavior is well approximated. +How- +ever, with the recent advent of discrete-time stochastic +algorithms that can produce correct statistics from in- +accurate, large time step trajectories [6–8], the details +of, e.g., stochastic noise correlation and distribution now +become essential for obtaining the advantages of the sta- +tistical accuracy for all meaningful time steps. The sig- +nificance of these details arises from the discretization of +time, as outlined in the following. +We are here concerned with the Langevin equation [9, +10] +m˙v + α ˙r = f + β , +(1) +where m is the mass of an object with spatial (configu- +rational) coordinate r and velocity (kinetic) coordinate +v = ˙r. The object is subjected to a force f = −∇Ep(r), +with Ep(r) being a potential energy surface for r, and lin- +ear friction that is represented by the non-negative con- +stant α. The fluctuation-dissipation relationship specifies +that the thermal fluctuations β can be represented by a +distribution for which the following first two moments +are given [11]: +⟨β(t)⟩ = 0 +(2a) +⟨β(t)β(t′)⟩ = 2α kBT δ(t − t′) , +(2b) + +2 +where δ(t) is Dirac’s delta function signifying that the +fluctuations are temporally uncorrelated, kB is Boltz- +mann’s constant, and T is the thermodynamic temper- +ature of the heat-bath, to which the system is coupled +through the friction constant α. A feature in statistical +mechanics is that the fluctuation-dissipation theorem for +the noise distribution β(t) only specifies the two moments +given in Eq. (2) [10–12]. Thus, there is a great deal of +freedom in choosing the distribution of the noise in the +Langevin equation such that correct thermodynamics is +represented. The reason that the physical (statistical) re- +sults do not depend on this freedom is that continuous- +time demands a new, uncorrelated noise value β(t) be +contributed and accumulated at every instant, resulting +in a Gaussian outcome at any non-zero time-scale due to +the central limit theorem (see, e.g., Ref. [13]). As a result, +the sampled (one-dimensional) Maxwell-Boltzmann ve- +locity distribution ρk,e(v) and the configurational Boltz- +mann distribution ρc,e(r) always become [10] +ρk,e(v) = +� +m +2πkBT exp +� +− +1 +2mv2 +kBT +� +(3a) +ρc,e(r) = Cc exp +� +−Ep(r) +kBT +� +, +(3b) +with the constant Cc determined by the normalization +of probability +� +all space +ρc,e(r) dr = 1 , +(3c) +regardless of which temporally uncorrelated distribution +β(t) is at play in Eq. (1), and for as long as Eq. (2) is +satisfied. Similarly, it is given that diffusive transport for +a flat surface is represented by a diffusion constant [1] +DE = 1 +d lim +t→∞ +⟨[r(t) − r(0)]2⟩ +2t +(4) += kBT +α +, +(5) +where d is the dimension of the configurational space. +Given that the central limit theorem ensures a Gaus- +sian noise distribution for any appreciable integration of +the instantaneous variable β(t), it is natural to choose +β(t) to be the Gaussian that conforms to Eq. (2). How- +ever, in discrete-time simulations it is not uncommon +to directly employ a uniform distribution that satisfies +Eq. (2). +The rationale is rooted in the computational +efficiency of uniformly distributed pseudo-random gener- +ators [14], compared with the additional computational +load required for transforming those variables into Gaus- +sian distributed variables before use. A thorough inves- +tigation of using non-Gaussian valuables in discrete time +was presented in the 1988 publication Ref. [15], where it +was concluded that, given the inherent inaccuracies of the +stochastic discrete-time integrators of the time, the scal- +ing in statistical moments upon the applied time step did +not change adversely when using non-Gaussian noise dis- +tributions, specifically the uniform. Building upon this +work, Ref. [16] extended the investigation in 1991 to a +broader class of noise distributions, which would yield +indistinguishable statistical results compared to the use +of Gaussian noise for vanishing time steps, where the +underlying systemic time step errors of the then con- +temporary methods would vanish as well. The studies +highlighted the benefits of computational efficiencies in +using simple, non-Gaussian noise given that the statisti- +cal differences were deemed to be insignificant for small +time steps, which in any case were necessary for the ac- +curacy of the stochastic integrators, even using Gaus- +sian noise. As pointed out in a recent article [17] that +compares the quality of contemporary stochastic integra- +tors, this practice has been widespread when using tra- +ditional algorithms, such as the Br¨unger-Brooks-Karplus +[18] (also analyzed in Ref. [19]) or the Schneider-Stoll [20] +method, for as long as small time steps are applied (for +a more expansive list of traditional and recent Langevin +integrators, please see Ref. [8]). For example, these two +methods are used with non-Gaussian noise in, e.g., the +wide-distribution molecular dynamics simulation suites, +LAMMPS [21–23] and ESPResSo [24]. +It is the aim of this work to revisit the discussion of +applying non-Gaussian noise in light of the most mod- +ern stochastic integrators, which have been designed to +produce accurate, time step independent statistics and +transport through the use of Gaussian noise. We demon- +strate that even if the low order moments of the config- +urational and kinetic distributions are seemingly unaf- +fected by the details of the noise distribution, there may +still be significant differences in the sampled phase space +that affect the statistical sampling of the system. +We +base the analysis exclusively on the recently formulated +GJ set of methods [8], since these methods have the most +optimized statistical properties that are invariant to the +time step for linear systems when using Gaussian noise. +This also allows us to investigate the large time step im- +plications of using non-Gaussian noise without being sub- +ject to other inherent time step errors of the integrators. +The presentation is organized as follows. Section II re- +views the basic features and structure of the GJ methods, +Sec. III analyzes analytically the first four statistical mo- +ments of the configurational and kinetic coordinates for +linear systems, Sec. IV presents companion simulations +to the analysis conducted in Sec. III, and Sec. V summa- +rizes the discussion with specific conclusions. + +3 +II. +REVIEW OF THE GJ METHODS +We adopt the complete set of GJ methods [8] for solv- +ing Eq. (1) in the form +un+ 1 +2 = √c1 vn + +√c3 ∆t +2m +f n + +√c3 +2m βn+1 +(6a) +rn+1 = rn + √c3 ∆t un+ 1 +2 +(6b) +vn+1 = +c2 +√c1 +un+ 1 +2 + +�c3 +c1 +∆t +2m f n+1 + +�c3 +c1 +1 +2m βn+1 , +(6c) +where a superscript n on the variables r, v, u, and f +pertains to the discrete time tn = t0 + n ∆t at which +the numerical approximation is given, with ∆t being the +discrete time step. Thus, f n = f(tn, rn). The two repre- +sented velocities are, respectively, the half-step velocity +un+ 1 +2 at time tn+ 1 +2 , and the on-site velocity vn at tn, +given by +un+ 1 +2 = rn+1 − rn +√c3 ∆t +(7a) +vn = rn+1 − (1 − c2)rn − c2rn−1 +2√c1c3 ∆t ++ 1 +4m(βn − βn+1) . +(7b) +The statistical definitions of what it means to be half- +step and on-site are given in Ref. [8]. +The functional +parameter c2 = c2( α∆t +m ) is the one-time-step attenuation, +which is a function of α∆t/m. The two other functional +parameters, c1 and c3, are given by +2c1 = 1 + c2 +(8a) +α∆t +m c3 = 1 − c2 . +(8b) +It is the choice of the function c2 that distinguishes +the GJ methods from each other (see Ref. [8] for sev- +eral choices of c2). The basic requirement for c2 is that +c2 → 1 − α∆t +m +for α∆t +m +→ 0, ensuring that also c1 → 1 +and c3 → 1 for +α∆t +m +→ 0 in accordance with the fric- +tionless Verlet method [25–29]. Finally, the discrete-time +stochastic noise variable is defined by +βn+1 = +� tn+1 +tn +β(t) dt , +(9) +such that the discrete-time fluctuation-dissipation theo- +rem follows from Eqs. (9) and (2) to become +⟨βn⟩ = 0 +(10a) +⟨βnβℓ⟩ = 2α∆t kBT δn,ℓ , +(10b) +where δn,ℓ is Kronecker’s delta function. +Eliminating the velocities from Eq. (6) yields the +purely configurational stochastic Verlet form +rn+1 = 2c1 rn − c2 rn−1 + c3∆t2 +m +f n + c3∆t +2m (βn + βn+1) , +(11) +which is the only stochastic Verlet form that 1) for a +harmonic potential Ep(r) = −κr (κ > 0) and Gaussian +fluctuations βn, will yield the correct Boltzmann distri- +bution ρc,e(rn) given by Eq. (3b), such that +ρc,e(rn) = +� +κ +2πkBT exp +� +− +1 +2κ(rn)2 +kBT +� +(12a) +⟨rnrn⟩ = kBT +κ +, +(12b) +for any time step within the stability range (Ω0∆t < +2 +� +c1/c3); 2) for a linear potential f = const, yield the +correct configurational drift velocity +vd = +�rn+1 − rn +∆t +� += f +α ; +(13) +and 3) for f = 0, yield the correct Einstein diffusion given +by Eq. (4), such that +DE = lim +n→∞ +� +(rn − r0)2� +2d n∆t += kBT +α +. +(14) +Similarly, it was found [8] that for a single Gaussian noise +value βn per time step, the harmonic potential Ep(r) = +−κr (κ ≥ 0) results in the unique half-step velocity un+ 1 +2 +in Eq. (7a) with correct velocity distribution function +ρk,e(un+ 1 +2 ) from Eq. (3a) such that +⟨un+ 1 +2 un+ 1 +2 ⟩ = kBT +m +. +(15) +In contrast, it was also found in Ref. [8] that is not possi- +ble to define an onsite velocity vn with the same time step +independent property. Given the statistical invariance of +these equations upon the time step when using Gaussian +noise, we will now investigate the statistical properties of +rn and un+ 1 +2 when using other distributions than Gaus- +sian for βn. +III. +MOMENT ANALYSIS FOR LINEAR +SYSTEMS +We will here investigate the relevant moments of the +configurational coordinate rn and half-step velocity un+ 1 +2 +as a function of the applied noise distribution ρ(β) of βn, +which satisfies Eq. (10). For this analysis we will mostly +assume a linear system with a harmonic potential +Ep(r) = 1 +2κ r2 +(16) +⇒ +f = −κr , +(17) +such that the linearized configurational stochastic GJ +methods in Eq. (11) can be written +rn+1 = 2c1X rn − c2 rn−1 + c3∆t +2m (βn + βn+1) . (18) +with +X = 1 − c3 +c1 +Ω2 +0∆t2 +2 +. +(19) + +4 +The natural frequency Ω0 of the oscillator is given by +Ω2 +0 = κ/m. +A. +First and Second Moments +For the first moments of the variables, it is obvious that +Eq. (10a) ensures both ⟨rn⟩ = 0 (from Eq. (18)) and +subsequently ⟨un+ 1 +2 ⟩ = 0 (from Eq. (7a)) regardless of +which distribution is chosen for βn. It similarly follows +that Eq. (10a) also ensures that the drift velocity in +Eq. (13) for f = const remains constant and correct +regardless of both time step and distribution βn. +For the second moments, ⟨rnrn⟩ and ⟨un+ 1 +2 un+ 1 +2 ⟩, we +first find ⟨rnrn⟩ as the solution to the system of equa- +tions given by the statistical averages of Eq. (18) mul- +tiplied by, respectively rn−1, rn, and rn+1. This linear +system of equations can be conveniently written + + +1 −2c1X +c2 +0 +2c1 +−2c1X +c2 −2c1X +1 + + + + +⟨rn−1rn+1⟩ +⟨rnrn+1⟩ +⟨rnrn⟩ + + = + + +0 +1 +2c1X + 2 + + +�c3∆t +2m +�2 +⟨βnβn⟩ . +(20) +We immediately observe that this equation is unaffected +by the details of the distribution of βn for as long as +Eq. (10) is satisfied. +Thus, also the second moment, +⟨rnrn⟩, is unaffected by the chosen distribution of βn, +regardless of the time step ∆t. The corresponding sec- +ond moment of the half-step velocity in Eq. (7a) is then +given by +⟨un+ 1 +2 un+ 1 +2 ⟩ = +2 +c3 ∆t2 (⟨rnrn⟩ − ⟨rnrn+1⟩) , +(21) +where both correlations on the right hand side are so- +lutions found from Eq. (20), which is unchanged for as +long as Eq. (10b) is satisfied. Thus, the second moment +of the half-step velocity is also unaffected by the chosen +distribution of βn, regardless of the time step ∆t. The +averages of potential (Eq. (16)) and kinetic energies, +⟨Ep(rn)⟩ = κ +2 ⟨rnrn⟩ = 1 +2kBT +(22) +� +Ek(un+ 1 +2 ) +� += 1 +2 +m +c3 ∆t2 (⟨rnrn⟩ − ⟨rnrn+1⟩) += 1 +2kBT , +(23) +are therefore unaffected for any applied stochastic vari- +able βn satisfying Eq. (10). +Since the second moments of the coordinates rn and +un+ 1 +2 of the GJ methods are invariant to the specific +choice of noise distribution that satisfies Eq. (10), we +conclude that the GJ methods display time-step invari- +ance in both potential and kinetic energy, and therefore +temperature, regardless of the applied βn distribution. +The invariance also extends to the above-mentioned dif- +fusion Eqs. (4) and (14), since this quantity is rooted in +only second moments of the noise variable. +Numerical validation of the time step independence of +the second moments is shown in Figs. 2ab and 6ab for a +couple of non-Gaussian noise variables, and simulations +are discussed in Sec. IV. While the invariance of the +second moments against the choice of βn distribution is +appealing, as it gives the impression that temperature +is correctly achieved, it may not be sufficient to ensure +proper thermodynamics sampling as we will see in the +following. +B. +Third Moments +The third moments of rn and un+ 1 +2 are generally not +needed for the calculations of thermodynamic quantities, +especially not if the system potentials are symmetric. +However, they may be relevant to study in case asym- +metric fluctuations are applied to the system for some +reason. Following the same procedure that was used for +calculating the second moments, we form the statistical +averages of the product of Eq. (18) with the six unique +combinations of rℓ1rℓ2, where ℓi = n − 1, n, n + 1. The +result is the linear system +A3 + + + + + + + +⟨rn−1rn−1rn+1⟩ +⟨rn−1rnrn+1⟩ +⟨rn−1rn+1rn+1⟩ +⟨rnrnrn+1⟩ +⟨rnrn+1rn+1⟩ +⟨rnrnrn⟩ + + + + + + + += +�kBT +κ +� 3 +2 +R = + + + + + + + +⟨rn−1rn−1βn⟩ +⟨rn−1rnβn⟩ +⟨rn−1rn+1(βn + βn+1)⟩ +⟨rnrnβn⟩ +⟨rnrn+1(βn + βn+1)⟩ +⟨rn+1rn+1(βn + βn+1)⟩ + + + + + + + +c3∆t +2m , +(24) + +5 +where +A3 = + + + + + + + +1 +0 +0 −2c1X +0 +c2 +0 +1 +0 +c2 +−2c1X +0 +c2 −2c1X +1 +0 +0 +0 +0 +0 +0 +1 +c2 +−2c1X +0 +c2 +0 −2c1X +1 +0 +0 +0 +c2 +0 +−2c1X +1 + + + + + + + +. +(25) +The right hand side R = {Ri} can after some calcula- +tions be expressed as +R1 = 0 +(26a) +R2 = 0 +(26b) +R3 = 0 +(26c) +R4 = (1 − c2)3 +�κ∆t +α +� 3 +2 +Γ3 +(26d) +R5 = (2c1X + 1)R4 +(26e) +R6 = (2c1X(2c1X + 2) + 2)R4 , +(26f) +where the fluctuations βn satisfy Eq. (10), and the skew- +ness is denoted by Γ3, +Γ3 = ⟨βnβnβn⟩ +⟨βnβn⟩ +3 +2 . +(27) +As is the case when using any symmetric distribution for +βn, all odd moments for the configurational coordinate +rn will be zero for a Gaussian distribution of βn applied +to a symmetric potential, Ep(r). +The third moment of the velocity un+ 1 +2 is given by +� +un+ 1 +2 un+ 1 +2 un+ 1 +2 +� += +1 +� +∆t √c3 +�3 +� +(rn+1 − rn)3� += +3 +� +∆t √c3 +�3 +�� +rnrnrn+1� +− +� +rnrn+1rn+1�� +, +(28) +where the two correlations on the right hand side are part +of the solution to Eq. (24). +For any given choice of GJ method (c2) the above +system of equations can be easily solved numerically for +given system parameters and skewness Γ3. +Examples +of this will be discussed in Sec. IV, and are shown +in Fig. 6cd along with the results of corresponding +numerical solution of Eqs. (6) and (17) for a particular +asymmetric noise distribution. Closed form expressions +are obtained in the following for a couple of special cases. +Special Case, X = 0: The matrix A3 and the vector R +simplify considerably with easy accessible solutions +⟨rnrnrn⟩ = +2 +1 + c3 +2 +�kBT +κ +� 3 +2 +R4 +(29a) +⟨rnrnrn+1⟩ = 1 − c2 +1 + c3 +2 +�kBT +κ +� 3 +2 +R4 +(29b) +⟨rnrn+1rn+1⟩ = 1 + c2 +2 +1 + c3 +2 +�kBT +κ +� 3 +2 +R4 , +(29c) +where we have used that this special case for +Ω2 +0∆t2 = 2 c3 +c1 offers R6 = 2R5 = 2R4, and where +R4 = +�1 − c2 +2 +2 +� 3 +2 +Γ3 . +(30) +Equations (28), (29b), and (29c) give +�� +un+ 1 +2 +�3� += − +3 +� +∆t√c3 +�3 +c2 +c2 +2 − c2 + 1 +�kBT +κ +� 3 +2 +R4 += − 3 +2 +√ +2 +�1 − c2 +2 +1 + c2 +� 3 +2 +c2 +c2 +2 − c2 + 1 +�kBT +m +� 3 +2 +Γ3 . +(31) +Examples of the use of these expressions are shown +in Fig. 6cd as open markers (◦) for a particular +asymmetric noise distribution. +Special Case, c2 = 0: In this case we have that c1 = 1 +2, +c3 = m/α∆t, and X = 1 − ∆tκ/α. The simplified +matrix A3 and vector R yield +⟨rnrnrn⟩ = 3X2 + 3X + 2 +1 − X3 +�kBT +κ +� 3 +2 +R4 +(32a) +⟨rnrnrn+1⟩ = 2X3 + 3X2 + 2X + 1 +1 − X3 +�kBT +κ +� 3 +2 +R4 +(32b) +⟨rnrn+1rn+1⟩ = X4 + 2X3 + 2X2 + 2X + 1 +1 − X3 +�kBT +κ +� 3 +2 +R4 , +(32c) +where +R4 = +�κ∆t +2α +� 3 +2 +Γ3 . +(33) + +6 +Equations (28), (32b), and (32c) give +�� +un+ 1 +2 +�3� += 3X2 +2 +√ +2 +1 − X2 +1 − X3 +�kBT +m +� 3 +2 +Γ3 . +(34) +Examples of the use of these expressions are shown +in Fig. 6cd as closed markers (•) for a particular +asymmetric noise distribution. +C. +Fourth Moments +The fourth moments of rn and un+ 1 +2 are needed for the +calculations of energy and pressure fluctuations, which +are the basis of several important thermodynamic quanti- +ties, such as heat capacity, thermal expansion, compress- +ibility, and the thermal pressure coefficient [1]. Fourth +moments are therefore critical for proper thermodynamic +characterization of a system. Following the same proce- +dure that was used for calculating the second and third +moments, we form the statistical averages of the product +of Eq. (18) with the ten unique combinations of rℓ1rℓ2rℓ3, +where ℓi = n− 1, n, n+ 1. The result is the linear system +A4 + + + + + + + + + + + + + + + +⟨rn−1rn−1rn−1rn+1⟩ +⟨rn−1rn−1rnrn+1⟩ +⟨rn−1rn−1rn+1rn+1⟩ +⟨rn−1rnrnrn+1⟩ +⟨rn−1rnrn+1rn+1⟩ +⟨rn−1rn+1rn+1rn+1⟩ +⟨rnrnrnrn+1⟩ +⟨rnrnrn+1rn+1⟩ +⟨rnrn+1rn+1rn+1⟩ +⟨rnrnrnrn⟩ + + + + + + + + + + + + + + + += +�kBT +κ +�2 +R = + + + + + + + + + + + + + + + +0 +⟨rn−1rn−1rnβn⟩ +⟨rn−1rn−1rn+1(βn + βn+1)⟩ +⟨rn−1rnrnβn⟩ +⟨rn−1rnrn+1(βn + βn+1)⟩ +⟨rn−1rn+1rn+1(βn + βn+1)⟩ +⟨rnrnrnβn⟩ +⟨rnrnrn+1(βn + βn+1)⟩ +⟨rnrn+1rn+1(βn + βn+1)⟩ +⟨rn+1rn+1rn+1(βn + βn+1)⟩ + + + + + + + + + + + + + + + +c3∆t +2m , +(35) +where +A4 = + + + + + + + + + + + + + + + +1 +0 +0 +0 +0 +0 −2c1X +0 +0 +c2 +0 +1 +0 +0 +0 +0 +c2 +−2c1X +0 +0 +c2 −2c1X +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +c2 +−2c1X +0 +0 +c2 +0 −2c1X +1 +0 +0 +0 +0 +0 +0 +0 +c2 +0 +−2c1X +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +c2 +−2c1X +0 +0 +0 +c2 +0 +0 −2c1X +1 +0 +0 +0 +0 +0 +0 +c2 +0 +0 +−2c1X +1 +0 +0 +0 +0 +0 +0 +c2 +0 +0 +−2c1X +1 + + + + + + + + + + + + + + + +. +(36) +The right hand side R = {Ri} can after some cumbersome work be expressed as +R1 = 0 +(37a) +R2 = 1 +2(1 − c2 +2)(1 − X) +(37b) +R3 = [2 + 2c1X]R2 +(37c) +R4 = [2c1X + 1 − c2]R2 +(37d) +R5 = [2c1X(2c1X + 2 − c2) + 1 − 2c2]R2 +(37e) +R6 = [(2c1X)3 + (2c1X)2(3 − c2) + 4c1X(1 − 2c2) − 4c2]R2 +(37f) +R7 = [3 + (Γ4 − 3)R2]R2 +(37g) +R8 = [4c1X + 3 − c2 + (2c1X + 1)(Γ4 − 3)R2]R2 +(37h) +R9 = [(2c1X)2 + 2c1X(3 − c2) + 3 − 2c2 + (1 + 2c1X)2(Γ4 − 3)R2]R2 +(37i) +R10 = [3(2c1X + 2) + (1 + (1 + 2c1X)3)(Γ4 − 3)R2]R2 , +(37j) +where we have used Eq. (10b). Notice that R7-R10 in +Eqs. (37g)-(37j) depend on the kurtosis Γ4 +Γ4 = ⟨βnβnβnβn⟩ +⟨βnβn⟩2 +, +(38) +which is not determined by the fluctuation-dissipation + +7 +relationship in Eq. (10). +While Eqs. (35)-(37) are not immediately informative, +they provide a straightforward path to calculating the +sought-after fluctuation σp in potential energy of Eq. (16) +σp = +�� +E2p(rn) +� +− ⟨Ep(rn)⟩2 += κ +2 +� +⟨rnrnrnrn⟩ − ⟨rnrn⟩2 . +(39) +Equations (35)-(37) also lead directly to the fluctuations +σk in kinetic energy of the velocity from Eq. (7a) +σk = +�� +E2 +k(un+ 1 +2 ) +� +− +� +Ek(un+ 1 +2 ) +�2 += m +2 +� +⟨un+ 1 +2 un+ 1 +2 un+ 1 +2 un+ 1 +2 ⟩ − ⟨un+ 1 +2 un+ 1 +2 ⟩2 , +(40) +where ⟨un+ 1 +2 un+ 1 +2 ⟩ is given by Eq. (15), and +⟨un+ 1 +2 un+ 1 +2 un+ 1 +2 un+ 1 +2 ⟩ = +1 +c2 +3∆t4 ⟨(rn+1 − rn)4⟩ += +2 +c2 +3∆t4 (⟨rnrnrnrn⟩ − 2⟨rnrnrnrn+1⟩ ++ 3⟨rnrnrn+1rn+1⟩ − 2⟨rnrn+1rn+1rn+1⟩) . +(41) +All the correlations on the right hand side are given from +the solution to Eqs. (35)-(37). +For any given choice of GJ method (c2) the above sys- +tem of equations can be easily solved numerically for +given system parameters and kurtosis Γ4. +Examples +of this will be discussed in Sec. IV, and are shown in +Figs. 2cd and 6ef along with the results of correspond- +ing numerical solution of Eqs. (6) and (17) for particular +noise distributions. Closed form expressions can be given +in the special cases of X = 0 and c2 = 0. +Special Case, X = 0: This particular time step is +Ω0∆t = +√ +2 +�c1 +c3 +. +(42) +Cumbersome, yet straightforward, algebra leads +to the closed expressions for ⟨rnrnrnrn⟩ and +⟨un+ 1 +2 un+ 1 +2 un+ 1 +2 un+ 1 +2 ⟩ for X = 0; i.e., for Ω0∆t = +√ +2 +� +c1/c3. The corresponding fluctuations in po- +tential and kinetic energies can be written +σp = κ +2 +� +⟨rnrnrnrn⟩ − ⟨rnrn⟩2 += kBT +√ +2 +� +1 +4 +1 + 7c2 +2 +1 + c2 +2 ++ 1 +4Γ4 +1 − c2 +2 +1 + c2 +2 +(43) +σk = m +2 +� +⟨un+ 1 +2 un+ 1 +2 un+ 1 +2 un+ 1 +2 ⟩ − ⟨un+ 1 +2 un+ 1 +2 ⟩2 += kBT +√ +2 +� +1 +2 +c3 +2 +1 + c2 +2 +[3 − Γ4] + 1 +4 [1 + Γ4] . +(44) +Notice that if βn is a Gaussian distributed vari- +able, where the kurtosis is Γ4 = 3, the above energy +fluctuations yield the expected values, σp = σk = +kBT/ +√ +2. +Examples of the use of these expressions are shown +in Figs. 2cd and 6ef as open markers (◦) for partic- +ular noise distributions. +Special Case, c2 = 0: This +special +case +(GJ-0, +see +Refs. [8]) corresponds to the first order difference +equation +rn+1 = rn + 1 +α +� +∆t f n + 1 +2(βn + βn+1) +� +, +(45) +which +can +approximate +the +solution +to +the +Langevin equation with no inertia; i.e., Brown- +ian motion. +The linearized, Hooke’s law system +(f = −κr) reads +rn+1 = Xrn + 1 +2α(βn + βn+1) , +(46) +with X = 1 − κ∆t/α, in correspondence with +Eqs. (18) and (19). This simplicity allows the last +four of the ten equations in Eq. (35) to decouple +from the rest, and the fourth moments can be cal- +culated to yield the average and fluctuations in po- +tential energy +⟨Ep(rn)⟩ = κ +2 ⟨rnrn⟩ = 1 +2kBT +(47) +σp = κ +2 +� +⟨rnrnrnrn⟩ − ⟨rnrn⟩2 += kBT +√ +2 +� +1 +2X4 [3 − Γ4] + 1 +4X3 [7 − Γ4] + 1 +4(X2 + X + 1) [1 + Γ4] +X3 + X2 + X + 1 +. +(48) +We also here notice that if βn is a Gaussian dis- +tributed variable, where the kurtosis is Γ4 = 3, + +8 +the potential energy fluctuations yield the expected +value, σp = kBT/ +√ +2. +Similarly, we can find the fluctuations σk in kinetic +energy from the correlation +�� +un+ 1 +2 +�4� += (1 − X)2 4σ2 +p + (kBT )2 +m2 ++ 6 +�kBT +m +�2 +X2(1 − X) − +�kBT +m +�2 +(3 + Γ4)X +� +1 − 3 +2X + X2 +� +, (49) +FIG. 1. The tree applied distributions, ρG(βn) (Gaussian), +ρU(βn) (Uniform), and ρP (βn) (Peaked), given in Eqs. (51), +(52), and (53), respectively. All three distributions satisfy the +discrete-time fluctuation-dissipation relationship, Eq. (10). +such that +σk = 1 +2m +��� +un+ 1 +2 +�4� +− +�kBT +m +�2 +, +(50) +where we have used Eq. (15). Thus, σk can be easily +computed also in this special case. We again notice +that for Gaussian fluctuations, where Γ4 = 3, the +kinetic energy fluctuations yield the expected value +σk = kBT/ +√ +2. +Examples of the use of these expressions are shown +in Figs. 2cd and 6ef as closed markers (•) for par- +ticular noise distributions. +IV. +SIMULATIONS +While the above expressions provide both direct and +indirect means for evaluating the thermodynamically im- +portant first four moments of the configurational and ki- +netic variables, we test these results against numerical +simulations such that both simulations and analysis can +be mutually verified. Further, the simulations provide +FIG. 2. Simulations and analyses of average potential and ki- +netic energies, ⟨Ep⟩ from Eq. (22) (a) and ⟨Ek⟩ from Eq. (23) +(b), and their respective fluctuations, σp from Eq. (39) (c) and +σk from Eq. (40) (d), as a function of reduced time step for the +harmonic system, Eqs. (6) and (17), with non-zero tempera- +ture T . Results are shown for Gaussian βn +G (Eq. (51a)) and +uniform βn +U (Eq. (52a)) noise distributions. Several friction +parameters α are used, as indicated in the figure. Simulation +results are indistinguishable from the analytical ones. Mark- +ers indicate special cases for c2 = 0 (• from Eqs. (48)-(50)) +and X = 0 (◦ from Eqs. (43) and (44)). +additional details on the actual distributions ρc(rn) and +ρk(un+ 1 +2 ), which in some cases reveal significant devia- +tions from Boltzmann statistics when the fluctuations βn +are non-Gaussian. +We show results of three different distributions for +the fluctuations βn, +all satisfying the discrete-time +fluctuation-dissipation balance that constrains the first +two moments given in Eq. (10), and thus, all resulting in +correct first and second moments of rn and un+ 1 +2 , lead- +ing to correct temperature as measured by the kinetic +and potential energies (see Sec. III A): + +9 +FIG. 3. Configurational and kinetic distributions ρc(rn) and +ρk(un+ 1 +2 ) for simulations with Gaussian and uniform noise ap- +plied to the linear system Eqs. (6) and (17) with α = 1mΩ0, +Ω0∆t = 0.5, kBT = E0. Shown curves are Eq. (3) (solid, +labeled exact), simulations data using Gaussian noise βn +G +(dashed, labeled Gaussian), and simulation data using uni- +form noise βn +U (dash-dot, labeled uniform). Distributions from +simulations using Gaussian noise are indistinguishable from +the exact references Eq. (3). (c) and (d) show the distribu- +tions, (a) and (b) show the deviations from the exact distri- +butions Eq. (3), and (e) and (f) show the effective potential +(PMF) and kinetic (MB) potentials (Eq. (56)) derived from +the distributions. +Gaussian,: βn = βn +G with distribution ρG(β): +ρG(β) = +1 +� +2π⟨βnβn⟩ +exp +� +− +ββ +2⟨βnβn⟩ +� +, +(51a) +with skewness and kurtosis +Γ3 = 0 +(51b) +Γ4 = 3 . +(51c) +Uniform,: βn = βn +U with distribution ρU(β): +ρU(β) = +1 +2 +� +3⟨βnβn⟩ +× +� +1 , − +√ +3 ≤ +β +√ +⟨βnβn⟩ < +√ +3 +0 , +otherwise +, +(52a) +with skewness and kurtosis +Γ3 = 0 +(52b) +Γ4 = 9 +5 . +(52c) +FIG. 4. Configurational and kinetic distributions ρc(rn) and +ρk(un+ 1 +2 ) for simulations with Gaussian and uniform noise ap- +plied to the linear system Eqs. (6) and (17) with α = 1mΩ0, +Ω0∆t = 1.5, kBT = E0. Shown curves are Eq. (3) (solid, +labeled exact), simulations data using Gaussian noise βn +G +(dashed, labeled Gaussian), and simulation data using uni- +form noise βn +U (dash-dot, labeled uniform). Distributions from +simulations using Gaussian noise are indistinguishable from +the exact references Eq. (3). (c) and (d) show the distribu- +tions, (a) and (b) show the deviations from the exact distri- +butions Eq. (3), and (e) and (f) show the effective potential +(PMF) and kinetic (MB) potentials (Eq. (56)) derived from +the distributions. +Peaked,: βn = βn +P with distribution ρP (β): +ρP (β) = 2 +3 δ +� +β +� +⟨βnβn⟩ ++ 1 +√ +2 +� ++ 1 +3 δ +� +β +� +⟨βnβn⟩ +− +√ +2 +� +, +(53a) +with skewness and kurtosis +Γ3 = +1 +√ +2 +(53b) +Γ4 = 12 +5 . +(53c) +The three distributions are sketched in Fig. 1, and +the realizations of the pseudo-random numbers βn are +produced from the uniformly distributed numbers in +[0, 1) generated by the RANMAR algorithm (used in +LAMMPS [30]). +In order to validate the analysis of the previous section, +we simulated the noisy harmonic oscillator Eqs. (6) and + +10 +FIG. 5. Configurational and kinetic distributions ρc(rn) and +ρk(un+ 1 +2 ) for simulations with Gaussian and uniform noise ap- +plied to the linear system Eqs. (6) and (17) with α = 1mΩ0, +Ω0∆t = 1.975, kBT = E0. Shown curves are Eq. (3) (solid, +labeled exact), simulations data using Gaussian noise βn +G +(dashed, labeled Gaussian), and simulation data using uni- +form noise βn +U (dash-dot, labeled uniform). Distributions from +simulations using Gaussian noise are indistinguishable from +the exact references Eq. (3). (c) and (d) show the distribu- +tions, (a) and (b) show the deviations from the exact distri- +butions Eq. (3), and (e) and (f) show the effective potential +(PMF) and kinetic (MB) potentials (Eq. (56)) derived from +the distributions. +(17) with several of the GJ methods, which all give the +same statistical results for Gaussian noise once the time +step is appropriately scaled according to the method. For +simplicity we limit the displayed results to those of the +GJ-I (GJF-2GJ) method [7, 8], for which +c2 = 1 − α∆t +2m +1 + α∆t +2m +(54a) +c1 = c3 = +1 +1 + α∆t +2m +. +(54b) +The GJF-2GJ method overlaps in the configurational co- +ordinate with the GJF method [6]. +We note that the +GJ-I (GJF-2GJ) method is available for use in LAMMPS +[30]. With time normalized by the characteristic unit t0 +given by the inverse of the natural frequency Ω0, the re- +duced damping is α/mΩ0, and the characteristic energy +is E0 = κr2 +0, where r0 is a chosen characteristic length to +which r is normalized. The simulations are conducted for +kBT = E0 for a variety of values of α/mΩ0 in the entire +stability range Ω0∆t < 2. Each statistical value is calcu- +lated from averages over 1,000 independent simulations, +each with 108 time steps. +FIG. 6. Simulations and analyses of average potential and ki- +netic energies, ⟨Ep⟩ from Eq. (22) (a) and ⟨Ek⟩ from Eq. (23) +(b), the respective third moments ⟨(rn)3⟩ from Eqs. (24)-(26) +(c) and ⟨(un+ 1 +2 )3⟩ from Eq. (28) (d), and the respective energy +fluctuations, σp from Eq. (39) (e) and σk from Eq. (40) (f), +as a function of reduced time step for the harmonic system, +Eqs. (6) and (17), with non-zero temperature T . Results are +shown for Gaussian βn +G (Eq. (51a)) and the asymmetrically +peaked βn +P (Eq. (53a)) noise distributions. +Several friction +parameters α are used, as indicated in the figure. Simulation +results are indistinguishable from the analytical ones. Mark- +ers indicate special cases for c2 = 0 (• from Eqs. (32a) and +(34), Eqs. (48)-(50)) and X = 0 (◦ from Eqs. (29a) and (31), +Eqs. (43) and (44)). +Figure 2 shows the results of using the uniform noise +distribution βn +U (Eq. (52a)) alongside the results of using +the Gaussian βn +G (Eq. (51a)) as reference. Figures 2ab, +respectively displaying results for configurational and ki- +netic variables, verify the general result from Sec. III A; +namely that a statistical property, which depends only on +the first and second moments of the noise distribution, +will be correctly evaluated for the GJ methods if the noise +satisfies Eq. (10). The shown data on each plot are for +damping parameters α/mΩ0=0.1, 0.25, 0.5, 1, 2.5, 5, and +the results for both Gaussian and uniform noise distribu- +tions are clearly in close agreement with the expected +values given in Eqs. (22) and (23). This has previously +been extensively validated for Gaussian noise [7, 8]. +Figures 2cd display simulation results and comparisons +to the expectations from the analysis in Sec. III C for the +configurational and kinetic energy fluctuations σp and +σk found in Eqs. (39) and (40). Since the Gaussian noise +βn +G from Eq. (51a) applied to the linear system produces +Gaussian distributions for rn and un+ 1 +2 with correct, and +time step independent, first and second moments (see +Sec. III A), it follows that the fourth moments are also + +11 +FIG. 7. Configurational and kinetic distributions ρc(rn) and +ρk(un+ 1 +2 ) for simulations with Gaussian and peaked noise ap- +plied to the linear system Eqs. (6) and (17) with α = 0.5mΩ0, +Ω0∆t = 0.5, kBT = E0. Shown curves are Eq. (3) (solid, +labeled exact), simulations data using Gaussian noise βn +G +(dashed, labeled Gaussian), and simulation data using peaked +noise βn +P (dash-dot, labeled uniform). Distributions from sim- +ulations using Gaussian noise are indistinguishable from the +exact references Eq. (3). (c) and (d) show the distributions, +(a) and (b) show the deviations from the exact distributions +Eq. (3), and (e) and (f) show the effective potential (PMF) +and kinetic (MB) potentials (Eq. (56)) derived from the dis- +tributions. +correct and time step independent. This is clearly ob- +served on the energy fluctuation figures, where both sim- +ulation results and the results of solving Eqs. (35)-(37) +for Γ4 = 3 are shown to be indistinguishable and con- +stant at the correct values σp = σk = kBT/ +√ +2. How- +ever, the uniformly distributed noise βn +U (Eq. (52a)) pro- +duces neither Gaussian nor uniform distributions for rn +and un+ 1 +2 , but instead produces distributions that de- +pend on the time step and the damping parameter, even +if Eq. (10) is satisfied. This can be seen in the figures +for the energy fluctuations, Figs. 2cd, where both simu- +lation results and the results of Eqs. (35)-(37) for Γ4 = 3 +and Γ4 = 9 +5, the latter being the kurtosis for the uniform +distribution, are shown. The simulation results are indis- +tinguishable from the analysis. It is obvious that when +using the uniform noise, the energy fluctuations, which +depend on the fourth moments of the variables, can de- +viate significantly from the correct value, given by the +Gaussian noise. +We also observe that the fluctuations +approach the correct values for small time steps or small +damping parameters. This is understandable, since the +damping per time step in those limits is very small, and +the fluctuations in rn and un+ 1 +2 therefore are composed +FIG. 8. Configurational and kinetic distributions ρc(rn) and +ρk(un+ 1 +2 ) for simulations with Gaussian and uniform noise ap- +plied to the linear system Eqs. (6) and (17) with α = 0.5mΩ0, +Ω0∆t = 1.5, kBT = E0. Shown curves are Eq. (3) (solid, +labeled exact), simulations data using Gaussian noise βn +G +(dashed, labeled Gaussian), and simulation data using peaked +noise βn +P (dash-dot, labeled uniform). Distributions from sim- +ulations using Gaussian noise are indistinguishable from the +exact references Eq. (3). (c) and (d) show the distributions, +(a) and (b) show the deviations from the exact distributions +Eq. (3), and (e) and (f) show the effective potential (PMF) +and kinetic (MB) potentials (Eq. (56)) derived from the dis- +tributions. +of βn +U contributions from many time steps, allowing the +integrated noise in rn and un+ 1 +2 to become near-Gaussian +by the central limit theorem. Conversely, if the damp- +ing per time step becomes appreciable, then the effective +noise in rn and un+ 1 +2 will be composed by only a few +βn +U contributions, which do not approximate a Gaussian +outcome very well. +In order to look at the details of the simulated distri- +butions, we have selected a few representative parame- +ter values as illustrations. Figures 3-5 show the simu- +lated configurational and kinetic distributions, ρc(r) and +ρk(u), for α = 1mΩ0 and Ω0∆t = 0.5, 1.5, 1.975, respec- +tively, for simulations using Gaussian (dashed) and uni- +form (dash-dotted) noise distributions. Also shown are +the results from the exact Gaussian distributions, ρk,e(u) +and ρc,e(r), (solid, from Eqs. (3a) and (12a)) that are ex- +pected from continuous-time Langevin dynamics. +The +distributions ρc(rn) and ρk(un+ 1 +2 ), are shown in Fig- +ures 3-5cd, the deviations +∆ρc(rn) = ρc(rn) − ρc,e(rn) +(55a) +∆ρk(un+ 1 +2 ) = ρk(un+ 1 +2 ) − ρk,e(un+ 1 +2 ) , +(55b) +from the expected (correct) distributions are shown in + +12 +FIG. 9. Configurational and kinetic distributions ρc(rn) and +ρk(un+ 1 +2 ) for simulations with Gaussian and uniform noise ap- +plied to the linear system Eqs. (6) and (17) with α = 0.5mΩ0, +Ω0∆t = 1.975, kBT = E0. Shown curves are Eq. (3) (solid, +labeled exact), simulations data using Gaussian noise βn +G +(dashed, labeled Gaussian), and simulation data using peaked +noise βn +P (dash-dot, labeled uniform). Distributions from sim- +ulations using Gaussian noise are indistinguishable from the +exact references Eq. (3). (c) and (d) show the distributions, +(a) and (b) show the deviations from the exact distributions +Eq. (3), and (e) and (f) show the effective potential (PMF) +and kinetic (MB) potentials (Eq. (56)) derived from the dis- +tributions. +Figures 3-5ab, and the effective configurational and ki- +netic potentials, +UP MF (rn) = −kBT ln ρc(rn) + ˜Cc +(56a) +UMB(un+ 1 +2 ) = −kBT ln ρk(un+ 1 +2 ) + ˜Ck , +(56b) +where +˜Cc +and +˜Ck +are +determined +such +that +min[UP MF (rn)] = min[UMB(un+ 1 +2 )] = 0, are shown in +Figures 3-5ef. For the harmonic potential, the statisti- +cally correct values of these potentials should be 1 +2κ(r)2 +and +1 +2m(u)2, which are indicated by thin solid curves. +For Ω0∆t = 0.5, Figure 3 shows that even a seemingly +modest deviation from a pure Gaussian distribution can +have rather large impacts on fourth-moment thermody- +namic measures, which for these parameters in Fig. 2cd +are seen to result in configurational and kinetic energy +fluctuations being depressed by about 7% and 14%, +respectively. +Increasing the time step to Ω0∆t = 1.5 +amplifies the deformation of the βn +U-generated distri- +butions away from the Gaussian. As seen in Figs. 4ef, +the increasingly noticeable difference is that the uniform +noise yields a more confined exploration of phase-space +than the Gaussian distribution does, consistent with +the depression in the fourth moment seen in Figs. 2cd. +Finally, in Fig. 5, we show the results for a time step, +Ω0∆t = 1.975, very close to the stability limit. +This +extreme case shows that the uniform distribution in +noise also can produce a near uniform distribution for +rn, while the distribution for un+ 1 +2 is near triangular, +consistent with a sum of two uniformly distributed num- +bers contributing to the kinetic fluctuations. Again, we +see from Figs. 5ef that the sampling of the phase space +is much more limited when using uniform noise than +when using Gaussian, even if the measured temperature, +configurational as well as kinetic, are measured to the +correct values. +Following the spirit of Refs. [15, 16] we explore the +application of a more challenging noise distribution βn +P +(the peaked distribution defined in Eq. (53a)), which is +both discrete and asymmetric (see Fig. 1). As was the +case for the uniform noise discussed above, we also here +reference the results next to those of Gaussian noise, +which give time step independent and correct behav- +ior. Conducting numerical simulations of Eqs. (6) and +(17) for different time steps and for normalized friction +α/mΩ0 = 0.1, 0.25, 0.5, 1, 2.5, 5 as described above, and +comparing to the evaluation of the moments from the +analyses of Sec. III, we obtain the data shown in Fig. 6, +where the simulation results are indistinguishable from +the results of the analyses for third and fourth moments +in Secs. III B and III C. As expected from the analysis, +the results of the second moments shown in Fig. 6ab of +rn and un+ 1 +2 , namely the potential and kinetic energies, +are perfectly aligned with statistical mechanics, since the +peaked distribution for βn +P satisfies the two moments in +Eq. (10). Since the applied noise is here asymmetric, the +resulting third moments of rn and un+ 1 +2 may also be- +come non-zero for non-zero time steps. This is seen in +Figures 6cd, where we observe rather complex behavior +as a function of the system parameters. We do, however, +see that for Ω0∆t → 0 the third moments approach zero, +consistent with the central limit theorem that guarantees +a Gaussian outcome when a very large number of noise +values contribute to the variables. Finally, in compari- +son with Fig. 2cd for uniform noise, we see very similar +behavior for the energy fluctuations (fourth moments) +for the application of the peaked distribution in Fig. 6ef. +Even if the uniform and peaked distributions are very dif- +ferent in appearance, the similarities between their out- +comes in their fourth moments are not surprising, given +that the analysis for the fourth moments in Sec. III C +shows that the difference between the two only depends +on the kurtoses, which have the values Γ4 = 9 +5 (uniform) +and Γ4 = +12 +5 (peaked). Again, we find that all signa- +tures of non-Gaussian noise vanish for Ω0∆t → 0, and +we find that non-zero time steps result in depressions of +the thermodynamically important energy fluctuations, as +derived from the fourth moments of the configurational +and kinetic coordinates. +The details of the simulated coordinate distributions +arising from the peaked noise are exemplified in Figs. 7- +9 for α/mΩ0 = 0.5 and Ω0∆t = 0.5, 1.5, 1.975, respec- + +13 +tively. For the smaller of the time steps, shown in Fig. 7, +we see the seemingly modest skewness and deformations +of the coordinate distributions arising from the peaked +noise. Yet, these modest deformations are what provide +the somewhat significant deviations from Gaussian char- +acteristics in third and fourth moments seen in Figs. 6cd +and Figs. 6ef for Ω0∆t = 0.5. More dramatic deviations +from Gaussian/Boltzmann characteristics are found in a +large range of Ω0∆t, including the value Ω0∆t = 1.5 +shown in Fig. 8. The seemingly discontinuous distribu- +tion is not a result of insufficient statistics. Rather, it +is the interference between the discrete nature of the ap- +plied noise distribution with the discrete time step that +happens to distinctively select certain preferred values of +rn and un+ 1 +2 over others. It is noticeable that these types +of peculiar distributions appear without any obvious or +abrupt signatures in the first four moments, as seen in +Fig. 6. As the time step Ω0∆t = 1.975 is pushed close to +the stability limit, we again find smooth coordinate dis- +tributions, seen in Fig. 9, where the applied peaked noise +is visible throughout the different displays. In analogy +with the visualization of the distributions from the uni- +form noise, this is the limit where the friction per time +step is relatively large, thereby making the behavior of +the coordinates rn and un+ 1 +2 subject to only a few noise +contributions at a time. +V. +DISCUSSION +In light of recent advances in stochastic thermostats +for simulating Langevin equations with accurate statis- +tics across the stability range of the applied time step +when using Gaussian noise [7, 8], we have revisited +the investigations of advantages and disadvantages of +using non-Gaussian thermal noise in discrete time. +Given the systemic first and second order time step +errors of the traditional methods (e.g., Refs. [18, 20]), +which necessitated rather small time steps for accurate +simulations, it was previously concluded [15, 16] that +other distributions, +including the desirable uniform +distribution, +would +be +efficient +substitutions +since +the central limit theorem would ensure near-Gaussian +outcomes for small time steps, thereby not further +significantly distort the simulation results due to the +imperfect noise. +This result has been a very useful +tool over the years when computational efficiency could +benefit from not converting stochastic variables from +uniform to Gaussian. However, as we have demonstrated +in this paper, when the time step becomes large, which +is allowed by the modern GJ methods when using +Gaussian noise, the application of non-Gaussian noise +does not retain the time step independent benefits of +these methods in thermodynamic measures that involve +moments higher than the second. We have found that, +given that the applied noise conforms to Eq. (10), +the GJ methods are invariant to the specific noise +distributions in measures of first and second moments, +such as configurational and kinetic temperatures. Thus, +it is deceiving to judge the quality of the thermostat +based on those moments alone. +As is evident from +the third and fourth moments, as well as the visual +impressions of the actual distributions of rn and un+ 1 +2 , +the sampling of the phase-space can be quite distorted +(non-Boltzmann) even if the measured temperatures +yield correct values. All of these results are consistent +with the significance of how the noise must be defined in +discrete-time; namely through the integral Eq. (9), which +ensures that any underlying distribution for β(t) will +yield a Gaussian outcome for βn due to the central limit +theorem. It follows that applying any other distribution +than Gaussian in discrete time is formally invalid, and +certainly leads to significant sampling errors unless +α∆t/m is small enough that the one-time-step attenua- +tion parameter c2 ≈ 1. It is therefore the conclusion of +this work that Gaussian noise must be applied to the +modern stochastic integrators if one wishes to take ad- +vantage of the large time step benefits of their properties. +VI. +ACKNOWLEDGMENTS +The author is grateful to Charlie Sievers for sharing +LAMMPS simulations that support the results of this +paper in more complex MD simulations using the GJF- +2GJ method, and to Lorenzo Mambretti for assistance +with the RANMAR random number generator. The au- +thor is also grateful for initial discussions with Chungho +Cheng on non-Gaussian noise. +[1] M.P. Allen, D.J. Tildesley, Computer Simulation of Liq- +uids, Oxford University Press, Inc., 1989. +[2] D. Frenkel and B. 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Parisi, Statistical Field Theory, (Addison-Wesley, +Menlo Park, 1988). +[12] Although not necessary, it is generally reasonable to ex- +pect that the distribution of β(t) is symmetric, yielding +all odd moments zero. +[13] A. Papoulis, Probability, Random Variables, and Stochas- +tic Processes, (McGraw-Hill, London, 1965). +[14] For a general description of pseudo-random generators, +see, e.g., W. H. Press, S. A. Teukolsky, W. T. Vetterling, +and B. P. Flannery, Numerical Recipes 3rd. ed. (Cam- +bridge University Press, Cambridge, 2007). +[15] A. Greiner, W. Strittmatter, and J. Honerkamp, J. Stat. +Phys. 51, 95 (1988). +[16] B. D¨unweg and W. Paul, Int. J. Mod. Phys. C 2, 817 +(1991). +[17] J. Finkelstein, G. Fiorin, and S. Seibold, Mol. Phys. 118, +e1649493 (2020). +[18] A. Br¨unger, C. L. Brooks, and M. Karplus, Chem. Phys. +Lett. 105, 495 (1984). +[19] R. W. Pastor, B. R. Brooks, and A. Szabo, Mol. Phys. 65, +1409 (1988). +[20] T. Schneider and E. Stoll, Phys. Rev. B 17, 1302 (1978). +[21] S. Plimpton, J. Comp. Phys. 117, 1 (1995). +[22] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolin- +tineanu, W. M. Brown, P. S. Crozier, P. J. in ′t Veld, +A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. +Stevens, J. Tranchida, C. Trott, S. J. Plimpton, Com- +pute. Phys. Commun. 271, 108171 (2022). +[23] See the LAMMPS documentation [30] for the use of +fix langevin using the Schneider and Stoll thermostat of +Ref. [20]. +[24] F. Weik, R. Weeber, K. Szuttor, K Breitsprecher, J. de +Graaf, M. Kuron, J. Landsgesell, H. Menke, D. Sean +and C. Holm., Euro. Phys. J. Special Topics 227, 1789 +(2019). See also https://espressomd.github.io/ documen- +tation for ESPResSo-4.2.0 Sec. 6.3.1. +[25] L. Verlet, Phys. Rev. 159, 98 (1967). +[26] W. C. Swope, H. C. Andersen, P. H. Berens, K. R. Wil- +son, J. Chem. Phys. 76, 637 (1982). +[27] D. Beeman, J. Comp. Phys. 20, 130 (1976). +[28] O. Bumeman, J. Comp. Phys. 1, 517 (1967). +[29] R. W. Hockney, Methods Comput. Phys. 9, 136 (1970). +[30] See http://lammps.sandia.gov/doc/Manual.pdf, for the +description of the “fix langevin” command. + diff --git a/uNE3T4oBgHgl3EQf-AtL/content/tmp_files/load_file.txt b/uNE3T4oBgHgl3EQf-AtL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5902c0e4844821caa4307a877a22923226fbadf --- /dev/null +++ b/uNE3T4oBgHgl3EQf-AtL/content/tmp_files/load_file.txt @@ -0,0 +1,782 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf,len=781 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='04821v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='stat-mech] 12 Jan 2023 January 13, 2023 On the Application of Non-Gaussian Noise in Stochastic Langevin Simulations Niels Grønbech-Jensen∗ Department of Mechanical & Aerospace Engineering Department of Mathematics University of California, Davis, CA 95616, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' In light of recent advances in time step independent stochastic integrators for Langevin equations, we revisit the considerations for using non-Gaussian distributions for the thermal noise term in discrete-time thermostats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We find that the desirable time step invariance of the modern methods is rooted in the Gaussian noise, and that deviations from this distribution will distort the Boltzmann statistics arising from the fluctuation-dissipation balance of the integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We use the GJ methods as the focus of our investigation since these methods are the ones that contain the most accurate thermodynamic measures of existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Within this set of methods we find that any distri- bution of applied noise which satisfies the two first moments given by the fluctuation-dissipation theorem, will result in correct, time step independent results that are generated by the first two moments of the system coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' However, if non-Gaussian noise is applied, undesired deviations in higher moments of the system coordinates will appear to the detriment of several important thermodynamic measures that depend especially on the fourth moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The deviations, induced by non-Gaussian noise, become significant with the one time step attenuation, thereby inhibiting the benefits of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, we conclude that the application of Gaussian noise is necessary for reliable thermodynamic results when using modern stochastic thermostats with large time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' INTRODUCTION Over the past several decades, discrete-time Langevin and Brownian simulations in computational statistical mechanics have been a centerpiece in many specific sim- ulation applications to molecular dynamics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', materi- als science, soft matter, and biomolecular modeling [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Inherent to simulations is the fundamental complication of time discretization, which on one hand is necessary for advancing time, but on the other hand inevitably cor- rupts the desired, continuous-time system features of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' An integral part of a simulation task is therefore to explore and optimize the balance between the two con- flicting core objectives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' namely simulation efficiency by increasing the time step, and simulation accuracy by de- creasing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Since discrete-time algorithms for time evolution pro- duce only approximate solutions to a continuous-time so- lution, and since the errors must vanish in the limit of infinitesimal time step, most considerations for a useful simulation environment have been considered in the limit of small time steps, where the simulated system variables change only little in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' In that limit, the discrete- time system behaves much like the continuous-time sys- tem, and algorithm development has therefore mostly been conducted by formulating objectives and measures in continuous-time whereafter time has been discretized and executed with relatively small steps in order to retain accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' ∗ ngjensen@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='ucdavis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='edu For stochastic differential equations, notably Langevin equations, an added complication is that it is not a specific trajectory which is of interest, but instead the statistics of trajectories that determines if the desired continuous-time behavior has been adequately repre- sented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The traditional approach to these problems has therefore been to acquire accurate statistics from accu- rate trajectories obtained from small time steps, where continuous-time behavior is well approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' How- ever, with the recent advent of discrete-time stochastic algorithms that can produce correct statistics from in- accurate, large time step trajectories [6–8], the details of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', stochastic noise correlation and distribution now become essential for obtaining the advantages of the sta- tistical accuracy for all meaningful time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The sig- nificance of these details arises from the discretization of time, as outlined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We are here concerned with the Langevin equation [9, 10] m˙v + α ˙r = f + β , (1) where m is the mass of an object with spatial (configu- rational) coordinate r and velocity (kinetic) coordinate v = ˙r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The object is subjected to a force f = −∇Ep(r), with Ep(r) being a potential energy surface for r, and lin- ear friction that is represented by the non-negative con- stant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The fluctuation-dissipation relationship specifies that the thermal fluctuations β can be represented by a distribution for which the following first two moments are given [11]: ⟨β(t)⟩ = 0 (2a) ⟨β(t)β(t′)⟩ = 2α kBT δ(t − t′) , (2b) 2 where δ(t) is Dirac’s delta function signifying that the fluctuations are temporally uncorrelated, kB is Boltz- mann’s constant, and T is the thermodynamic temper- ature of the heat-bath, to which the system is coupled through the friction constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' A feature in statistical mechanics is that the fluctuation-dissipation theorem for the noise distribution β(t) only specifies the two moments given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (2) [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, there is a great deal of freedom in choosing the distribution of the noise in the Langevin equation such that correct thermodynamics is represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The reason that the physical (statistical) re- sults do not depend on this freedom is that continuous- time demands a new, uncorrelated noise value β(t) be contributed and accumulated at every instant, resulting in a Gaussian outcome at any non-zero time-scale due to the central limit theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As a result, the sampled (one-dimensional) Maxwell-Boltzmann ve- locity distribution ρk,e(v) and the configurational Boltz- mann distribution ρc,e(r) always become [10] ρk,e(v) = � m 2πkBT exp � − 1 2mv2 kBT � (3a) ρc,e(r) = Cc exp � −Ep(r) kBT � , (3b) with the constant Cc determined by the normalization of probability � all space ρc,e(r) dr = 1 , (3c) regardless of which temporally uncorrelated distribution β(t) is at play in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (1), and for as long as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Similarly, it is given that diffusive transport for a flat surface is represented by a diffusion constant [1] DE = 1 d lim t→∞ ⟨[r(t) − r(0)]2⟩ 2t (4) = kBT α , (5) where d is the dimension of the configurational space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Given that the central limit theorem ensures a Gaus- sian noise distribution for any appreciable integration of the instantaneous variable β(t), it is natural to choose β(t) to be the Gaussian that conforms to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' How- ever, in discrete-time simulations it is not uncommon to directly employ a uniform distribution that satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The rationale is rooted in the computational efficiency of uniformly distributed pseudo-random gener- ators [14], compared with the additional computational load required for transforming those variables into Gaus- sian distributed variables before use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' A thorough inves- tigation of using non-Gaussian valuables in discrete time was presented in the 1988 publication Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [15], where it was concluded that, given the inherent inaccuracies of the stochastic discrete-time integrators of the time, the scal- ing in statistical moments upon the applied time step did not change adversely when using non-Gaussian noise dis- tributions, specifically the uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Building upon this work, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [16] extended the investigation in 1991 to a broader class of noise distributions, which would yield indistinguishable statistical results compared to the use of Gaussian noise for vanishing time steps, where the underlying systemic time step errors of the then con- temporary methods would vanish as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The studies highlighted the benefits of computational efficiencies in using simple, non-Gaussian noise given that the statisti- cal differences were deemed to be insignificant for small time steps, which in any case were necessary for the ac- curacy of the stochastic integrators, even using Gaus- sian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As pointed out in a recent article [17] that compares the quality of contemporary stochastic integra- tors, this practice has been widespread when using tra- ditional algorithms, such as the Br¨unger-Brooks-Karplus [18] (also analyzed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [19]) or the Schneider-Stoll [20] method, for as long as small time steps are applied (for a more expansive list of traditional and recent Langevin integrators, please see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For example, these two methods are used with non-Gaussian noise in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', the wide-distribution molecular dynamics simulation suites, LAMMPS [21–23] and ESPResSo [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' It is the aim of this work to revisit the discussion of applying non-Gaussian noise in light of the most mod- ern stochastic integrators, which have been designed to produce accurate, time step independent statistics and transport through the use of Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We demon- strate that even if the low order moments of the config- urational and kinetic distributions are seemingly unaf- fected by the details of the noise distribution, there may still be significant differences in the sampled phase space that affect the statistical sampling of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We base the analysis exclusively on the recently formulated GJ set of methods [8], since these methods have the most optimized statistical properties that are invariant to the time step for linear systems when using Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This also allows us to investigate the large time step im- plications of using non-Gaussian noise without being sub- ject to other inherent time step errors of the integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The presentation is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Section II re- views the basic features and structure of the GJ methods, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III analyzes analytically the first four statistical mo- ments of the configurational and kinetic coordinates for linear systems, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' IV presents companion simulations to the analysis conducted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' V summa- rizes the discussion with specific conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 3 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' REVIEW OF THE GJ METHODS We adopt the complete set of GJ methods [8] for solv- ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (1) in the form un+ 1 2 = √c1 vn + √c3 ∆t 2m f n + √c3 2m βn+1 (6a) rn+1 = rn + √c3 ∆t un+ 1 2 (6b) vn+1 = c2 √c1 un+ 1 2 + �c3 c1 ∆t 2m f n+1 + �c3 c1 1 2m βn+1 , (6c) where a superscript n on the variables r, v, u, and f pertains to the discrete time tn = t0 + n ∆t at which the numerical approximation is given, with ∆t being the discrete time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, f n = f(tn, rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The two repre- sented velocities are, respectively, the half-step velocity un+ 1 2 at time tn+ 1 2 , and the on-site velocity vn at tn, given by un+ 1 2 = rn+1 − rn √c3 ∆t (7a) vn = rn+1 − (1 − c2)rn − c2rn−1 2√c1c3 ∆t + 1 4m(βn − βn+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (7b) The statistical definitions of what it means to be half- step and on-site are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The functional parameter c2 = c2( α∆t m ) is the one-time-step attenuation, which is a function of α∆t/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The two other functional parameters, c1 and c3, are given by 2c1 = 1 + c2 (8a) α∆t m c3 = 1 − c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (8b) It is the choice of the function c2 that distinguishes the GJ methods from each other (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [8] for sev- eral choices of c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The basic requirement for c2 is that c2 → 1 − α∆t m for α∆t m → 0, ensuring that also c1 → 1 and c3 → 1 for α∆t m → 0 in accordance with the fric- tionless Verlet method [25–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Finally, the discrete-time stochastic noise variable is defined by βn+1 = � tn+1 tn β(t) dt , (9) such that the discrete-time fluctuation-dissipation theo- rem follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (9) and (2) to become ⟨βn⟩ = 0 (10a) ⟨βnβℓ⟩ = 2α∆t kBT δn,ℓ , (10b) where δn,ℓ is Kronecker’s delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Eliminating the velocities from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) yields the purely configurational stochastic Verlet form rn+1 = 2c1 rn − c2 rn−1 + c3∆t2 m f n + c3∆t 2m (βn + βn+1) , (11) which is the only stochastic Verlet form that 1) for a harmonic potential Ep(r) = −κr (κ > 0) and Gaussian fluctuations βn, will yield the correct Boltzmann distri- bution ρc,e(rn) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3b), such that ρc,e(rn) = � κ 2πkBT exp � − 1 2κ(rn)2 kBT � (12a) ⟨rnrn⟩ = kBT κ , (12b) for any time step within the stability range (Ω0∆t < 2 � c1/c3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2) for a linear potential f = const, yield the correct configurational drift velocity vd = �rn+1 − rn ∆t � = f α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (13) and 3) for f = 0, yield the correct Einstein diffusion given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (4), such that DE = lim n→∞ � (rn − r0)2� 2d n∆t = kBT α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (14) Similarly, it was found [8] that for a single Gaussian noise value βn per time step, the harmonic potential Ep(r) = −κr (κ ≥ 0) results in the unique half-step velocity un+ 1 2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (7a) with correct velocity distribution function ρk,e(un+ 1 2 ) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3a) such that ⟨un+ 1 2 un+ 1 2 ⟩ = kBT m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (15) In contrast, it was also found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [8] that is not possi- ble to define an onsite velocity vn with the same time step independent property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Given the statistical invariance of these equations upon the time step when using Gaussian noise, we will now investigate the statistical properties of rn and un+ 1 2 when using other distributions than Gaus- sian for βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' MOMENT ANALYSIS FOR LINEAR SYSTEMS We will here investigate the relevant moments of the configurational coordinate rn and half-step velocity un+ 1 2 as a function of the applied noise distribution ρ(β) of βn, which satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For this analysis we will mostly assume a linear system with a harmonic potential Ep(r) = 1 2κ r2 (16) ⇒ f = −κr , (17) such that the linearized configurational stochastic GJ methods in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (11) can be written rn+1 = 2c1X rn − c2 rn−1 + c3∆t 2m (βn + βn+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (18) with X = 1 − c3 c1 Ω2 0∆t2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (19) 4 The natural frequency Ω0 of the oscillator is given by Ω2 0 = κ/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' First and Second Moments For the first moments of the variables, it is obvious that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10a) ensures both ⟨rn⟩ = 0 (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (18)) and subsequently ⟨un+ 1 2 ⟩ = 0 (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (7a)) regardless of which distribution is chosen for βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' It similarly follows that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10a) also ensures that the drift velocity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (13) for f = const remains constant and correct regardless of both time step and distribution βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For the second moments, ⟨rnrn⟩ and ⟨un+ 1 2 un+ 1 2 ⟩, we first find ⟨rnrn⟩ as the solution to the system of equa- tions given by the statistical averages of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (18) mul- tiplied by, respectively rn−1, rn, and rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This linear system of equations can be conveniently written \uf8eb \uf8ed 1 −2c1X c2 0 2c1 −2c1X c2 −2c1X 1 \uf8f6 \uf8f8 \uf8eb \uf8ed ⟨rn−1rn+1⟩ ⟨rnrn+1⟩ ⟨rnrn⟩ \uf8f6 \uf8f8 = \uf8eb \uf8ed 0 1 2c1X + 2 \uf8f6 \uf8f8 �c3∆t 2m �2 ⟨βnβn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (20) We immediately observe that this equation is unaffected by the details of the distribution of βn for as long as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, also the second moment, ⟨rnrn⟩, is unaffected by the chosen distribution of βn, regardless of the time step ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The corresponding sec- ond moment of the half-step velocity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (7a) is then given by ⟨un+ 1 2 un+ 1 2 ⟩ = 2 c3 ∆t2 (⟨rnrn⟩ − ⟨rnrn+1⟩) , (21) where both correlations on the right hand side are so- lutions found from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (20), which is unchanged for as long as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10b) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, the second moment of the half-step velocity is also unaffected by the chosen distribution of βn, regardless of the time step ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The averages of potential (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (16)) and kinetic energies, ⟨Ep(rn)⟩ = κ 2 ⟨rnrn⟩ = 1 2kBT (22) � Ek(un+ 1 2 ) � = 1 2 m c3 ∆t2 (⟨rnrn⟩ − ⟨rnrn+1⟩) = 1 2kBT , (23) are therefore unaffected for any applied stochastic vari- able βn satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Since the second moments of the coordinates rn and un+ 1 2 of the GJ methods are invariant to the specific choice of noise distribution that satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10), we conclude that the GJ methods display time-step invari- ance in both potential and kinetic energy, and therefore temperature, regardless of the applied βn distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The invariance also extends to the above-mentioned dif- fusion Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (4) and (14), since this quantity is rooted in only second moments of the noise variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Numerical validation of the time step independence of the second moments is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2ab and 6ab for a couple of non-Gaussian noise variables, and simulations are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' While the invariance of the second moments against the choice of βn distribution is appealing, as it gives the impression that temperature is correctly achieved, it may not be sufficient to ensure proper thermodynamics sampling as we will see in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Third Moments The third moments of rn and un+ 1 2 are generally not needed for the calculations of thermodynamic quantities, especially not if the system potentials are symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' However, they may be relevant to study in case asym- metric fluctuations are applied to the system for some reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Following the same procedure that was used for calculating the second moments, we form the statistical averages of the product of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (18) with the six unique combinations of rℓ1rℓ2, where ℓi = n − 1, n, n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The result is the linear system A3 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ⟨rn−1rn−1rn+1⟩ ⟨rn−1rnrn+1⟩ ⟨rn−1rn+1rn+1⟩ ⟨rnrnrn+1⟩ ⟨rnrn+1rn+1⟩ ⟨rnrnrn⟩ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = �kBT κ � 3 2 R = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed ⟨rn−1rn−1βn⟩ ⟨rn−1rnβn⟩ ⟨rn−1rn+1(βn + βn+1)⟩ ⟨rnrnβn⟩ ⟨rnrn+1(βn + βn+1)⟩ ⟨rn+1rn+1(βn + βn+1)⟩ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 c3∆t 2m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (24) 5 where A3 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 −2c1X 0 c2 0 1 0 c2 −2c1X 0 c2 −2c1X 1 0 0 0 0 0 0 1 c2 −2c1X 0 c2 0 −2c1X 1 0 0 0 c2 0 −2c1X 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (25) The right hand side R = {Ri} can after some calcula- tions be expressed as R1 = 0 (26a) R2 = 0 (26b) R3 = 0 (26c) R4 = (1 − c2)3 �κ∆t α � 3 2 Γ3 (26d) R5 = (2c1X + 1)R4 (26e) R6 = (2c1X(2c1X + 2) + 2)R4 , (26f) where the fluctuations βn satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10), and the skew- ness is denoted by Γ3, Γ3 = ⟨βnβnβn⟩ ⟨βnβn⟩ 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (27) As is the case when using any symmetric distribution for βn, all odd moments for the configurational coordinate rn will be zero for a Gaussian distribution of βn applied to a symmetric potential, Ep(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The third moment of the velocity un+ 1 2 is given by � un+ 1 2 un+ 1 2 un+ 1 2 � = 1 � ∆t √c3 �3 � (rn+1 − rn)3� = 3 � ∆t √c3 �3 �� rnrnrn+1� − � rnrn+1rn+1�� , (28) where the two correlations on the right hand side are part of the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For any given choice of GJ method (c2) the above system of equations can be easily solved numerically for given system parameters and skewness Γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Examples of this will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' IV, and are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6cd along with the results of corresponding numerical solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) for a particular asymmetric noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Closed form expressions are obtained in the following for a couple of special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Special Case, X = 0: The matrix A3 and the vector R simplify considerably with easy accessible solutions ⟨rnrnrn⟩ = 2 1 + c3 2 �kBT κ � 3 2 R4 (29a) ⟨rnrnrn+1⟩ = 1 − c2 1 + c3 2 �kBT κ � 3 2 R4 (29b) ⟨rnrn+1rn+1⟩ = 1 + c2 2 1 + c3 2 �kBT κ � 3 2 R4 , (29c) where we have used that this special case for Ω2 0∆t2 = 2 c3 c1 offers R6 = 2R5 = 2R4, and where R4 = �1 − c2 2 2 � 3 2 Γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (30) Equations (28), (29b), and (29c) give �� un+ 1 2 �3� = − 3 � ∆t√c3 �3 c2 c2 2 − c2 + 1 �kBT κ � 3 2 R4 = − 3 2 √ 2 �1 − c2 2 1 + c2 � 3 2 c2 c2 2 − c2 + 1 �kBT m � 3 2 Γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (31) Examples of the use of these expressions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6cd as open markers (◦) for a particular asymmetric noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Special Case, c2 = 0: In this case we have that c1 = 1 2, c3 = m/α∆t, and X = 1 − ∆tκ/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The simplified matrix A3 and vector R yield ⟨rnrnrn⟩ = 3X2 + 3X + 2 1 − X3 �kBT κ � 3 2 R4 (32a) ⟨rnrnrn+1⟩ = 2X3 + 3X2 + 2X + 1 1 − X3 �kBT κ � 3 2 R4 (32b) ⟨rnrn+1rn+1⟩ = X4 + 2X3 + 2X2 + 2X + 1 1 − X3 �kBT κ � 3 2 R4 , (32c) where R4 = �κ∆t 2α � 3 2 Γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (33) 6 Equations (28), (32b), and (32c) give �� un+ 1 2 �3� = 3X2 2 √ 2 1 − X2 1 − X3 �kBT m � 3 2 Γ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (34) Examples of the use of these expressions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6cd as closed markers (•) for a particular asymmetric noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Fourth Moments The fourth moments of rn and un+ 1 2 are needed for the calculations of energy and pressure fluctuations, which are the basis of several important thermodynamic quanti- ties, such as heat capacity, thermal expansion, compress- ibility, and the thermal pressure coefficient [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Fourth moments are therefore critical for proper thermodynamic characterization of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Following the same proce- dure that was used for calculating the second and third moments, we form the statistical averages of the product of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (18) with the ten unique combinations of rℓ1rℓ2rℓ3, where ℓi = n− 1, n, n+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The result is the linear system ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='A4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn−1rn−1rn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn−1rnrn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn−1rn+1rn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rnrnrn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rnrn+1rn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn+1rn+1rn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrnrnrn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrnrn+1rn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrn+1rn+1rn+1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrnrnrn⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='�kBT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8eb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn−1rnβn⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn−1rn+1(βn + βn+1)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rnrnβn⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rnrn+1(βn + βn+1)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn−1rn+1rn+1(βn + βn+1)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrnrnβn⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrnrn+1(βn + βn+1)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rnrn+1rn+1(βn + βn+1)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='⟨rn+1rn+1rn+1(βn + βn+1)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='\uf8f8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='c3∆t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='2m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (35) where A4 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 1 0 0 0 0 0 −2c1X 0 0 c2 0 1 0 0 0 0 c2 −2c1X 0 0 c2 −2c1X 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 c2 −2c1X 0 0 c2 0 −2c1X 1 0 0 0 0 0 0 0 c2 0 −2c1X 1 0 0 0 0 0 0 0 0 0 0 1 0 c2 −2c1X 0 0 0 c2 0 0 −2c1X 1 0 0 0 0 0 0 c2 0 0 −2c1X 1 0 0 0 0 0 0 c2 0 0 −2c1X 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='The right hand side R = {Ri} can after some cumbersome work be expressed as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R1 = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R2 = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='2(1 − c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='2)(1 − X) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R3 = [2 + 2c1X]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R4 = [2c1X + 1 − c2]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R5 = [2c1X(2c1X + 2 − c2) + 1 − 2c2]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R6 = [(2c1X)3 + (2c1X)2(3 − c2) + 4c1X(1 − 2c2) − 4c2]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R7 = [3 + (Γ4 − 3)R2]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R8 = [4c1X + 3 − c2 + (2c1X + 1)(Γ4 − 3)R2]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R9 = [(2c1X)2 + 2c1X(3 − c2) + 3 − 2c2 + (1 + 2c1X)2(Γ4 − 3)R2]R2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='(37i) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='R10 = [3(2c1X + 2) + (1 + (1 + 2c1X)3)(Γ4 − 3)R2]R2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (37j) where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Notice that R7-R10 in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (37g)-(37j) depend on the kurtosis Γ4 Γ4 = ⟨βnβnβnβn⟩ ⟨βnβn⟩2 , (38) which is not determined by the fluctuation-dissipation 7 relationship in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' While Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (35)-(37) are not immediately informative, they provide a straightforward path to calculating the sought-after fluctuation σp in potential energy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (16) σp = �� E2p(rn) � − ⟨Ep(rn)⟩2 = κ 2 � ⟨rnrnrnrn⟩ − ⟨rnrn⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (39) Equations (35)-(37) also lead directly to the fluctuations σk in kinetic energy of the velocity from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (7a) σk = �� E2 k(un+ 1 2 ) � − � Ek(un+ 1 2 ) �2 = m 2 � ⟨un+ 1 2 un+ 1 2 un+ 1 2 un+ 1 2 ⟩ − ⟨un+ 1 2 un+ 1 2 ⟩2 , (40) where ⟨un+ 1 2 un+ 1 2 ⟩ is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (15), and ⟨un+ 1 2 un+ 1 2 un+ 1 2 un+ 1 2 ⟩ = 1 c2 3∆t4 ⟨(rn+1 − rn)4⟩ = 2 c2 3∆t4 (⟨rnrnrnrn⟩ − 2⟨rnrnrnrn+1⟩ + 3⟨rnrnrn+1rn+1⟩ − 2⟨rnrn+1rn+1rn+1⟩) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (41) All the correlations on the right hand side are given from the solution to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (35)-(37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For any given choice of GJ method (c2) the above sys- tem of equations can be easily solved numerically for given system parameters and kurtosis Γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Examples of this will be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' IV, and are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd and 6ef along with the results of correspond- ing numerical solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) for particular noise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Closed form expressions can be given in the special cases of X = 0 and c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Special Case, X = 0: This particular time step is Ω0∆t = √ 2 �c1 c3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (42) Cumbersome, yet straightforward, algebra leads to the closed expressions for ⟨rnrnrnrn⟩ and ⟨un+ 1 2 un+ 1 2 un+ 1 2 un+ 1 2 ⟩ for X = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', for Ω0∆t = √ 2 � c1/c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The corresponding fluctuations in po- tential and kinetic energies can be written σp = κ 2 � ⟨rnrnrnrn⟩ − ⟨rnrn⟩2 = kBT √ 2 � 1 4 1 + 7c2 2 1 + c2 2 + 1 4Γ4 1 − c2 2 1 + c2 2 (43) σk = m 2 � ⟨un+ 1 2 un+ 1 2 un+ 1 2 un+ 1 2 ⟩ − ⟨un+ 1 2 un+ 1 2 ⟩2 = kBT √ 2 � 1 2 c3 2 1 + c2 2 [3 − Γ4] + 1 4 [1 + Γ4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (44) Notice that if βn is a Gaussian distributed vari- able, where the kurtosis is Γ4 = 3, the above energy fluctuations yield the expected values, σp = σk = kBT/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Examples of the use of these expressions are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd and 6ef as open markers (◦) for partic- ular noise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Special Case, c2 = 0: This special case (GJ-0, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [8]) corresponds to the first order difference equation rn+1 = rn + 1 α � ∆t f n + 1 2(βn + βn+1) � , (45) which can approximate the solution to the Langevin equation with no inertia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', Brown- ian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The linearized, Hooke’s law system (f = −κr) reads rn+1 = Xrn + 1 2α(βn + βn+1) , (46) with X = 1 − κ∆t/α, in correspondence with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This simplicity allows the last four of the ten equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (35) to decouple from the rest, and the fourth moments can be cal- culated to yield the average and fluctuations in po- tential energy ⟨Ep(rn)⟩ = κ 2 ⟨rnrn⟩ = 1 2kBT (47) σp = κ 2 � ⟨rnrnrnrn⟩ − ⟨rnrn⟩2 = kBT √ 2 � 1 2X4 [3 − Γ4] + 1 4X3 [7 − Γ4] + 1 4(X2 + X + 1) [1 + Γ4] X3 + X2 + X + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (48) We also here notice that if βn is a Gaussian dis- tributed variable, where the kurtosis is Γ4 = 3, 8 the potential energy fluctuations yield the expected value, σp = kBT/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Similarly, we can find the fluctuations σk in kinetic energy from the correlation �� un+ 1 2 �4� = (1 − X)2 4σ2 p + (kBT )2 m2 + 6 �kBT m �2 X2(1 − X) − �kBT m �2 (3 + Γ4)X � 1 − 3 2X + X2 � , (49) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The tree applied distributions, ρG(βn) (Gaussian), ρU(βn) (Uniform), and ρP (βn) (Peaked), given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (51), (52), and (53), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' All three distributions satisfy the discrete-time fluctuation-dissipation relationship, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' such that σk = 1 2m ��� un+ 1 2 �4� − �kBT m �2 , (50) where we have used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, σk can be easily computed also in this special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We again notice that for Gaussian fluctuations, where Γ4 = 3, the kinetic energy fluctuations yield the expected value σk = kBT/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Examples of the use of these expressions are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd and 6ef as closed markers (•) for par- ticular noise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' SIMULATIONS While the above expressions provide both direct and indirect means for evaluating the thermodynamically im- portant first four moments of the configurational and ki- netic variables, we test these results against numerical simulations such that both simulations and analysis can be mutually verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Further, the simulations provide FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Simulations and analyses of average potential and ki- netic energies, ⟨Ep⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (22) (a) and ⟨Ek⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (23) (b), and their respective fluctuations, σp from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (39) (c) and σk from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (40) (d), as a function of reduced time step for the harmonic system, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17), with non-zero tempera- ture T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Results are shown for Gaussian βn G (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (51a)) and uniform βn U (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (52a)) noise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Several friction parameters α are used, as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Simulation results are indistinguishable from the analytical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Mark- ers indicate special cases for c2 = 0 (• from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (48)-(50)) and X = 0 (◦ from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (43) and (44)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' additional details on the actual distributions ρc(rn) and ρk(un+ 1 2 ), which in some cases reveal significant devia- tions from Boltzmann statistics when the fluctuations βn are non-Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We show results of three different distributions for the fluctuations βn, all satisfying the discrete-time fluctuation-dissipation balance that constrains the first two moments given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10), and thus, all resulting in correct first and second moments of rn and un+ 1 2 , lead- ing to correct temperature as measured by the kinetic and potential energies (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III A): 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Configurational and kinetic distributions ρc(rn) and ρk(un+ 1 2 ) for simulations with Gaussian and uniform noise ap- plied to the linear system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) with α = 1mΩ0, Ω0∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, kBT = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Shown curves are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3) (solid, labeled exact), simulations data using Gaussian noise βn G (dashed, labeled Gaussian), and simulation data using uni- form noise βn U (dash-dot, labeled uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Distributions from simulations using Gaussian noise are indistinguishable from the exact references Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (c) and (d) show the distribu- tions, (a) and (b) show the deviations from the exact distri- butions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3), and (e) and (f) show the effective potential (PMF) and kinetic (MB) potentials (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (56)) derived from the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Gaussian,: βn = βn G with distribution ρG(β): ρG(β) = 1 � 2π⟨βnβn⟩ exp � − ββ 2⟨βnβn⟩ � , (51a) with skewness and kurtosis Γ3 = 0 (51b) Γ4 = 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (51c) Uniform,: βn = βn U with distribution ρU(β): ρU(β) = 1 2 � 3⟨βnβn⟩ × � 1 , − √ 3 ≤ β √ ⟨βnβn⟩ < √ 3 0 , otherwise , (52a) with skewness and kurtosis Γ3 = 0 (52b) Γ4 = 9 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (52c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Configurational and kinetic distributions ρc(rn) and ρk(un+ 1 2 ) for simulations with Gaussian and uniform noise ap- plied to the linear system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) with α = 1mΩ0, Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, kBT = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Shown curves are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3) (solid, labeled exact), simulations data using Gaussian noise βn G (dashed, labeled Gaussian), and simulation data using uni- form noise βn U (dash-dot, labeled uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Distributions from simulations using Gaussian noise are indistinguishable from the exact references Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (c) and (d) show the distribu- tions, (a) and (b) show the deviations from the exact distri- butions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3), and (e) and (f) show the effective potential (PMF) and kinetic (MB) potentials (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (56)) derived from the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Peaked,: βn = βn P with distribution ρP (β): ρP (β) = 2 3 δ � β � ⟨βnβn⟩ + 1 √ 2 � + 1 3 δ � β � ⟨βnβn⟩ − √ 2 � , (53a) with skewness and kurtosis Γ3 = 1 √ 2 (53b) Γ4 = 12 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (53c) The three distributions are sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 1, and the realizations of the pseudo-random numbers βn are produced from the uniformly distributed numbers in [0, 1) generated by the RANMAR algorithm (used in LAMMPS [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' In order to validate the analysis of the previous section, we simulated the noisy harmonic oscillator Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Configurational and kinetic distributions ρc(rn) and ρk(un+ 1 2 ) for simulations with Gaussian and uniform noise ap- plied to the linear system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) with α = 1mΩ0, Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='975, kBT = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Shown curves are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3) (solid, labeled exact), simulations data using Gaussian noise βn G (dashed, labeled Gaussian), and simulation data using uni- form noise βn U (dash-dot, labeled uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Distributions from simulations using Gaussian noise are indistinguishable from the exact references Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (c) and (d) show the distribu- tions, (a) and (b) show the deviations from the exact distri- butions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3), and (e) and (f) show the effective potential (PMF) and kinetic (MB) potentials (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (56)) derived from the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (17) with several of the GJ methods, which all give the same statistical results for Gaussian noise once the time step is appropriately scaled according to the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For simplicity we limit the displayed results to those of the GJ-I (GJF-2GJ) method [7, 8], for which c2 = 1 − α∆t 2m 1 + α∆t 2m (54a) c1 = c3 = 1 1 + α∆t 2m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (54b) The GJF-2GJ method overlaps in the configurational co- ordinate with the GJF method [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We note that the GJ-I (GJF-2GJ) method is available for use in LAMMPS [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' With time normalized by the characteristic unit t0 given by the inverse of the natural frequency Ω0, the re- duced damping is α/mΩ0, and the characteristic energy is E0 = κr2 0, where r0 is a chosen characteristic length to which r is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The simulations are conducted for kBT = E0 for a variety of values of α/mΩ0 in the entire stability range Ω0∆t < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Each statistical value is calcu- lated from averages over 1,000 independent simulations, each with 108 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Simulations and analyses of average potential and ki- netic energies, ⟨Ep⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (22) (a) and ⟨Ek⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (23) (b), the respective third moments ⟨(rn)3⟩ from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (24)-(26) (c) and ⟨(un+ 1 2 )3⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (28) (d), and the respective energy fluctuations, σp from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (39) (e) and σk from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (40) (f), as a function of reduced time step for the harmonic system, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17), with non-zero temperature T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Results are shown for Gaussian βn G (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (51a)) and the asymmetrically peaked βn P (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (53a)) noise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Several friction parameters α are used, as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Simulation results are indistinguishable from the analytical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Mark- ers indicate special cases for c2 = 0 (• from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (32a) and (34), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (48)-(50)) and X = 0 (◦ from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (29a) and (31), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (43) and (44)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Figure 2 shows the results of using the uniform noise distribution βn U (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (52a)) alongside the results of using the Gaussian βn G (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (51a)) as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Figures 2ab, respectively displaying results for configurational and ki- netic variables, verify the general result from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' namely that a statistical property, which depends only on the first and second moments of the noise distribution, will be correctly evaluated for the GJ methods if the noise satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The shown data on each plot are for damping parameters α/mΩ0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 5, and the results for both Gaussian and uniform noise distribu- tions are clearly in close agreement with the expected values given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (22) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This has previously been extensively validated for Gaussian noise [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Figures 2cd display simulation results and comparisons to the expectations from the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III C for the configurational and kinetic energy fluctuations σp and σk found in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Since the Gaussian noise βn G from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (51a) applied to the linear system produces Gaussian distributions for rn and un+ 1 2 with correct, and time step independent, first and second moments (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III A), it follows that the fourth moments are also 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Configurational and kinetic distributions ρc(rn) and ρk(un+ 1 2 ) for simulations with Gaussian and peaked noise ap- plied to the linear system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5mΩ0, Ω0∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, kBT = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Shown curves are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3) (solid, labeled exact), simulations data using Gaussian noise βn G (dashed, labeled Gaussian), and simulation data using peaked noise βn P (dash-dot, labeled uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Distributions from sim- ulations using Gaussian noise are indistinguishable from the exact references Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (c) and (d) show the distributions, (a) and (b) show the deviations from the exact distributions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3), and (e) and (f) show the effective potential (PMF) and kinetic (MB) potentials (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (56)) derived from the dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' correct and time step independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This is clearly ob- served on the energy fluctuation figures, where both sim- ulation results and the results of solving Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (35)-(37) for Γ4 = 3 are shown to be indistinguishable and con- stant at the correct values σp = σk = kBT/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' How- ever, the uniformly distributed noise βn U (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (52a)) pro- duces neither Gaussian nor uniform distributions for rn and un+ 1 2 , but instead produces distributions that de- pend on the time step and the damping parameter, even if Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This can be seen in the figures for the energy fluctuations, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd, where both simu- lation results and the results of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (35)-(37) for Γ4 = 3 and Γ4 = 9 5, the latter being the kurtosis for the uniform distribution, are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The simulation results are indis- tinguishable from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' It is obvious that when using the uniform noise, the energy fluctuations, which depend on the fourth moments of the variables, can de- viate significantly from the correct value, given by the Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We also observe that the fluctuations approach the correct values for small time steps or small damping parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This is understandable, since the damping per time step in those limits is very small, and the fluctuations in rn and un+ 1 2 therefore are composed FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Configurational and kinetic distributions ρc(rn) and ρk(un+ 1 2 ) for simulations with Gaussian and uniform noise ap- plied to the linear system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5mΩ0, Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, kBT = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Shown curves are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3) (solid, labeled exact), simulations data using Gaussian noise βn G (dashed, labeled Gaussian), and simulation data using peaked noise βn P (dash-dot, labeled uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Distributions from sim- ulations using Gaussian noise are indistinguishable from the exact references Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (c) and (d) show the distributions, (a) and (b) show the deviations from the exact distributions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3), and (e) and (f) show the effective potential (PMF) and kinetic (MB) potentials (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (56)) derived from the dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' of βn U contributions from many time steps, allowing the integrated noise in rn and un+ 1 2 to become near-Gaussian by the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Conversely, if the damp- ing per time step becomes appreciable, then the effective noise in rn and un+ 1 2 will be composed by only a few βn U contributions, which do not approximate a Gaussian outcome very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' In order to look at the details of the simulated distri- butions, we have selected a few representative parame- ter values as illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Figures 3-5 show the simu- lated configurational and kinetic distributions, ρc(r) and ρk(u), for α = 1mΩ0 and Ω0∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='975, respec- tively, for simulations using Gaussian (dashed) and uni- form (dash-dotted) noise distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Also shown are the results from the exact Gaussian distributions, ρk,e(u) and ρc,e(r), (solid, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3a) and (12a)) that are ex- pected from continuous-time Langevin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The distributions ρc(rn) and ρk(un+ 1 2 ), are shown in Fig- ures 3-5cd, the deviations ∆ρc(rn) = ρc(rn) − ρc,e(rn) (55a) ∆ρk(un+ 1 2 ) = ρk(un+ 1 2 ) − ρk,e(un+ 1 2 ) , (55b) from the expected (correct) distributions are shown in 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Configurational and kinetic distributions ρc(rn) and ρk(un+ 1 2 ) for simulations with Gaussian and uniform noise ap- plied to the linear system Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5mΩ0, Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='975, kBT = E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Shown curves are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3) (solid, labeled exact), simulations data using Gaussian noise βn G (dashed, labeled Gaussian), and simulation data using peaked noise βn P (dash-dot, labeled uniform).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Distributions from sim- ulations using Gaussian noise are indistinguishable from the exact references Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (c) and (d) show the distributions, (a) and (b) show the deviations from the exact distributions Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (3), and (e) and (f) show the effective potential (PMF) and kinetic (MB) potentials (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (56)) derived from the dis- tributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Figures 3-5ab, and the effective configurational and ki- netic potentials, UP MF (rn) = −kBT ln ρc(rn) + ˜Cc (56a) UMB(un+ 1 2 ) = −kBT ln ρk(un+ 1 2 ) + ˜Ck , (56b) where ˜Cc and ˜Ck are determined such that min[UP MF (rn)] = min[UMB(un+ 1 2 )] = 0, are shown in Figures 3-5ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For the harmonic potential, the statisti- cally correct values of these potentials should be 1 2κ(r)2 and 1 2m(u)2, which are indicated by thin solid curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For Ω0∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, Figure 3 shows that even a seemingly modest deviation from a pure Gaussian distribution can have rather large impacts on fourth-moment thermody- namic measures, which for these parameters in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd are seen to result in configurational and kinetic energy fluctuations being depressed by about 7% and 14%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Increasing the time step to Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5 amplifies the deformation of the βn U-generated distri- butions away from the Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 4ef, the increasingly noticeable difference is that the uniform noise yields a more confined exploration of phase-space than the Gaussian distribution does, consistent with the depression in the fourth moment seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 5, we show the results for a time step, Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='975, very close to the stability limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This extreme case shows that the uniform distribution in noise also can produce a near uniform distribution for rn, while the distribution for un+ 1 2 is near triangular, consistent with a sum of two uniformly distributed num- bers contributing to the kinetic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Again, we see from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 5ef that the sampling of the phase space is much more limited when using uniform noise than when using Gaussian, even if the measured temperature, configurational as well as kinetic, are measured to the correct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Following the spirit of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [15, 16] we explore the application of a more challenging noise distribution βn P (the peaked distribution defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (53a)), which is both discrete and asymmetric (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As was the case for the uniform noise discussed above, we also here reference the results next to those of Gaussian noise, which give time step independent and correct behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Conducting numerical simulations of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (6) and (17) for different time steps and for normalized friction α/mΩ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 5 as described above, and comparing to the evaluation of the moments from the analyses of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III, we obtain the data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6, where the simulation results are indistinguishable from the results of the analyses for third and fourth moments in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III B and III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As expected from the analysis, the results of the second moments shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6ab of rn and un+ 1 2 , namely the potential and kinetic energies, are perfectly aligned with statistical mechanics, since the peaked distribution for βn P satisfies the two moments in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Since the applied noise is here asymmetric, the resulting third moments of rn and un+ 1 2 may also be- come non-zero for non-zero time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This is seen in Figures 6cd, where we observe rather complex behavior as a function of the system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We do, however, see that for Ω0∆t → 0 the third moments approach zero, consistent with the central limit theorem that guarantees a Gaussian outcome when a very large number of noise values contribute to the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Finally, in compari- son with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 2cd for uniform noise, we see very similar behavior for the energy fluctuations (fourth moments) for the application of the peaked distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Even if the uniform and peaked distributions are very dif- ferent in appearance, the similarities between their out- comes in their fourth moments are not surprising, given that the analysis for the fourth moments in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' III C shows that the difference between the two only depends on the kurtoses, which have the values Γ4 = 9 5 (uniform) and Γ4 = 12 5 (peaked).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Again, we find that all signa- tures of non-Gaussian noise vanish for Ω0∆t → 0, and we find that non-zero time steps result in depressions of the thermodynamically important energy fluctuations, as derived from the fourth moments of the configurational and kinetic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The details of the simulated coordinate distributions arising from the peaked noise are exemplified in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 7- 9 for α/mΩ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5 and Ω0∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='975, respec- 13 tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' For the smaller of the time steps, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 7, we see the seemingly modest skewness and deformations of the coordinate distributions arising from the peaked noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Yet, these modest deformations are what provide the somewhat significant deviations from Gaussian char- acteristics in third and fourth moments seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6cd and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6ef for Ω0∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' More dramatic deviations from Gaussian/Boltzmann characteristics are found in a large range of Ω0∆t, including the value Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='5 shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The seemingly discontinuous distribu- tion is not a result of insufficient statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Rather, it is the interference between the discrete nature of the ap- plied noise distribution with the discrete time step that happens to distinctively select certain preferred values of rn and un+ 1 2 over others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' It is noticeable that these types of peculiar distributions appear without any obvious or abrupt signatures in the first four moments, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As the time step Ω0∆t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='975 is pushed close to the stability limit, we again find smooth coordinate dis- tributions, seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 9, where the applied peaked noise is visible throughout the different displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' In analogy with the visualization of the distributions from the uni- form noise, this is the limit where the friction per time step is relatively large, thereby making the behavior of the coordinates rn and un+ 1 2 subject to only a few noise contributions at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' DISCUSSION In light of recent advances in stochastic thermostats for simulating Langevin equations with accurate statis- tics across the stability range of the applied time step when using Gaussian noise [7, 8], we have revisited the investigations of advantages and disadvantages of using non-Gaussian thermal noise in discrete time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Given the systemic first and second order time step errors of the traditional methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [18, 20]), which necessitated rather small time steps for accurate simulations, it was previously concluded [15, 16] that other distributions, including the desirable uniform distribution, would be efficient substitutions since the central limit theorem would ensure near-Gaussian outcomes for small time steps, thereby not further significantly distort the simulation results due to the imperfect noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' This result has been a very useful tool over the years when computational efficiency could benefit from not converting stochastic variables from uniform to Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' However, as we have demonstrated in this paper, when the time step becomes large, which is allowed by the modern GJ methods when using Gaussian noise, the application of non-Gaussian noise does not retain the time step independent benefits of these methods in thermodynamic measures that involve moments higher than the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' We have found that, given that the applied noise conforms to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (10), the GJ methods are invariant to the specific noise distributions in measures of first and second moments, such as configurational and kinetic temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Thus, it is deceiving to judge the quality of the thermostat based on those moments alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' As is evident from the third and fourth moments, as well as the visual impressions of the actual distributions of rn and un+ 1 2 , the sampling of the phase-space can be quite distorted (non-Boltzmann) even if the measured temperatures yield correct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' All of these results are consistent with the significance of how the noise must be defined in discrete-time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' namely through the integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' (9), which ensures that any underlying distribution for β(t) will yield a Gaussian outcome for βn due to the central limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' It follows that applying any other distribution than Gaussian in discrete time is formally invalid, and certainly leads to significant sampling errors unless α∆t/m is small enough that the one-time-step attenua- tion parameter c2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' It is therefore the conclusion of this work that Gaussian noise must be applied to the modern stochastic integrators if one wishes to take ad- vantage of the large time step benefits of their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' ACKNOWLEDGMENTS The author is grateful to Charlie Sievers for sharing LAMMPS simulations that support the results of this paper in more complex MD simulations using the GJF- 2GJ method, and to Lorenzo Mambretti for assistance with the RANMAR random number generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' The au- thor is also grateful for initial discussions with Chungho Cheng on non-Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Allen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Tildesley, Computer 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Plimpton, Com- pute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' 271, 108171 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [23] See the LAMMPS documentation [30] for the use of fix langevin using the Schneider and Stoll thermostat of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' [24] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Weik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Weeber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Szuttor, K Breitsprecher, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' de Graaf, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Kuron, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Landsgesell, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Menke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Sean and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Holm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=', Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' Special Topics 227, 1789 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content=' See also https://espressomd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='io/ documen- tation for ESPResSo-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='0 Sec.' metadata={'source': 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http://lammps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='sandia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='gov/doc/Manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} +page_content='pdf, for the description of the “fix langevin” command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE3T4oBgHgl3EQf-AtL/content/2301.04821v1.pdf'} diff --git a/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf b/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f7fa01a3778ba74e6341a8ded3aaa58d51d2ce26 --- /dev/null +++ b/utAzT4oBgHgl3EQfBvrw/content/2301.00949v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c959edfcf47ac9678126ebb56a9f59c4deee3b353b29c4bd094b455c85368f4 +size 331615 diff --git a/utAzT4oBgHgl3EQfBvrw/vector_store/index.pkl b/utAzT4oBgHgl3EQfBvrw/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7c71f8bc2771ed4ee27528771033d4f6e7c3877b --- /dev/null +++ b/utAzT4oBgHgl3EQfBvrw/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:19ae8d37b0f45fc1a8d521b0039876e1db6b91971806a57871a0ecd610c63e6d +size 40244 diff --git a/vdE0T4oBgHgl3EQfsgFD/content/tmp_files/2301.02579v1.pdf.txt b/vdE0T4oBgHgl3EQfsgFD/content/tmp_files/2301.02579v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ded5e2e73bdd3a1fcfb6c8d51955102b5eb68e7c --- /dev/null +++ b/vdE0T4oBgHgl3EQfsgFD/content/tmp_files/2301.02579v1.pdf.txt @@ -0,0 +1,616 @@ +arXiv:2301.02579v1 [gr-qc] 6 Jan 2023 +Singularities from Hyperentropic regions using the Quantum Expansion +Vaibhav Kalvakota∗ +Turito Institute, 500081, Hyderabad, India +(Dated: January 9, 2023) +A recent paper [1] put forward a theorem showing that hyperentropic surface would result in +incomplete null generators for a null hypersurface emanating from the surface provided it satisfies +the null curvature condition and the spacetime is globally hyperbolic. In this paper, we will put +forward a version of this theorem using the quantum expansion in place of the classical expansion, +and we will discuss the quantum focusing conjecture in this regard. +We propose a quantum version of the theorem [1], +which can be stated as: +Theorem 1. Let (M, g) be a globally hyperbolic space- +time with a codimension 2 surface I with a compact +boundary ∂I. If S(I) > +A(∂I) +4Gℏ +and if the expansion is +negative along the future ingoing null congruence that is +orthogonal to ∂I under the satisfaction of the null curva- +ture condition, then at least one null generator is incom- +plete. +The Bousso bound [2, 3] is the statement that for a +surface with at least one of the four principle null surface- +orthogonal congruences having a negative expansion, the +entropy bound on the surface is given by the area of the +boundary of the surface (when there exist more than one +light-sheet, we simply add in those factors – the maximal +form of the inequality is with S ≤ A): +S(σ) ≤ A(∂σ) +4Gℏ . +(1) +This is interpreted as the entropy that is bound through +the light-sheet, and since the light-sheet is compact (ter- +minating either at a focal point or a caustic) [2], the +boundary of the light-sheet is the boundary of the sur- +face itself. In the paper [1], Bousso and Moghaddam in- +troduced a classical singularity theorem that states that +if a surface has an entropy greater than what the Bousso +bound allows, i.e. if +S(I) > A(∂I) +4Gℏ , +(2) +then under theorem 1, the region (called a hyperentropic +region) must necessarily have at least one incomplete +null generator provided the surface is at least future +marginally trapped. This is an interesting result particu- +larly since this can be interpreted as a singularity arising +because of too much information in a given surface. A +possible converse of this theorem will be discussed in an +upcoming paper, which would hint at the existence of a +hyperentropic region in a spacetime with a null geodesic +incompleteness. As we will state and prove in this paper, +the result in theorem 1 has implications from the gener- +alized second law, which can be stated as the following +variation condition +δSgen(σ, Σ) > 0, +(3) +where Sgen is the generalized entropy for a codimension 2 +surface σ in a Cauchy slice Σ. Throughout this paper, by +a surface I we necessarily imply a codimension 2 surface +that has at least one congruence with a negative expan- +sion, while by σ we will imply a codimension 2 surface +without such restriction. +∂I +I +L +The theorem we wish to prove in relation to theorem +1 is as follows, where Θ is the quantum expansion: +Theorem 2. Let a surface I have a boundary ∂I that is +a compact codimension 2 submanifold with splitting, and +let Θ+ +l < 0. If the surface is hyperentropic, there exists +at least one null incomplete generator. +We will now discuss some definitions and propositions +that will help us in reviewing this theorem and discuss +the implications of the proof [1]. +Convention 1. A spacetime (M, g) is said to be globally +hyperbolic if J+(p) ∩ J−(q) is compact for p, q ∈ M, and +if there exists a Cauchy surface Σ in (M, g). +We will +assume throughout this paper that (M, g) is globally hy- +perbolic. To indicate the boundary of a surface I in Σ, +we use ∂I, and to indicate the boundary of a surface in +(M, g), we will use ˙I [4]. +Convention 2. By a boundary ∂σ we mean a compact +Cauchy splitting codimension 2 submanifold for a surface +σ that defines an interior and exterior for the Cauchy +slice Σ and σ − ∂σ ̸= ∅. For σ = I, we would mean a +surface for which at least the future ingoing congruence of +the principle null I−orthogonal directions has a negative +expansion. +Definition 1. By the future (resp. past) domain of depen- +dence of a set A ⊂ M, we mean the set of points p ∈ M + +2 +such that every past (resp. future) inextendible timelike +curve γ passing through p also necessarily intersects A. +The domain of dependence D(A) = D+(A) ∪ D−(A). +Definition 2. By a light-sheet L for a surface I, we mean a +null hypersurface formed by all null geodesics originating +orthogonal to I such that d±θ± +L ≤ 0 and for which I ⊂ +∂L. +Definition 3. (Null curvature condition): If (M, g) is such +that for all null vectors k we have +Rµνkµkν ≥ 0, +(4) +then we say that M satisfies the null curvature condition. +Note that, in itself the null curvature condition does +not necessarily imply the Bousso bound – rather, the role +of this condition is to ensure that given a non-positive +expansion θ0, under the Raychaudari equation +dθ +dλ = −1 +2θ2 − 8πGTµνkµkν − σµνσµν + ωµνωµν, +(5) +(where the vorticity tensor vanishes since we are consider- +ing orthogonal null congruences) the variation of the ex- +pansion will be non-positive throughout the light-sheet. +Definition 4. The Bousso bound states that for a surface I +with a light-sheet L such that L does not terminate before +a focal point (i.e. there are no dense regions contributing +to a closing or terminating light-sheet before a point b on +L as defined below), if ∂I = ∂L, then the entropy bound +by the light-sheet in terms of the boundary ∂I is given by +(1). +Lemma 3. If a point p ∈ M is on the future boundary of +a surface I, then the following conditions must hold [5]: +1. there are no intersecting null geodesics before p, +2. p lies on a null orthogonal geodesic γ emanating +from ∂I, and +3. there are no conjugate points on γ before p. +In this paper, we will assume that there are no caus- +tics interfering with the light-sheet. Therefore, the light- +sheet would normally close off if the region were non- +hyperentropic. We now turn our attention towards some +lemmas that will prove theorem 1, which will further help +us in stating and proving theorem 2. +Lemma 4. Let X denote the complement of I in Σ. +Then, D+(I) = D+(Σ) − I+(X) − X. Further, D(I) = +M − I(X). +Proof. Let a point p ∈ M be p ∈ D+(Σ) − I+(X) − X. +Then, a past inextendible timelike curve passing through +p would necessarily be in D+(Σ), while the subtracted +I+(X) − X would mean that such a curve would not +intersect X. Due to this, such curves would intersect I, +and therefore p ∈ D+(I). +Similarly, the same can be +said for the past domain of dependence. Due to this, we +can state that the domain of dependence of I is M − +I(X). +Lemma 5. X has the property that ˙I+(X) − X = +˙I+(∂X) − I+(X) − X. +The proof of this lemma can be read from [1], which +we will not reproduce here. We will instead use this to +state the following lemma: +Lemma 6. ˙I+(X) − X is the future outgoing null con- +gruence from I and terminates at either a caustic or at +a point where neighbouring null geodesics intersect. +Convention 3. We will adopt the following convention: +future (resp. past) outgoing null orthogonal congruence +will be labelled as K±, while the future (resp. past) ingo- +ing null orthogonal congruence will be labelled as L±. +Proof. This follows from [5], where naturally the null +geodesics that compose the future outgoing congruence +do not enter I+(X) (i.e. exist the future boundary of +K+(I)) unless we encounter a caustic or self-intersection. +Due to this and the previous lemma, we can state that +the set ˙I+(X) − X is generated by the future null or- +thogonal geodesics originating from ∂I. +The proof of theorem 1 is as follows: +Proof. The light-sheet L+ = ˙I+(X)−X is a surface such +that the boundary ∂L = ∂I, and from lemma 6 this +would be a null hypersurface. Further, since the expan- +sion condition θ < 0 holds on ∂I along L, due to the +null curvature condition it must be negative throughout +the hypersurface (before encountering a caustic), imply- +ing L is a light-sheet. If we assume that all generators +of L are necessarily complete, it must also be true that +the light-sheet is compact, since the light-sheet gener- +ators must leave L after a fixed rescaled affine length, +implying a closed L. Now, since ∂L = ∂I, the domain +of dependence D(L) = D(I) (from proposition 10 in [1]) +and therefore the Bousso bound would be contradicted, +since by assumption the region I is hyperentropic, and +therefore the light-sheet must not close to preserve the +Bousso bound, contradicting our assumption that all the +null generators of L are necessarily complete. +It should be possible to extend this result into the +quantum limit, where the expansion takes a slightly dif- +ferent meaning. In the classical limit, one would expect +the expansion to be the measure of the rate of change of +area via deformations, while in the quantum regimes, one +would expect this to be a measure of the rate of change +of generalized entropy [6] via deformations: +Θ = lim +A→0 +4Gℏ +A +dSgen(σ, Σ) +dλ +|x∈σ. +(6) + +3 +In general, surfaces are defined as: +θ+ +k θ+ +l +Q Trapped +− +− +Q marginally trapped 0 +− +Q Untrapped ++ +− +Q Extremal +0 +0 +Before we prove the theorem 2, we will first look at +Aron Wall’s result [7] from generalized entropy, which +goes analogously to that of Penrose’s theorem. In fact, +Penrose’s statement can be written into stating that if +one has a non-compact region with a compact boundary +with the expansion along at least one future null surface +orthogonal congruence being negative, due to global hy- +perbolicity of (M, g) a non-compact slice cannot evolve +into a compact slice, which is a topological constraint +using which we state that there has to exist some null +geodesic incompleteness. That is, if we have a compact +splitting boundary with the surface compact and if the +expansion condition holds, then we can state that the +light-sheet would form L = ∂+D(I), and then at least +one generator of this must be incomplete. This is essen- +tially the backing to the theorems discussed previously +– we can consider L to be homeomorphically identified +to I, and therefore if L is compact, I must be compact +as well. If all the congruences have a negative expan- +sion, we call the surface I a trapped surface. Similarly, +one defines a quantum trapped surface as follows: let a +compact I split the Cauchy slice Σ into an interior and +an exterior region. Then, the future boundary of the do- +main of dependence of the exterior Ext(I) is defined as +a null surface K+ constructed from future outgoing light +rays from I. If there is a way to evolve the slice so that +the fine-grained generalized entropy is decreasing along +L, then we say that I has a negative quantum expan- +sion throughout L under the quantum focusing conjec- +ture. Wall’s semiclassical theorem [7] can be stated as +the following for the future ingoing null congruence with +a negative expansion: +Theorem 7. Let I be a non-compact surface with a com- +pact boundary ∂I as considered previously. Then, if the +quantum expansion Θ(l) < 0, then there exists at least +one incomplete null generator. +Proof. This is similar to the classical Penrose’s theorem +discussed above. Since the generalized entropy is decreas- +ing, the compactness of L in terms of the generators can +be found. Defining a timelike vector field in (M, g), every +curve intersecting L would also intersect I under a con- +tinuous 1 − 1 mapping ϕ : L → I (this can be replaced +by a non-compact Cauchy surface to get Penrose’s theo- +rem). Since the image ϕ(L(I)) would also be compact as +L is compact, this would not cover all of I, since this is +non-compact. Due to this, there must exist at least one +incomplete null generator. +We now prove theorem 2 with respect to the quantum +focusing condition. +Proof. Since the generalized entropy on the null hyper- +surface L is decreasing, the generator segments on L must +terminate at some finite affine value. Here on, the proof +will take up a similar structure as of the original theorem +proof, the similarity being that in the classical version +the classical expansion is negative, while in this case the +quantum expansion being negative implies a decreasing +generalized entropy. Note that this would imply a vio- +lation of the generalized second law when the surface in +question is a horizon, in which case we would have to +reconsider the entire scheme. However, we are consid- +ering this surface to simply have the future ingoing null +orthogonal congruence expansion to be negative, and we +do not make statements about the outgoing congruence. +Similarly to the classical expansion, the quantum ex- +pansion being negative implies that the generators must +leave L after some finite affine length, which would again +imply a closed L. Since the boundaries of L as well as +that of I are the same, the domains of dependence be- +come D(L) = D(I) [1], and due to this the entropies +must satisfy S(L) = S(I). Since S(I) was assumed to be +hyperentropic, this would imply a violation of the Bousso +bound (in a sense we will discuss below), and therefore +the light-sheet cannot close off. +From this result one could argue that the entropy con- +tained in a surface with a compact boundary has to be +in satisfaction with the Bousso bound, and the ”over- +flow” of information in a region of spacetime affects the +geodesics passing through the surface. The generalized +entropy perspective reduces to the classical expansion, +stating that the null hypersurface L is compact. +The implications of this theorem are straightforward +– let g0 be a point on a null generator of the null hy- +persurface L in consideration. Then, if the semiclassical +approximation is valid around g0, and if the surface I is +hyperentropic, then by evolving I in time so that the gen- +eralized entropy on the null hypersurface L is decreasing, +the generalized second law would require that at least one +null generator of L is incomplete. If this null hypersur- +face were a horizon, the GSL would be violated [7]. This +can be viewed in the cosmological perspective by iden- +tifying a compact region such that the relative scaling +between A(I) and S(I) is such that the region becomes +hyperentropic, such as in the case of an flat expanding +cosmology (refer to figure below). If I were non-compact, +this would imply a null geodesic incompleteness directly +from theorem 7, and the past light-sheet would run into a +big bang singularity. In the semiclassical result theorem +2 we get a similar result, for a case when the semiclassical +approximation holds for a generator point g0. + +4 +I2 +∂I2 +L− (∂I2) +∂I1 +I1 +L−(∂I1) +It must be noted that the quantum expansion is sat- +isfying the quantum focusing condition [8], namely that +analogously to that of the classical expansion, under in- +finitesimal deformations of a surface along a null ray the +quantum expansion cannot increase: +δ +δV (y)Θ [V (y); y1] ≤ 0. +(7) +Here, the positive function V (y) defines the null hyper- +surface L with y ∈ I. Then, we can define the quan- +tum expansion Θ[V (y); y1] as in equation (6), in terms +of another point y1 around which we are considering the +”patch” area A. Then, the quantum Bousso bound [9, 10] +is the statement that along a light-sheet, the generalized +entropy must be monotonically decreasing if the quantum +expansion is negative. Finding a surface σ′, we require +that the quantum expansion does not become positive, +and that the generalized entropy is decreasing through- +out the hypersurface. Then, defining the generalized en- +tropy Sgen(σ, Σ) as [8]: +Sgen(σ, Σ) = A(σ) +4Gℏ + Sext(σ, Σ), +(8) +we can say that at σ′, we would have +∆Sext(σ, σ′) ≤ ∆A(σ, σ′) +4Gℏ +. +(9) +This would imply a quantum null energy condition, which +states that following holds: +⟨Tkk⟩ ≥ lim +A→0 +ℏ +2πA +d2Sext +dλ2 . +(10) +It must be noted here that the quantum focusing condi- +tion in itself makes not much difference to the proof in +the theorem 2 than the assumption that the generalized +entropy must be decreasing and must terminate the null +generator segments at some affine value. However, in un- +derstanding the effect of the generalized second law as a +constraint on the topological conditions on (M, g), this +has a very significant role. For instance, one would expect +the generalized entropy to be in terms of the QFC, which +has been shown in the case of holographic screens (or +marginally trapped tubes) in [11–13]. We have not consid- +ered the future outgoing null congruence in a strict sense +– we have only made statements about the lµ congruence. +If we consider the outgoing congruence to have a vanish- +ing expansion, we would refer to a marginally trapped +surface. In this case, we would expect the spacetime to +contain a horizon, under which we can construct a holo- +graphic screen (quantum holographic screen if the sur- +faces are quantum marginally trapped surfaces), which +would equip the codimension 1 surface formed by the fo- +liation of these marginally trapped surfaces with an area +law that implies a generalized second law. This is based +off the QFC, which ensures the GSL is implied through- +out the foliation of marginally trapped surfaces (called +”leaves”). In a paper on the use of holographic screens +in cosmological evolution [14], it was shown that the holo- +graphic screens form of the area law predicts that the late +evolved states of cosmologies can be found to be that of +de Sitter spacetime, implying the cosmic no-hair theo- +rem. Further, in the case of a quantum trapped surface, +we can invoke Wall’s theorem [7] to find out null geodesic +incompleteness in the spacetime. One can define Wall’s +version directly in terms of Θ, which would have the same +effect as stating that the fine-grained Sgen(σ, Σ) is de- +creasing, but with the constraint that this variation is +monotonic, which we can impose via the QFC. In fact, +Wall’s semiclassical generalization can be extended from +the non-compact case of I to a compact case under the +assumption that along the K congruence the expansion +is also negative. Then, we have the following theorem [7]: +Theorem 8. If a globally hyperbolic spacetime contains a +surface K as mentioned above and there is a non-compact +Cauchy surface H, then there exists a null geodesic in- +completeness. +Proof. The proof of this theorem is rather straightfor- +ward, with mostly the natural proof of the Penrose theo- +rem proving that such a spacetime contains a geodesic in- +completeness. Since the fine-grained generalized entropy +is decreasing on the null hypersurface, the null genera- +tor segments must terminate at some affine value λ. The +next parts of the proof are directly that of Penrose’s, and +the topological constraint is that a non-compact slice Σ +must not evolve into a compact K due to global hyper- +bolicity. Since I is a trapped surface both the future null +orthogonal congruence hypersurfaces must be compact +(one can define this in terms of the quantum expansion, +as we will see in the quantum expansion discussion), and +by defining a timelike vector field on (M, g), one can see +that the corresponding continuous 1−1 mapping defined +by the boundary of the causal domain J(I) would have +to be compact, but this would mean that there would be +a boundary of the image ϕ( ˙J+(I)) in H, since H is a +non-compact Cauchy surface [4]. Due to this, there must +exist an incomplete null geodesic. + +5 +Under the quantum Bousso bound, +we have the +condition that the variation of the exterior entropy +∆Sext(σ, σ′) is always less than or equal to one-fourth +of the variation of the interior entropy. If the light-sheet +was allowed to terminate when I is a hyperentropic re- +gion, we would have a violation of the Bousso bound. +Naturally, the quantum focusing condition can be inte- +grated to get an entropy bound on the light-sheet. Due +to this, it must be necessary to condition singularities +in cases of hyperentropic regions so as to prevent a vio- +lation of the Bousso bound. The null energy condition +was implied to ensure that the classical expansion is non- +positive throughout L when θ0(∂I) ≤ 0, so that the only +cases where the light-sheet can terminate are at caustics +or singularities. +Remarks: In this paper, we have put forward a quan- +tum version of the theorem introduced in [1]. We have +discussed and proved the classical and quantum versions +of the theorem. We have shown that the result by Bousso +and Moghaddam puts forward geodesic incompleteness +as a response of spacetime to hyperentropic regions, and +discussed the generalized entropy as an analogous tool +in showing the compactness and the equivalence of the +light-sheet entropy bound to the entropy intersecting the +surface. If the light-sheet closes off, the curves intersect- +ing the surface would also intersect L, and by a homeo- +morphic mapping one could say that the timelike curves +would also intersect L, which would violate the Bousso +bound. Furthermore, the role of the quantum focusing +conjecture in singularity theorems is an interesting one, +where the role of the classical expansion is replaced by +a stronger condition on the generalized entropy of the +surface. It will be of significance to see how the exterior +entropy affects the overall geometry in such singularity +theorems, and how the quantum focusing condition gov- +erns scenarios like non-locality. +∗ vaibhavkalvakota@icloud.com +[1] R. Bousso and A. Shahbazi-Moghaddam, Singulari- +ties from Entropy, Phys. Rev. Lett. 128, 231301 (2022), +arXiv:2201.11132 [hep-th]. +[2] R. +Bousso, +A +Covariant +entropy +conjecture, +JHEP 07, 004, arXiv:hep-th/9905177. +[3] E. E. Flanagan, D. Marolf, and R. M. Wald, Proof of +classical versions of the Bousso entropy bound and of the +generalized second law, Phys. Rev. D 62, 084035 (2000), +arXiv:hep-th/9908070. +[4] S. +W. +Hawking +and +G. +F. +R. +Ellis, +The Large Scale Structure of Space-Time, +Cambridge +Monographs +on +Mathematical +Physics +(Cambridge +University Press, 2011). +[5] C. +Akers, +R. +Bousso, +I. +F. +Halpern, +and +G. +N. +Remmen, +Boundary +of +the +future +of +a +surface, +Phys. Rev. D 97, 024018 (2018), +arXiv:1711.06689 [hep-th]. +[6] J. +D. +Bekenstein, +Generalized +second +law +of +thermodynamics +in +black +hole +physics, +Phys. Rev. D 9, 3292 (1974). +[7] A. +C. +Wall, +The +Generalized +Second +Law +implies +a +Quantum +Singularity +Theorem, +Class. Quant. Grav. 30, 165003 (2013), +[Erra- +tum: +Class.Quant.Grav. +30, +199501 +(2013)], +arXiv:1010.5513 [gr-qc]. +[8] R. +Bousso, +Z. +Fisher, +S. +Leichenauer, +and +A. +C. +Wall, +Quantum +focusing +con- +jecture, +Phys. Rev. D 93, 064044 (2016), +arXiv:1506.02669 [hep-th]. +[9] A. +Strominger +and +D. +M. +Thompson, +A +Quan- +tum +Bousso +bound, +Phys. Rev. D 70, 044007 (2004), +arXiv:hep-th/0303067. +[10] R. +Bousso, +H. +Casini, +Z. +Fisher, +and +J. +Maldacena, +Proof +of +a +Quantum +Bousso +Bound, +Phys. Rev. D 90, 044002 (2014), +arXiv:1404.5635 [hep-th]. +[11] R. Bousso and N. Engelhardt, New Area Law in +General Relativity, Phys. Rev. Lett. 115, 081301 (2015), +arXiv:1504.07627 [hep-th]. +[12] R. Bousso and N. Engelhardt, Proof of a New Area Law +in General Relativity, Phys. Rev. D 92, 044031 (2015), +arXiv:1504.07660 [gr-qc]. +[13] R. Bousso and N. Engelhardt, Generalized +Second +Law +for +Cosmology, +Phys. Rev. D 93, 024025 (2016), +arXiv:1510.02099 [hep-th]. +[14] S. M. Carroll and A. Chatwin-Davies, Cosmic Equilibra- +tion: A Holographic No-Hair Theorem from the Gen- +eralized Second Law, Phys. Rev. D 97, 046012 (2018), +arXiv:1703.09241 [hep-th]. + diff --git a/vdE0T4oBgHgl3EQfsgFD/content/tmp_files/load_file.txt b/vdE0T4oBgHgl3EQfsgFD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b01eb428b684d3e058db16f763dc745a9121bb02 --- /dev/null +++ b/vdE0T4oBgHgl3EQfsgFD/content/tmp_files/load_file.txt @@ -0,0 +1,257 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf,len=256 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content='02579v1 [gr-qc] 6 Jan 2023 Singularities from Hyperentropic regions using the Quantum Expansion Vaibhav Kalvakota∗ Turito Institute, 500081, Hyderabad, India (Dated: January 9, 2023) A recent paper [1] put forward a theorem showing that hyperentropic surface would result in incomplete null generators for a null hypersurface emanating from the surface provided it satisfies the null curvature condition and the spacetime is globally hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In this paper, we will put forward a version of this theorem using the quantum expansion in place of the classical expansion, and we will discuss the quantum focusing conjecture in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We propose a quantum version of the theorem [1], which can be stated as: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Let (M, g) be a globally hyperbolic space- time with a codimension 2 surface I with a compact boundary ∂I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If S(I) > A(∂I) 4Gℏ and if the expansion is negative along the future ingoing null congruence that is orthogonal to ∂I under the satisfaction of the null curva- ture condition, then at least one null generator is incom- plete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The Bousso bound [2, 3] is the statement that for a surface with at least one of the four principle null surface- orthogonal congruences having a negative expansion, the entropy bound on the surface is given by the area of the boundary of the surface (when there exist more than one light-sheet, we simply add in those factors – the maximal form of the inequality is with S ≤ A): S(σ) ≤ A(∂σ) 4Gℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' (1) This is interpreted as the entropy that is bound through the light-sheet, and since the light-sheet is compact (ter- minating either at a focal point or a caustic) [2], the boundary of the light-sheet is the boundary of the sur- face itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In the paper [1], Bousso and Moghaddam in- troduced a classical singularity theorem that states that if a surface has an entropy greater than what the Bousso bound allows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' if S(I) > A(∂I) 4Gℏ , (2) then under theorem 1, the region (called a hyperentropic region) must necessarily have at least one incomplete null generator provided the surface is at least future marginally trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' This is an interesting result particu- larly since this can be interpreted as a singularity arising because of too much information in a given surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' A possible converse of this theorem will be discussed in an upcoming paper, which would hint at the existence of a hyperentropic region in a spacetime with a null geodesic incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' As we will state and prove in this paper, the result in theorem 1 has implications from the gener- alized second law, which can be stated as the following variation condition δSgen(σ, Σ) > 0, (3) where Sgen is the generalized entropy for a codimension 2 surface σ in a Cauchy slice Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Throughout this paper, by a surface I we necessarily imply a codimension 2 surface that has at least one congruence with a negative expan- sion, while by σ we will imply a codimension 2 surface without such restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' ∂I I L The theorem we wish to prove in relation to theorem 1 is as follows, where Θ is the quantum expansion: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Let a surface I have a boundary ∂I that is a compact codimension 2 submanifold with splitting, and let Θ+ l < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If the surface is hyperentropic, there exists at least one null incomplete generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We will now discuss some definitions and propositions that will help us in reviewing this theorem and discuss the implications of the proof [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Convention 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' A spacetime (M, g) is said to be globally hyperbolic if J+(p) ∩ J−(q) is compact for p, q ∈ M, and if there exists a Cauchy surface Σ in (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We will assume throughout this paper that (M, g) is globally hy- perbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' To indicate the boundary of a surface I in Σ, we use ∂I, and to indicate the boundary of a surface in (M, g), we will use ˙I [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Convention 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' By a boundary ∂σ we mean a compact Cauchy splitting codimension 2 submanifold for a surface σ that defines an interior and exterior for the Cauchy slice Σ and σ − ∂σ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' For σ = I, we would mean a surface for which at least the future ingoing congruence of the principle null I−orthogonal directions has a negative expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' By the future (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' past) domain of depen- dence of a set A ⊂ M, we mean the set of points p ∈ M 2 such that every past (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' future) inextendible timelike curve γ passing through p also necessarily intersects A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The domain of dependence D(A) = D+(A) ∪ D−(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' By a light-sheet L for a surface I, we mean a null hypersurface formed by all null geodesics originating orthogonal to I such that d±θ± L ≤ 0 and for which I ⊂ ∂L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' (Null curvature condition): If (M, g) is such that for all null vectors k we have Rµνkµkν ≥ 0, (4) then we say that M satisfies the null curvature condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Note that, in itself the null curvature condition does not necessarily imply the Bousso bound – rather, the role of this condition is to ensure that given a non-positive expansion θ0, under the Raychaudari equation dθ dλ = −1 2θ2 − 8πGTµνkµkν − σµνσµν + ωµνωµν, (5) (where the vorticity tensor vanishes since we are consider- ing orthogonal null congruences) the variation of the ex- pansion will be non-positive throughout the light-sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The Bousso bound states that for a surface I with a light-sheet L such that L does not terminate before a focal point (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' there are no dense regions contributing to a closing or terminating light-sheet before a point b on L as defined below), if ∂I = ∂L, then the entropy bound by the light-sheet in terms of the boundary ∂I is given by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If a point p ∈ M is on the future boundary of a surface I, then the following conditions must hold [5]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' there are no intersecting null geodesics before p, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' p lies on a null orthogonal geodesic γ emanating from ∂I, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' there are no conjugate points on γ before p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In this paper, we will assume that there are no caus- tics interfering with the light-sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Therefore, the light- sheet would normally close off if the region were non- hyperentropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We now turn our attention towards some lemmas that will prove theorem 1, which will further help us in stating and proving theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Let X denote the complement of I in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, D+(I) = D+(Σ) − I+(X) − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Further, D(I) = M − I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Let a point p ∈ M be p ∈ D+(Σ) − I+(X) − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, a past inextendible timelike curve passing through p would necessarily be in D+(Σ), while the subtracted I+(X) − X would mean that such a curve would not intersect X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Due to this, such curves would intersect I, and therefore p ∈ D+(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Similarly, the same can be said for the past domain of dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Due to this, we can state that the domain of dependence of I is M − I(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' X has the property that ˙I+(X) − X = ˙I+(∂X) − I+(X) − X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The proof of this lemma can be read from [1], which we will not reproduce here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We will instead use this to state the following lemma: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' ˙I+(X) − X is the future outgoing null con- gruence from I and terminates at either a caustic or at a point where neighbouring null geodesics intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We will adopt the following convention: future (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' past) outgoing null orthogonal congruence will be labelled as K±, while the future (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' past) ingo- ing null orthogonal congruence will be labelled as L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' This follows from [5], where naturally the null geodesics that compose the future outgoing congruence do not enter I+(X) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' exist the future boundary of K+(I)) unless we encounter a caustic or self-intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Due to this and the previous lemma, we can state that the set ˙I+(X) − X is generated by the future null or- thogonal geodesics originating from ∂I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The proof of theorem 1 is as follows: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The light-sheet L+ = ˙I+(X)−X is a surface such that the boundary ∂L = ∂I, and from lemma 6 this would be a null hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Further, since the expan- sion condition θ < 0 holds on ∂I along L, due to the null curvature condition it must be negative throughout the hypersurface (before encountering a caustic), imply- ing L is a light-sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If we assume that all generators of L are necessarily complete, it must also be true that the light-sheet is compact, since the light-sheet gener- ators must leave L after a fixed rescaled affine length, implying a closed L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Now, since ∂L = ∂I, the domain of dependence D(L) = D(I) (from proposition 10 in [1]) and therefore the Bousso bound would be contradicted, since by assumption the region I is hyperentropic, and therefore the light-sheet must not close to preserve the Bousso bound, contradicting our assumption that all the null generators of L are necessarily complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' It should be possible to extend this result into the quantum limit, where the expansion takes a slightly dif- ferent meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In the classical limit, one would expect the expansion to be the measure of the rate of change of area via deformations, while in the quantum regimes, one would expect this to be a measure of the rate of change of generalized entropy [6] via deformations: Θ = lim A→0 4Gℏ A dSgen(σ, Σ) dλ |x∈σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' (6) 3 In general, surfaces are defined as: θ+ k θ+ l Q Trapped − − Q marginally trapped 0 − Q Untrapped + − Q Extremal 0 0 Before we prove the theorem 2, we will first look at Aron Wall’s result [7] from generalized entropy, which goes analogously to that of Penrose’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In fact, Penrose’s statement can be written into stating that if one has a non-compact region with a compact boundary with the expansion along at least one future null surface orthogonal congruence being negative, due to global hy- perbolicity of (M, g) a non-compact slice cannot evolve into a compact slice, which is a topological constraint using which we state that there has to exist some null geodesic incompleteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' That is, if we have a compact splitting boundary with the surface compact and if the expansion condition holds, then we can state that the light-sheet would form L = ∂+D(I), and then at least one generator of this must be incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' This is essen- tially the backing to the theorems discussed previously – we can consider L to be homeomorphically identified to I, and therefore if L is compact, I must be compact as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If all the congruences have a negative expan- sion, we call the surface I a trapped surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Similarly, one defines a quantum trapped surface as follows: let a compact I split the Cauchy slice Σ into an interior and an exterior region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, the future boundary of the do- main of dependence of the exterior Ext(I) is defined as a null surface K+ constructed from future outgoing light rays from I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If there is a way to evolve the slice so that the fine-grained generalized entropy is decreasing along L, then we say that I has a negative quantum expan- sion throughout L under the quantum focusing conjec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Wall’s semiclassical theorem [7] can be stated as the following for the future ingoing null congruence with a negative expansion: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Let I be a non-compact surface with a com- pact boundary ∂I as considered previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, if the quantum expansion Θ(l) < 0, then there exists at least one incomplete null generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' This is similar to the classical Penrose’s theorem discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since the generalized entropy is decreas- ing, the compactness of L in terms of the generators can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Defining a timelike vector field in (M, g), every curve intersecting L would also intersect I under a con- tinuous 1 − 1 mapping ϕ : L → I (this can be replaced by a non-compact Cauchy surface to get Penrose’s theo- rem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since the image ϕ(L(I)) would also be compact as L is compact, this would not cover all of I, since this is non-compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Due to this, there must exist at least one incomplete null generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We now prove theorem 2 with respect to the quantum focusing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since the generalized entropy on the null hyper- surface L is decreasing, the generator segments on L must terminate at some finite affine value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Here on, the proof will take up a similar structure as of the original theorem proof, the similarity being that in the classical version the classical expansion is negative, while in this case the quantum expansion being negative implies a decreasing generalized entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Note that this would imply a vio- lation of the generalized second law when the surface in question is a horizon, in which case we would have to reconsider the entire scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' However, we are consid- ering this surface to simply have the future ingoing null orthogonal congruence expansion to be negative, and we do not make statements about the outgoing congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Similarly to the classical expansion, the quantum ex- pansion being negative implies that the generators must leave L after some finite affine length, which would again imply a closed L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since the boundaries of L as well as that of I are the same, the domains of dependence be- come D(L) = D(I) [1], and due to this the entropies must satisfy S(L) = S(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since S(I) was assumed to be hyperentropic, this would imply a violation of the Bousso bound (in a sense we will discuss below), and therefore the light-sheet cannot close off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' From this result one could argue that the entropy con- tained in a surface with a compact boundary has to be in satisfaction with the Bousso bound, and the ”over- flow” of information in a region of spacetime affects the geodesics passing through the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The generalized entropy perspective reduces to the classical expansion, stating that the null hypersurface L is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The implications of this theorem are straightforward – let g0 be a point on a null generator of the null hy- persurface L in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, if the semiclassical approximation is valid around g0, and if the surface I is hyperentropic, then by evolving I in time so that the gen- eralized entropy on the null hypersurface L is decreasing, the generalized second law would require that at least one null generator of L is incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If this null hypersur- face were a horizon, the GSL would be violated [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' This can be viewed in the cosmological perspective by iden- tifying a compact region such that the relative scaling between A(I) and S(I) is such that the region becomes hyperentropic, such as in the case of an flat expanding cosmology (refer to figure below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If I were non-compact, this would imply a null geodesic incompleteness directly from theorem 7, and the past light-sheet would run into a big bang singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In the semiclassical result theorem 2 we get a similar result, for a case when the semiclassical approximation holds for a generator point g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' 4 I2 ∂I2 L− (∂I2) ∂I1 I1 L−(∂I1) It must be noted that the quantum expansion is sat- isfying the quantum focusing condition [8], namely that analogously to that of the classical expansion, under in- finitesimal deformations of a surface along a null ray the quantum expansion cannot increase: δ δV (y)Θ [V (y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' y1] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' (7) Here, the positive function V (y) defines the null hyper- surface L with y ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, we can define the quan- tum expansion Θ[V (y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' y1] as in equation (6), in terms of another point y1 around which we are considering the ”patch” area A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, the quantum Bousso bound [9, 10] is the statement that along a light-sheet, the generalized entropy must be monotonically decreasing if the quantum expansion is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Finding a surface σ′, we require that the quantum expansion does not become positive, and that the generalized entropy is decreasing through- out the hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, defining the generalized en- tropy Sgen(σ, Σ) as [8]: Sgen(σ, Σ) = A(σ) 4Gℏ + Sext(σ, Σ), (8) we can say that at σ′, we would have ∆Sext(σ, σ′) ≤ ∆A(σ, σ′) 4Gℏ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' (9) This would imply a quantum null energy condition, which states that following holds: ⟨Tkk⟩ ≥ lim A→0 ℏ 2πA d2Sext dλ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' (10) It must be noted here that the quantum focusing condi- tion in itself makes not much difference to the proof in the theorem 2 than the assumption that the generalized entropy must be decreasing and must terminate the null generator segments at some affine value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' However, in un- derstanding the effect of the generalized second law as a constraint on the topological conditions on (M, g), this has a very significant role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' For instance, one would expect the generalized entropy to be in terms of the QFC, which has been shown in the case of holographic screens (or marginally trapped tubes) in [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We have not consid- ered the future outgoing null congruence in a strict sense – we have only made statements about the lµ congruence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If we consider the outgoing congruence to have a vanish- ing expansion, we would refer to a marginally trapped surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In this case, we would expect the spacetime to contain a horizon, under which we can construct a holo- graphic screen (quantum holographic screen if the sur- faces are quantum marginally trapped surfaces), which would equip the codimension 1 surface formed by the fo- liation of these marginally trapped surfaces with an area law that implies a generalized second law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' This is based off the QFC, which ensures the GSL is implied through- out the foliation of marginally trapped surfaces (called ”leaves”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In a paper on the use of holographic screens in cosmological evolution [14], it was shown that the holo- graphic screens form of the area law predicts that the late evolved states of cosmologies can be found to be that of de Sitter spacetime, implying the cosmic no-hair theo- rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Further, in the case of a quantum trapped surface, we can invoke Wall’s theorem [7] to find out null geodesic incompleteness in the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' One can define Wall’s version directly in terms of Θ, which would have the same effect as stating that the fine-grained Sgen(σ, Σ) is de- creasing, but with the constraint that this variation is monotonic, which we can impose via the QFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' In fact, Wall’s semiclassical generalization can be extended from the non-compact case of I to a compact case under the assumption that along the K congruence the expansion is also negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Then, we have the following theorem [7]: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If a globally hyperbolic spacetime contains a surface K as mentioned above and there is a non-compact Cauchy surface H, then there exists a null geodesic in- completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The proof of this theorem is rather straightfor- ward, with mostly the natural proof of the Penrose theo- rem proving that such a spacetime contains a geodesic in- completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since the fine-grained generalized entropy is decreasing on the null hypersurface, the null genera- tor segments must terminate at some affine value λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The next parts of the proof are directly that of Penrose’s, and the topological constraint is that a non-compact slice Σ must not evolve into a compact K due to global hyper- bolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Since I is a trapped surface both the future null orthogonal congruence hypersurfaces must be compact (one can define this in terms of the quantum expansion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' as we will see in the quantum expansion discussion),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' and by defining a timelike vector field on (M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' g),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' one can see that the corresponding continuous 1−1 mapping defined by the boundary of the causal domain J(I) would have to be compact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' but this would mean that there would be a boundary of the image ϕ( ˙J+(I)) in H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' since H is a non-compact Cauchy surface [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Due to this, there must exist an incomplete null geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' 5 Under the quantum Bousso bound, we have the condition that the variation of the exterior entropy ∆Sext(σ, σ′) is always less than or equal to one-fourth of the variation of the interior entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If the light-sheet was allowed to terminate when I is a hyperentropic re- gion, we would have a violation of the Bousso bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Naturally, the quantum focusing condition can be inte- grated to get an entropy bound on the light-sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Due to this, it must be necessary to condition singularities in cases of hyperentropic regions so as to prevent a vio- lation of the Bousso bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' The null energy condition was implied to ensure that the classical expansion is non- positive throughout L when θ0(∂I) ≤ 0, so that the only cases where the light-sheet can terminate are at caustics or singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Remarks: In this paper, we have put forward a quan- tum version of the theorem introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We have discussed and proved the classical and quantum versions of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' We have shown that the result by Bousso and Moghaddam puts forward geodesic incompleteness as a response of spacetime to hyperentropic regions, and discussed the generalized entropy as an analogous tool in showing the compactness and the equivalence of the light-sheet entropy bound to the entropy intersecting the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' If the light-sheet closes off, the curves intersect- ing the surface would also intersect L, and by a homeo- morphic mapping one could say that the timelike curves would also intersect L, which would violate the Bousso bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Furthermore, the role of the quantum focusing conjecture in singularity theorems is an interesting one, where the role of the classical expansion is replaced by a stronger condition on the generalized entropy of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' It will be of significance to see how the exterior entropy affects the overall geometry in such singularity theorems, and how the quantum focusing condition gov- erns scenarios like non-locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' ∗ vaibhavkalvakota@icloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content='com [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdE0T4oBgHgl3EQfsgFD/content/2301.02579v1.pdf'} +page_content=' Bousso and A.' metadata={'source': 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Zihuai Lin, Senior Member, IEEE, Zhiyun Lin, and Branka Vucetic, Life Fellow, IEEE +Abstract—Emotion recognition or detection is broadly utilized +in patient-doctor interactions for diseases such as schizophrenia +and autism and the most typical techniques are speech detection +and facial recognition. However, features extracted from these +behavior-based emotion recognitions are not reliable since hu- +mans can disguise their emotions. Recording voices or tracking +facial expressions for a long term is also not efficient. Therefore, +our aim is to find a reliable and efficient emotion recognition +scheme, which can be used for non-behavior-based emotion +recognition in real-time. This can be solved by implement- +ing a single-channel electrocardiogram (ECG) based emotion +recognition scheme in a lightweight embedded system. However, +existing schemes have relatively low accuracy. For instance, the +accuracy is about 82.78% by using a least squares support vector +machine (SVM). Therefore, we propose a reliable and efficient +emotion recognition scheme—exploitative and explorative grey +wolf optimizer based SVM (X-GWO-SVM) for ECG-based emo- +tion recognition. Two datasets, one raw self-collected iRealcare +dataset, and the widely-used benchmark WESAD dataset are +used in the X-GWO-SVM algorithm for emotion recognition. +Leave-single-subject-out cross-validation yields a mean accuracy +of 93.37% for the iRealcare dataset and a mean accuracy of +95.93% for the WESAD dataset. This work demonstrates that +the X-GWO-SVM algorithm can be used for emotion recognition +and the algorithm exhibits superior performance in reliability +compared to the use of other supervised machine learning +methods in earlier works. It can be implemented in a lightweight +embedded system, which is much more efficient than existing +solutions based on deep neural networks. +Index Terms—Emotion recognition, IoT, Smart health, ECG +signals, GWO, SVM. +I. INTRODUCTION +T +HE use of the Internet of Things (IoT) is growing steadily +over the years. It is expected that by 2025, there will +be approximately 27 billion connected IoT devices [1]. At +present, the IoT is one of the main promoters of technological +innovation and one of the areas with greater potential for social +and economic transformation [2]–[4]. Through a network of +sensors and actuators connected to a wireless network [5]– +[14], the operator has the power to remotely gather data. +Alternatively, actuators could be programmed to actuate au- +tomatically according to values reported by the sensor. +Emotion recognition or detection based on IoT wireless +sensing and networking has gained lots of attention since it +Xucun +Yan, +Zihuai +Lin +and +Branka +Vucetic +are +with +School +of +Electrical +and +Information +Engineering, +University +of +Sydney, +New +South +Wales +2006, +Australia +(e-mail: +xucun.yan@sydney.edu.au; +zi- +huai.lin@sydney.edu.au; branka.vucetic@sydney.edu.au). +Zhiyun Lin is with the Department of Electrical and Electronic Engi- +neering, Southern University of Science and Technology, Shenzhen 518055, +China, and Peng Cheng Laboratory, Shenzhen 518066, China (email: +linzy@sustech.edu.cn). Corresponding author: Zhiyun Lin +can be broadly utilized in interfaces between humans and +computers and patient-doctor interactions for diseases such as +schizophrenia and autism. Most emotion detection methods +are based on behaviors such as speech detection and face +recognition [15], [16]. However, features extracted from the +abovementioned behavior-based emotion recognition are not +adequate for identifying emotions, because the behavior in- +duced by emotion can be disguised by artifacts of human +social masking [17]. For example, emotion recognition based +on facial expressions can be easily misled by a poker face. +Using physiological signals, such as electroencephalograms +(EEGs) [18]–[20], electromyograms (EMGs), and electrocar- +diograms (ECGs) [21], is an alternative to identify emotions +since physiological signals are one of the most notable means +to manifest the central nervous system in which emotions are +processed [22]. +Using physiological cues for emotion identification has two +advantages over prior approaches to emotion recognition. The +first is that physiological signals generated from automatic +reactions are difficult to disguise. The second is that wearable +emotion monitoring can continually record physiological in- +formation. This differs from the instance of voice recognition +where data may only be recorded when individuals are speak- +ing. However, using multi-channel biosignals to recognize hu- +man emotions is not suitable for practical applications because +subjects may be hindered during daily life activities [23]. It +has been proved that ECG signals are a suitable physiological +channel with acceptable recognition abilities [17]. +However, according to [24]–[27], the accuracy of emotion +detection based on a single ECG channel fluctuates a lot for +various datasets compared to that of other approaches such +as facial emotion recognition. On the one hand, recent efforts +in emotion recognition using ECG signals have largely relied +on relatively simple supervised learning techniques [28], such +as random forest (RF), support vector machine (SVM), K- +nearest neighbor (K-NN), etc. However, these methods have +relatively low accuracy (for instance, the accuracy is about +82.78% +[17] by using least squares SVM). On the other +hand, the current maximum level of single ECG channel- +based emotion recognition accuracy reaches 96.9% [29] for +Wearable Stress and Affect Detection (WESAD) database +and 88.2% [29] for a dataset for multi-modal research of +affect, personality traits, and mood in individuals and groups +(AMIGOS) [27], which utilizes self-supervised convolutional +neural network (CNN) model. Facial emotion recognition +accuracy achieves 92.07% [30] for MMI Facial Expression +Database and 94.91% [30] for the Japanese Female Facial +Expression Database, which uses CNN embedded with re- +arXiv:2301.01887v1 [eess.SP] 5 Jan 2023 + +2 +current neural network (RNN) [26]. Nevertheless, these deep +neural network-based techniques, e.g., CNN, RNN, etc, tend to +achieve high accuracy but are complex with low computation +efficiency, which cannot be implemented in a lightweight +embedded system operating in real-time. Therefore, seeking +a simple supervised learning scheme to accurately, stably, and +efficiently recognize emotions based on a single ECG channel +in a lightweight embedded system is necessary. +Towards this objective, this paper aims to develop a novel +exploitative and explorative GWO-SVM (X-GWO-SVM) for +ECG-based emotion recognition. The goal is to achieve good +classification accuracy (as high as utilizing complex neural +networks) while simultaneously reducing computation so that +it can be implemented in a lightweight embedded system. The +idea is motivated from the fact that the SVM algorithm can +be used to solve single-channel ECG-based emotion recogni- +tion issue with lightweight embedded system implementation. +However, the existing SVM works do not offer a good classi- +fication accuracy performance for ECG-based recognition due +to difficulties in finding appropriate hyper-parameters while +preventing overfitting of the training data. +In general, the selection of hyperparameters is a non-convex +optimization issue. Therefore, many heuristic algorithms such +as genetic algorithm (GA), particle swarm optimization (PSO), +and grey wolf optimizer (GWO) [31]–[34] are introduced to +tackle it. Compared with PSO and a set of search algorithms, +GWO provides better performance in computation reduction +(e.g., in feature subset selection [35]). Moreover, the GWO +approach has been demonstrated to be more stable against +initialization than PSO and GA [35]. However, as discussed +in [36], conventional GWO-based SVM (GWO-SVM) tech- +niques are still easy to fall into local solutions. +In this work, an improved method, the X-GWO-SVM +method, is proposed. The proposed X-GWO-SVM method +is the first to apply GWO-SVM idea to solve ECG-based +recognition, and as shown in this paper, this method has +higher recognition accuracy than existing SVM and PSO-SVM +techniques for ECG emotion recognition use. It can effectively +avoid the algorithm falling into a local solution by increasing +the exploration ability, and speed up the convergence by +increasing the exploitation ability. In this paper, two datasets, +one raw self-collected iRealcare dataset, and the widely used +benchmark WESAD dataset are used in the X-GWO-SVM +algorithm for emotion recognition. Leave-single-subject-out +cross-validation yields a mean accuracy of 93.37% and an F1- +score of 93.38% for the iRealcare dataset and a mean accuracy +of 95.93% and an F1-score of 95.56% for the WESAD dataset. +The main contributions of this paper are summarized as +follows: +1) We use a self-built wearable IoT ECG patch with only +one ECG channel to collect four emotions, i.e., happi- +ness, tension, peacefulness and excitement, by playing +different videos. +2) We designed a novel X-GWO-SVM algorithm to inter- +nally learn hyperparameters on SVM. It can effectively +avoid the algorithm falling into a local solution by +increasing the exploration ability, and speed up the +convergence by increasing the exploitation ability. +3) This novel X-GWO-SVM algorithm can accurately and +efficiently recognize emotions for single-channel ECG- +based signals and be implemented in a lightweight +embedded system operating in real-time. It improves +accuracy compared to existing simple machine learning +methods and dramatically reduces complexity compared +to some novel deep neural networks. Thus, the efficiency +is also increased compared to other time-consuming +emotion recognition methods. +The outline of the rest of the paper is given below. Section II +introduces our database and an expanded dataset. Our model +formulation is described in Section III. In Sections IV and Sec- +tionV, we present results and discussions, respectively, before +concluding with a discussion of potential future directions in +Section VI. +II. DATASET +ECG signals are composed of the P wave, T wave, and +QRS complex, which represent the three phases of an ECG +pulse. In atrial systole, the P wave is the contraction pulse. +The QRS complex signifies ventricular depolarization. The +T wave represents ventricular re-polarization [37]. An ECG +device records the electrical changes caused by the activities +of the heart, which are collected by electrodes over the skin +for a period of time. It has been proved that ECG signals are +a suitable physiological channel with acceptable recognition +abilities [17] to identify emotions. Therefore, single-channel +ECG signals are used in this study. In order to verify the +general representation ability of X-GWO-SVM, two datasets +of ECG signals are used, one raw self-collected iRealcare +dataset with 5 subjects and the other widely-used benchmark +WESAD dataset with 15 subjects. +A. Description of iRealcare dataset +Data collection is one of the most important steps for +emotion detection. The definition of different emotions must +be explicit in this phase. If the definition is not clear, confusion +may occur among different emotions in the classification +phase and the classification performance will be influenced +negatively. However, emotions normally instantaneously occur +and hold for a short period. The longer the period is, the more +irrelevant data is included in ECG signals. Thus, it is hard to +properly label the corresponding emotion class. +To avoid the aforementioned issue, we self-collect a dataset +with high quality and a short period for each emotion, making +sure accurate data collection and labeling processes. The +ECG signals are recorded by a low-cost wearable IoT ECG +patch, called iRealcare [38]–[42] with 128 Hz sampling rates. +The data collected by the iRealcare IoT ECG sensor can be +transmitted to a smartphone application (APP) via Bluetooth +Low Energy (BLE) and then to a cloud. From the cloud, we +can acquire the raw ECG signals. Signals are recorded for +four emotions including happiness, tension, peacefulness, and +excitement. Except for peacefulness, each emotion is generated +based on an external environmental stimulus, which is similar +to the published datasets stimulating subjects through audio +or video [43]. The peacefulness describes the normal state, + +3 +for which the ECG signals are recorded without any external +stimulus. Signals for happiness, tension, and excitement are +recorded when subjects watch comedies, watch thriller movies +and do exercises, respectively. Generally, the record duration +should be short as we discussed before. Therefore, the record +time is in a range of 3.22-6.16 minutes for each emotion. +It should be noticed that we only record the period that +subjects are actually in that emotion condition and ignore the +transition period. Clearly, the definition of different emotions +under this external stimulus setting is clear and subjects are +easy to get into a specific emotion. Taking into account +differences among different subjects, 5 subjects are involved +and each subject is recorded with four emotion types. For +each subject, there are 192-229 samples for peacefulness, 99- +141 samples for excitement, 156-236 samples for happiness +and 166-205 samples for tension. More information on the +iRealcare database is shown in Table I and segmentation +details are described in Section III-A2. +B. Description of WESAD dataset +The dataset, accessible in [43], is comprised of recordings +of 15 subjects (aged 24–35) watching video clips and doing +public speaking and mental arithmetic tasks. The dataset is +recorded with a wrist-based device (including the following +sensors: photoplethysmography, accelerometer, electrodermal +activity, and body temperature) and a chest-based device +(including the following sensors: ECG, accelerometer, elec- +tromyogram, respiration, and body temperature). This dataset +offers a fusion of physiological parameters to efficiently +identify human emotions, as these represent the body’s in- +stinctive reactions. However, it is not suitable for practical +applications, and it may hinder subjects during daily life +activities [23]. Therefore, in this paper, we only study single +ECG channel signals for this dataset. The ECG signal is +acquired from a RespiBAN Professional using a three-lead +configuration with 700Hz sampling rates. Three types of +emotions (baseline, stress and amusement) are annotated by +subjects [43]. Amusement condition signals are collected when +subjects watch funny video clips. Stress condition signals are +collected when subjects are asked to provide public speaking. +Baseline condition signals are collected when subjects sit/stand +at a table and read magazines. For each subject, there are 9 +samples for amusement, 15-18 samples for stress, and 28-29 +samples for baseline. The segmentation details are described +in Section III-A2. +III. METHOD +A. Preprocessing +Normally, ECG signals are non-linear with low signal am- +plitudes. The frequency range of ECG signals is from 0.05Hz +to 100Hz and the dynamic range is below 4mV [44]. Thus, +the collected ECG signals are susceptible to being disturbed by +external factors such as interference. To acquire ECG signals +with low interference, we conduct the pre-processing of the +raw ECG signals. During the data collection and transmission +stage, ECG signals are mainly affected by baseline drift, power +line interference, and electrode contact noise. The baseline +drift is caused by body movement and breathing. It can make +the entire ECG signal shift down or up at the horizontal +axis. The frequency of baseline drift is around 0.5Hz and +it will influence the analysis of ECG signals. The power- +line interference is characterized by 50 or 60Hz, which can +be caused by the electromagnetic field of nearby facilities +and electromagnetic interference of the power lines. Since the +iRealcare sensor used BLE instead of cables, the power-line +interference will not affect the collected ECG signals from the +sensor. The electrode contact noise is caused by the variance +of impedance when the skin is stretched. This frequency is +typically between 1 and 10Hz [45]. +1) Filtering: The finite impulse response (FIR) filter is used +to filter the aforementioned noises. It is a reliable and simple +filter. Moreover, the output of a FIR filter is not distorted +because it is a linear filter [46]. FIR filters are created utilizing +window-based techniques, such as the Hamming window, +Rectangular window, Hanning window, and the Blackman +window. These different windows are used to design the low +pass filter and high pass filter with cut-off frequencies. For our +band-pass FIR filter, the cut-off frequencies are set to 3Hz and +100Hz, respectively. +2) Segmentation and splitting: For the iRealcare dataset, +the aforementioned 20 groups are denoised, non-overlapping +segmented with 200 data points (1.56s), and then split into +training and test sets. Non-overlapping is designated between +segments to avoid any potential data leakage between training +and test data. It should be noticed that the selection of the +window size (200 data points) is empirical. Prior research +employing these datasets utilized a broad variety of window +sizes. For instance, [43] has chosen 5-second windows for +WESAD whereas [47] has used 1-second windows for the +same dataset. Specifically, the training set consists of 16 +groups, each of which has four emotions, whereas the test +set consists of 4 groups. Similar to the iRealcare dataset, the +WESAD dataset is also filtered by a FIR filter, non-overlapping +segmented with 14000 data points (20s), and then 12 subjects +are treated as a training set while the rest 3 subjects form a +test set. +Fig. 1 depicts four emotion segments with 200 randomly +chosen ECG signal data samples from the iRealcare dataset. +We can see that for the emotion of peacefulness, the subject’s +heart rate is comparatively sluggish. However, it is hard to +identify the other three emotions based on the original ECG +signals. As a result, the design of an efficient feature extraction +approach is necessitated. +3) Discrete cosine transform (DCT): In this paper, we use +the DCT methods to extract the main information of ECG +signals in the frequency domain [48]. It is computed for a +compressed version of input ECG signals containing signifi- +cant information, and only a small subset of the coefficients +is maintained as a feature vector. The main merit of the +DCT is its high computational speed which is suitable for +data compression [49]. To improve performance, the Z-score +normalization technique is invoked prior to recognition to +account for small perturbations in motion artifacts caused by +electrodes’ movement on the skin surface. +The DCT uses a sum of N cosine functions at different + +4 +TABLE I +DATA INFORMATION OF IREALCARE DATABASE. +ID +Peacefulness duration/min +Excitement duration/min +Happiness duration/min +Tension duration/min +Total duration/min +(Segment number) +(Segment number) +(Segment number) +(Segment number) +(Segment number) +1 +5 ( 192 ) +3.22 ( 123 ) +6.16 ( 236 ) +4.33 ( 166 ) +18.71 ( 717 ) +2 +5.23 ( 200 ) +3.63 ( 139 ) +5.42 ( 208 ) +4.48 ( 172 ) +18.76 ( 719 ) +3 +5.37 ( 206 ) +3.69 ( 141 ) +4.91 ( 188 ) +5.35 ( 205 ) +19.32 ( 740 ) +4 +5.98 ( 229 ) +3.65 ( 140 ) +4.51 ( 173 ) +4.7 ( 180 ) +18.84 ( 722 ) +5 +5.06 ( 194 ) +2.59 ( 99 ) +4.07 ( 156 ) +4.66 ( 178 ) +16.39 ( 627 ) +Sum +26.63 ( 1021 ) +16.78 ( 642 ) +25.08 ( 961 ) +23.52 ( 901 ) +92.01 ( 3525 ) +0 +0.5 +1 +1.5 +180 +200 +220 +Peacefulness +0 +0.5 +1 +1.5 +180 +200 +220 +Excitment +0 +0.5 +1 +1.5 +180 +200 +220 +Happiness +0 +0.5 +1 +1.5 +Time(s) +180 +200 +220 +Voltage(mV) +Tension +Fig. 1. The ECG segments with 200 points (1.56s) from four emotions +frequencies to express finite data samples. It converts temporal +signals into spectral signals. Eq. (1) defines the DCT formula +for a data sequence x(n), which is a Fourier transform without +the conjugate portion. +y(k) = w(k) �N +n=1 x(n) cos[ π +2N (2n − 1)(k − 1)], +(1) +k = 1, ..., N, +where +w(k) = +� +� +� +1 +√ +N +, k = 1 +� +2 +N +, 2 ≤ k ≤ N +(2) +and N is the length of the data sequence [50]. +During DCT, data samples from each ECG segment are +translated into the frequency domain, generating a series of +DCT coefficients with length N. Then, the generated DCT +coefficients are arranged in a decreasing order based on +their absolute values. DCT coefficients with larger absolute +values are treated as significant features which will be fed +into the proposed X-GWO-SVM scheme. Descending DCT +coefficients with dimension u (u ≤ N) can be selected as +the extracted features. The determination of a proper dimen- +sion u of extracted features will be discussed in Section IV +by comparing classification performances at different values. +Fig. 2 shows corresponding extracted features with dimension +u = 95, i.e., coefficients with the largest 95 absolute values, +from the aforementioned ECG segments (plotted in Fig. 1). +It should be noticed that the first coefficient takes the highest +energy (highlighted with red color), which stores the most +significant features. To observe details on the rest coefficients, +we zoom in the rest of coefficients (the 2nd to 95th coef- +ficients) for each emotion. Compared to the original ECG +signals, the four emotions are clearly differentiated between +segments following feature extraction. +0 +20 +40 +60 +80 +0 +1000 +2000 +Peacefulness +1 +10 +20 +30 +40 +50 +60 +70 +80 +90 +-20 +0 +20 +40 +0 +20 +40 +60 +80 +0 +1000 +2000 +Excitment +1 +10 +20 +30 +40 +50 +60 +70 +80 +90 +-20 +0 +20 +0 +20 +40 +60 +80 +0 +1000 +2000 +Happiness +1 +10 +20 +30 +40 +50 +60 +70 +80 +90 +-20 +0 +20 +40 +0 +20 +40 +60 +80 +Feature dimension u +0 +1000 +2000 +DCT coefficient +Tension +1 +10 +20 +30 +40 +50 +60 +70 +80 +90 +-20 +0 +20 +40 +Fig. 2. The extracted features from ECG segments with dimension of 95 +B. Exploitative and explorative grey wolf optimizer based +support vector machine +For the first time, the X-GWO-SVM approach is pro- +posed for ECG emotion identification in this work. The +hyperparameter-free property of the proposed method provides +a new way for radial basis function-based SVM (RBF-SVM) +learning. In general, classifying the non-linearly separable +data with RBF-SVM requires two hyperparameter which are + +5 +𝑡 < 𝐿 ? +Yes +Yes +Update positions of 𝜉1, 𝜉2, and +𝜉3 based on the fitness values +SVM +Input +training set +Trained SVM +model +Input +test set +Output Fitness value +𝑖 = 𝑖 + 1 +Update 𝜙(𝑡) = +𝟒 +𝟏+𝒆𝒕−𝑳 − 𝟐 +No +Update position of +search agents 𝜂𝑗 based +on 𝜉1, 𝜉2, and 𝜉3 +Update 𝑏𝑗(𝑡) +and 𝑐𝑗(𝑡) +𝑗 ≤ 𝑛 ? +Initialize +1) Maximum iteration L and number of search agents n +2) Positions of search agents {𝜂1, 𝜂2, … , 𝜂𝑖, … 𝜂𝑛}, where +d-dimensional 𝜂𝑖 contains {𝐶𝑖, 𝛾𝑖} when 𝑑 = 2. +3) Positions of alpha 𝜉1, beta 𝜉2, and delta 𝜉3 +No +Output 𝜉1 = (𝐶𝑜, 𝛾𝑜) +and trained model +Yes +No +𝑖 ≤ 𝑛 ? +𝑡 = 𝑡 + 1 +𝑗 = 𝑗 + 1 +Fig. 3. +Flow chart of the X-GWO-SVM algorithm. Total number of +search agents (wolves) is represented by n. Co and γo are components +stored in the final fittest solution ξ1. +considered—a penalty coefficient C and a spacial parameter γ. +The objective function of RBF-SVM with C and γ introduced +is expressed in Eqs. (3) and (4): +min 1 +2||w||2 + C �P +i=1 ϵi, +(3) +s.t. +yi(wT Ψ(xi) + b) − 1 + ϵi ≥ 0 +∀i = 1, 2, ..., P, +K(xi, xj) = e−γ∥xi−xj∥2 = Ψ(xi)T Ψ(xj), +(4) +where P is the number of training samples; ϵi is a slack +variable which is added to relax the constraints of linear +SVM; wT Ψ(xi) + b is the decision function; yi is the class +label; xi is the sample; C is the penalty parameter and it +controls the trade-off between the size of the margin and the +slack variable penalty; γ is a spacial parameter which controls +data distribution in a new feature space [51], [52]. Obviously, +hyperparameter (C and γ) tuning for RBF-SVM is necessary +but complex. Thus, the proposed method can internally learn +hyperparameters by emphasizing the importance of the α +wolf and non-linearly updating coefficient vectors used in +GWO. Moreover, this method has higher recognition accuracy +than the existing GWO-SVM and PSO-SVM techniques for +ECG emotion recognition use, and it can effectively avoid +the algorithm falling into a local solution by increasing the +exploration ability and speed up the convergence ability by +increasing the exploitation ability. +Fig. 3 demonstrates our X-GWO-SVM method, which is +inspired by the activity of grey wolves. There are 4 types +of grey wolves, named alpha (α), beta (β), delta (δ), and +omega (ω), simulating the leadership hierarchy. These wolves +continuously search for prey, the optimal solution in our case, +and hunting (optimization) is guided by the fittest solution, +second and third best solutions, α, β and δ, respectively. +The ω wolves follow these three wolves. A total number of +search agents (wolves) is represented by n. Co and γo are two +elements of the searched optimal solution ξ1. The X-GWO- +SVM method has 10 steps as described below. +1) The X-GWO-SVM related parameters are initialized, +i.e., maximum iteration L; the number of search agents +n; positions of α (ξ1), β (ξ2) and δ (ξ3); positions of +search agents (wolves) η1, η2, ..., ηi, ..., ηn. ηi ∈ Rd and +ξi ∈ Rd are d-dimensional vectors. In this case, d is +equal to 2, representing two optimal hyperparameters +(C and γ) required for search. +2) If the current iteration time t is less than the maximum +iteration L, go to the subsequent steps; otherwise, pro- +ceed directly to step 9). +3) For each agent, train RBF-SVM with current position +elements ηi = (Ci, γi). +4) Predict trained RBF-SVM with the test set for each agent +and output its loss as a fitness value based on Eq. (5): +loss(ηi) = 1 +M +M +� +i=1 +(yi − hi)2, +(5) +where M is the number of test samples and hi represents +the predicted value for the ith test sample. +5) Sort all fitness values in ascending order and assign +positions which have the corresponding top three fitness +values as ξ1, ξ2 and ξ3, respectively. The mathematical +expressions are +ξ1(t) = +arg max +ηi(t),i=1,...,n +loss(ηi(t)), +(6) +ξ2(t) = +arg max +ηi(t);ηi(t)̸=ξ1(t) +loss(ηi(t)), +(7) +ξ3(t) = +arg max +ηi(t);ηi(t)̸=ξ1(t),ξ2(t) +loss(ηi(t)). +(8) +6) Update exploration-exploitation regulation function φ(t) +based on Eq. (9): +φ(t) = −2t − L + 1 +−t + L . +(9) +7) For each search agent, update its position ηi based on +following equations: +ηi(t + 1) = 1 +4ξ1(t) + 1 +4 +3 +� +i=1 +[ξi(t) +(10) +− bi(t) ⊙ |ci(t) ⊙ ξi(t) − ηi(t)|], +bi(t) = 2φ(t)ri(t) − φ(t)1, +ci(t) = 2si(t), +(11) +where ⊙ and | · | represent Hadamard product operation +and element wise absolute value operations, respec- +tively; t is the iteration number; 1 ∈ R2 and its elements +are all ones; bi ∈ R2 and ci ∈ R2 are coefficient vectors. + +6 +The coefficients ri ∈ R2 and si ∈ R2 are random +vectors, where elements are in the range 0 to 1. +8) Accumulate iterative time and go back to step 2). +9) Output the optimal parameters ξ1 = (Co, γo) and the +trained SVM model. +10) Calculate the classification accuracy of the model based +on the test set and end the X-GWO-SVM algorithm. +We demonstrate improvements of the proposed X-GWO- +SVM algorithm with respect to its exploration and exploitation +ability in following two subsections. +1) Exploration: Conventionally, components of φ(t) are +linearly decreased from 2 to 0 over the course of itera- +tions [53], which models wolves approaching the prey. In our +design, we set components of φ(t) non-linearly decrease from +2 to 0 with slower declining rate near 2 and faster declining +rate near 0 (referring to Eq. (9)). Fig. 4 (a) demonstrates +the components of φ(t) linearly (blue stars) and non-linearly +(black circles) decreased from 2 to 0 over the course of +iterations when the maximum iteration time L is set to 100. +Clearly, for the designed nonlinear decreasing method, we can +observe that there is slow declining at the left side of the black +dash line (iteration time = 94), aiming to explore a larger range +and increase exploration compared to the conventional linear +way. +As discussed in [53], bi(t) with random values greater +than 1 or less than -1 is used to oblige the search agent +to diverge from the prey, which emphasizes exploration and +allows the X-GWO-SVM algorithm to search globally. It +should be noticed that the fluctuation range of bi(t) is also +decreased under an effect of φ(t). Components of bi(t) are +random values in the interval [−φ(t), φ(t)], where components +of φ(t) are non-linearly decreased from 2 to 0 over the course +of iterations [53]. Fig. 4 (b) shows a variation of bi(t) when +linear and nonlinear φ(t) are applied. The blue and grey +shadows indicate variation trends for bi(t) when linear φ(t) +and nonlinear φ(t) are applied, respectively. Obviously, the +value of bi(t) (black circles) for nonlinear φ(t) applied has +a larger range compared with the value of bi(t) (blue stars) +for linear φ(t) at the left side of the black dash line, i.e., +|bi(t)| > 1. In other words, the next search range for the +fittest position in the nonlinear case smoothly attenuates before +iteration reaches a threshold—94 in this figure, making sure a +large exploration range. +2) Exploitation (convergence): In [53], when updating the +positions, the weights for α, β, and δ wolves are all the +same. While for our proposed approach, when updating the +positions, we assign more weight to the α wolf (referring to +Eq. (10)), which emphasizes the importance of the α wolf. +Consequently, the fittest solution from the previous iteration +can be retained and continually influences the subsequent +updating step, ensuring a faster convergence. +We can observe from Fig. 4 (a) that, for the designed +nonlinear decreasing method, there is a much faster decay +at the right side of the black dash line. To clearly track +the convergence of φ(t), red-filled circles are utilized for the +nonlinear case after the 94th iteration. The convergence tends +to speed up as the iteration continuously increases, whereas, +0 +20 +40 +60 +80 +100 +Iteration time t +0 +0.5 +1 +1.5 +2 +Range of (t) +Linear decreasing (t) +Nonlinear decreasing (t) +(a) +0 +20 +40 +60 +80 +100 +Iteration time t +-3 +-2 +-1 +0 +1 +2 +Range of bi(t) +Linear decreasing bi(t) +Nonlinear decreasing bi(t) +(b) +Fig. 4. (a) Components of φ(t) linearly (blue circles) and non-linearly +(black circles) decreased from 2 to 0 over the course of iterations. (b) +The corresponding variations for components of bi(t) based on the +components of φ(t) linearly (blue circles) and non-linearly (black circles) +decreased over the course of iterations. The maximum iteration time L is +set to 100 for both cases. The black dash line lies at the 94th iteration. The +red-filled circles aim to clearly indicate variations of φ(t) and bi(t) for the +nonlinear case after the 94th iteration. +for the conventional linear decreasing method, the components +of φ(t) just evenly decrease from 2 to 0. +As we discussed before, the value of bi(t) is influenced by +the value of φ(t). Therefore, a similar phenomenon can be +observed in Fig. 4 (b), where the value of bi(t) converges +much faster than the linear case at the right side of the black +dash line, i.e., |bi(t)| < 1. In other words, the next search +range for the fittest position in the nonlinear case dramatically +decreases after iteration time reaches 94. +To sum up, the proposed X-GWO-SVM algorithm does +not require hyperparameter tuning on SVM in order to get +good accuracy. Additionally, it improves the way of updating +position by involving the fittest position α, which emphasizes +the importance of the α wolf and keeps the effect of the fittest + +7 +solution for the next iteration. We also enhance the ability of +exploitation by nonlinearly decreasing the value of φ(t). The +algorithm improves its global search ability by increasing the +exploration ability and speeds up the convergence ability by +increasing the exploitation ability. +C. Measurements +The classification performance of various methods can +be evaluated by standard statistical measurements: accuracy +(ACC) and F1-score (F1), defined as +ACC = +TP + TN +TP + FP + TN + FN, +(12) +F1 = 2PRE × REC +PRE + REC, +(13) +REC = +TP +TP + FN, +PRE = +TP +TP + FP, +(14) +where TP (true positive) is the number of samples correctly +predicted as the current class; TN (true negative) means the +number of correctly predicted as other classes; FP (false +positive) indicates the number of samples incorrectly detected +as the current class; FN (false negative) denotes the number +of samples incorrectly detected as other classes. Accuracy is +the general measurement of the correctly predicted ratio of the +total testing samples for each dataset, indicating the method’s +capability to classify emotions correctly. The F1-score, on the +other hand, more accurately captures the ideal model for the +unbalanced class distribution. The goal is to maximize these +two measures as representations of effective models. +IV. RESULTS +A. Feature dimension selection +As aforementioned in Section III-A3, determination of a +proper number of extracted features is necessary. Therefore, +we perform feature importance selection in the range of 20 +to 135 with step size 5 under the X-GWO-SVM method for +iRealcare dataset. Each simulation result is repeated 10 times +for random selection of training and test samples. +Fig. 7 illustrates the accuracy versus the dimension of the +feature under the X-GWO-SVM algorithm with 10-fold cross- +validation for the iRealcare dataset. Clearly, the recognition +accuracy displays the tendency to rise up at the beginning and +decline in late. The highest mean accuracy is 93.37% located at +the feature dimension equal to 95. Moreover, its corresponding +box plot (filled with orange color) has relatively low variance, +indicating the stability of this feature dimension. After getting +the most discriminative result with feature dimension 95, we +apply it to other comparison methods. +B. Exploration-exploitation regulation function selection +As +we +demonstrated +the +significance +of +exploration- +exploitation regulation function φ(t) in Section III-B, various +exploration-exploitation regulation functions are used in our +experiments here to demonstrate that our choice of φ(t) used +in the X-GWO-SVM algorithm is the best. Expressions on +them are shown in Eqs. (15) to (19) and these exploration- +exploitation regulation functions are plotted in Fig 6. It should +20 +25 +30 +35 +40 +45 +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +105 +110 +115 +120 +125 +130 +135 +Dimension of feature +0.8 +0.85 +0.9 +0.95 +Accuracy +75 +80 +85 +90 +95 +100 +105 +110 +115 +0.92 +0.94 +Fig. 5. The accuracy versus the dimension of the feature under the X- +GWO-SVM algorithm with 10-cross validation for the iRealcare dataset. +The blue shadow indicates a trend for mean accuracy values of different +dimensions of features under the X-GWO-SVM algorithm with 10-cross +validation. Box plot is employed with the box top and bottom denoting the +75th and 25th percentiles respectively for the results of 10-cross validation; +The red straight line inside the box denotes the median value, while the red +dot denotes the mean value; The blue star denotes the outlier value. The most +discriminative result is filled with orange color. +be noticed that the conventional linear exploration-exploitation +regulation function, a benchmark, is expressed in Eq. (15). +Additionally, the one we proposed in the X-GWO-SVM in +Eq. (9) is rewritten as fφ4(t) in Eq. (18). +fφ1(t) = 2 − 2t +L , +(15) +fφ2(t) = +4 +1 + et−L − 2, +(16) +fφ3(t) = +−4 +1 + e−t + 4, +(17) +fφ4(t) = −2t − L + 1 +−t + L += φ(t), +(18) +fφ5(t) = 2 cos( π +2tL). +(19) +Based on Fig. 6, we can observe that both fφ2 and fφ3 +are deformed from the Sigmoid function, which dramatically +decrease from 2 to 0 at the beginning and the end of the +iteration, respectively. The function fφ4 and function fφ5 +successively alleviate this decreasing trend on a basis of +function 1 +x and cos, respectively. +To evaluate the effects of exploration-exploitation regulation +function, we apply 10-fold cross-validation to the proposed X- +GWO-SVM, varying exploration-exploitation regulation func- +tions based on the aforementioned five functions. The eval- +uated results on exploration-exploitation regulation functions +are shown in Table II for iRealcare dataset and Table III for +WESAD dataset. Clearly, for iRealcare dataset, X-GWO-SVM +combined with fφ4 has the highest accuracy (93.37%) and F1- +score (93.38%) among others. Moreover, the lowest variance +and pretty low training time indicate its stability with low + +8 +0 +20 +40 +60 +80 +100 +Iteration time t +0 +0.5 +1 +1.5 +2 +Range of various (t) +f 1 +f 2 +f 3 +f 4 +f 5 +Fig. 6. +Variations of components of φ(t) for different exploration- +exploitation regulation functions. +computation time. A similar conclusion can be derived for +results on the WESAD dataset, where the highest accuracy +(95.93%) and F1-score (95.56%) are from the combination of +X-GWO-SVM with fφ4. To this end, we have determined the +optimal feature dimension—95, and exploration-exploitation +regulation function—fφ4. Therefore, later evaluations of the +proposed X-GWO-SVM are based on these two settings. +TABLE II +RESULTS OF X-GWO-SVM INVOLVED WITH FIVE +EXPLORATION-EXPLOITATION REGULATION FUNCTIONS ON IREALCARE +DATASET. +Exploration-exploitation regulation function +Mean(ACC) +Var(ACC) +Mean(F1)/% +Training time/s +fφ1 +92.90 +9.66E-05 +92.91 +429.98 +fφ2 +93.05 +6.10E-05 +93.08 +402.80 +fφ3 +83.26 +1.40E-02 +83.50 +509.66 +fφ4 +93.37 +2.84E-05 +93.38 +380.16 +fφ5 +92.93 +3.31E-05 +92.94 +368.14 +TABLE III +RESULTS OF X-GWO-SVM INVOLVED WITH FIVE +EXPLORATION-EXPLOITATION REGULATION FUNCTIONS ON WESAD +DATASET. +Exploration-exploitation regulation function +Mean(ACC) +Var(ACC) +Mean(F1)/% +Training time/s +fφ1 +95.29 +1.62E-04 +95.07 +856.31 +fφ2 +95.79 +7.48E-05 +95.44 +828.31 +fφ3 +95.29 +1.62E-04 +95.07 +825.14 +fφ4 +95.93 +5.61E-05 +95.56 +813.58 +fφ5 +95.79 +7.48E-05 +95.44 +835.40 +C. Classification Performance of Proposed Model +1) Classification Performance for of iRealcare dataset: +One may suspect that only one of the improvements on X- +GWO-SVM can achieve a considerable performance. Thus, we +investigate the other three methods: 1) using the GWO-SVM +method, where none of the improvement on GWO is applied; +2) using the nonlinear φ(t) based grey wolf optimizer (N- +GWO-SVM) method, where only the nonlinearly decreasing +value of φ(t) is used; 3) using PSO-SVM method, where +the conventional PSO algorithm is used for searching optimal +hyperparameters. +Tables IV to VII show the classification performance for the +hyperparameter optimizer-based techniques stated above. The +following metrics are reported: accuracy, F1-score, variation +of accuracy, and training duration of the schemes. All of them +are calculated from 10 repeated classification trials for each +scheme (rows in Tabs. IV to VII). +TABLE IV +THE MEAN ACC OF FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER +BASED SCHEMES EVALUATED ON IREALCARE DATASET. +Scheme +Peacefulness/% +Excitement/% +Happiness/% +Tension/% +PSO-SVM +83.20 +84.87 +95.40 +91.13 +GWO-SVM +84.07 +97.67 +94.73 +91.40 +N-GWO-SVM +84.60 +97.93 +94.33 +91.27 +X-GWO-SVM +86.03 +98.03 +95.07 +94.33 +Table IV shows that GWO-SVM performs significantly +better than PSO-SVM for peacefulness (84.07% vs. 83.20%), +excitement (97.67% vs. 84.87%), and tension (94.73% vs. +91.13%), but N-GWO-SVM only slightly improved perfor- +mance on peacefulness (84.60%) and excitement (97.93%). +Except for a slightly lower performance on happiness com- +pared to the PSO-SVM scheme (95.07% vs 95.40%), the +proposed X-GWO-SVM scheme provides a significant per- +formance boost over others. It has the highest accuracy for +peacefulness, excitement, and tension of 86.03%, 98.03%, and +94.33%, respectively. Table V presents similar results for the +mean F1 score. The proposed X-GWO-SVM scheme provides +a significant performance boost over others. +TABLE V +THE MEAN F1 OF FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER +BASED SCHEMES EVALUATED ON IREALCARE DATASET. +Scheme +Peacefulness/% +Excitement/% +Happiness/% +Tension/% +PSO-SVM +85.29 +82.46 +91.07 +94.45 +GWO-SVM +84.19 +97.87 +90.77 +93.57 +N-GWO-SVM +84.59 +97.99 +91.03 +93.36 +X-GWO-SVM +87.73 +98.31 +91.23 +96.24 +TABLE VI +THE MEAN VARIANCE OF ACC FOR FOUR EMOTIONS WITH +HYPERPARAMETER OPTIMIZER BASED SCHEMES EVALUATED ON +IREALCARE DATASET. +Scheme +Peacefulness +Excitement +Happiness +Tension +PSO-SVM +6.00E-04 +7.17E-02 +3.00E-04 +3.00E-04 +GWO-SVM +5.00E-04 +2.00E-04 +4.00E-04 +1.10E-03 +N-GWO-SVM +1.60E-03 +2.00E-04 +1.4E-04 +3.00E-04 +X-GWO-SVM +6.01E-04 +6.04E-05 +1.38E-04 +7.65E-05 +The variance result in Table VI suggests a similar con- +clusion. Compared to the conventional PSO-SVM scheme or +GWO-SVM scheme, except for the variance on peacefulness +(6.01E-04 vs 6.00E-04), our proposed method also shows +the lowest variance of accuracy on excitement (6.04E-05), +happiness (1.38E-04) and tension (7.65E-05), indicating its +stability. +Table +VII shows the mean accuracy, the mean variance +and the mean training time of 10-fold cross-validation results. +Note that, the proposed X-GWO-SVM scheme has the highest + +9 +TABLE VII +THE MEAN VALUES OF ACC, F1, VARIANCE AND TRAINING TIME FOR +FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER BASED SCHEMES +EVALUATED ON IREALCARE DATASET. +Scheme +Mean(ACC)/% +Mean(F1)/% +Mean(Var) +Training time/s +PSO-SVM +88.65 +88.32 +1.82E-02 +8824.80 +GWO-SVM +91.97 +91.60 +5.50E-04 +446.24 +N-GWO-SVM +92.03 +91.74 +5.25E-04 +446.42 +X-GWO-SVM +93.37 +93.38 +2.19E-04 +380.16 +mean of class accuracy (93.37%), the highest mean of class +f1-score (93.38%), the lowest mean of class variance (2.19E- +04), and the shortest hyperparameter training time (380.16s). +The results demonstrated its high reliability, stability, and +efficiency. +TABLE VIII +THE MEAN ACC, MEAN F1 AND VARIANCE OF NONPARAMETRIC +CLASSIFICATION METHODS AND THE PROPOSED METHOD EVALUATED ON +IREALCARE DATASET. +Algorithm +Mean(ACC)/% +Mean(F1)/% +Mean(Var) +RF +81.71 +82.35 +1.56E-04 +K-NN +82.48 +81.22 +7.35E-04 +X-GWO-SVM +93.37 +93.38 +2.19E-04 +To demonstrate the the significance of our X-GWO-SVM +scheme, we also examine other nonparametric classification +methods using similar features on the iRealcare dataset, such +as RF and K-NN. The results are shown in Table VIII. All +of them are calculated from 10 repeated classification trials +for each scheme. It is apparent from the results that the X- +GWO-SVM scheme has the highest accuracy performance. +Actually, RF is more stable than the X-GWO-SVM, while +its accuracy and F1-score are much lower than the proposed +scheme (81.71% vs 93.37%, 82.35% vs 93.38%, respectively). +In addition, the X-GWO-SVM completely outperforms K- +NN in terms of reliability and stability. Overall, the proposed +X-GWO-SVM strategy is more stable and more effective at +achieving high mean accuracy on the iRealcare dataset than +the existing methods. +2) Classification Performance for WESAD dataset: +To +demonstrate the reliability and stability of the proposed X- +GWO-SVM method, we further examine it on the WESAD +dataset in terms of accuracy and F1-score, compared with other +existing methods. By applying the feature dimension selection +in the range of 4000 to 10000 with step size 500 under the +X-GWO-SVM approach as we described in Section IV-B, the +best feature dimension is found to be 5000. +Fig. 7 illustrates the accuracy versus the dimension of +the feature under the X-GWO-SVM algorithm with 10-cross +validation for the WESAD dataset. Clearly, the recognition +accuracy has a similar pattern to Fig. 6, in which it increases +initially and decreases afterwards. The highest mean accuracy +is 95.93% located at the feature dimension equal to 5000. The +corresponding box plot has the highest accuracy—96.30%, the +lowest accuracy—94.44%, and the mean F1-score—95.56%. +Moreover, its corresponding box plot also has a relatively +low mean variance (5.49E-05), indicating the stability of this +feature dimension. The mean training time is 113.07s, in this +case. +Table IX presents a comparison between the proposed +classification scheme and the state-of-the-art methods pub- +lished for single channel ECG-based emotion recognition +methods on the WESAD dataset. The same testing dataset +ensures that the comparison is persuasive and feasible. It can +be observed from Table IX that the proposed classification +approach outperforms all simple machine learning methods, +such as RF, K-NN, linear discriminant analysis, and decision +tree. Though slightly inferior to that of self-supervised CNN, +the proposed X-GWO-SVM technique exhibits comparable +classification performance among deep neural networks. How- +ever, considering the computation complexity, the proposed +method is much simpler and more efficient than the self- +supervised CNN. Our algorithm has successfully been loaded +into a lightweight embedded system with a prediction time +of 2.659ms per 200-points iRealcare sample and 4.648ms +per 14000-points WESAD sample. Details on these results +will be discussed in Section V-C3. Overall, the proposed X- +GWO-SVM method achieves comparable accuracy and F1- +score (Fig. 7 and Table IX ) among neural network-based +deep learning classifiers on the WESAD dataset and has an +overwhelming performance on other existing techniques. +4000 +4500 +5000 +5500 +6000 +6500 +7000 +7500 +8000 +8500 +9000 +9500 +10000 +Number of features +0.85 +0.9 +0.95 +Accuracy +Fig. 7. The accuracy versus the dimension of the feature under the X- +GWO-SVM-SVM algorithm with 10-cross validation on WESAD dataset. +TABLE IX +COMPARISON OF VARIOUS SINGLE CHANNEL ECG-BASED EMOTION +RECOGNITION METHODS ON WESAD DATASET +Reference +Year +Method +ACC/% +F1/% +RF +82.78 +79.64 +K-NN +79.19 +75.39 +Schmidt et al. [43] +2018 +Linear discriminant analysis +85.44 +81.31 +AdaBoost Decision Tree +83.37 +80.20 +Decision Tree +80.17 +77.01 +Lin et al. [47] +2019 +CNN +83.00 +81.00 +Sarkar et al. [29] +2020 +Self-supervised CNN +96.90 +96.30 +Fully-supervised CNN +93.20 +91.20 +Proposed work +—— +X-GWO-SVM +95.93 +95.56 +V. DISCUSSION +The X-GWO-SVM algorithm, for the first time, is proposed +and also the first time used in single channel ECG-based +emotion recognition. By designing a suitable exploration- +exploitation regulation function and updating technique, we +are able to increase the exploration ability and exploitation +ability with the proposed approach. Two ECG datasets are + +10 +used: one raw self-collected iRealcare dataset and one credible +WESAD dataset. The X-GWO-SVM technique effectively +avoids the algorithm from falling into a local solution; hence, +it has a greater recognition accuracy than the existing GWO- +SVM and PSO-SVM techniques for ECG emotion recognition. +The algorithm enables accurate, stable, and efficient emotion +recognition based on single-channel ECG-based signals, which +fills a gap for GWO-SVM research on ECG-based emotion +recognition and also has the potential for clinical use. +A. Evaluation of datasets +Despite the restricted number of subjects in the iRealcare +dataset, the number of samples for each subject is sufficient +since the time of data collection for each emotion is sufficient. +It is true that the WESAD dataset contains a larger number of +subjects; however, the sample length required for this dataset +to achieve high accuracy, which is 14000, drastically reduces +the actual number of samples, for example, 9 samples for +each subject on amusement, 15-18 samples for each subject +on stress, and 28-29 samples for each subject on the baseline. +On the contrary, the sample length required for the iRealcare +dataset to achieve high accuracy, which is only 200. Therefore, +the number of samples for the iRealcare dataset is much larger +than the one in the WESAD dataset. +The reason for the caused aforementioned situation might +come from the way of giving external stimulus and recording +data. For the iRealcare dataset, ECG signals for happiness, +tension, and excitement are recorded when subjects watch +comedies, watch thriller movies and do exercises, respectively. +It should be noticed that emotions normally instantaneously +occur and hold for a short period. Therefore, we only record +the period that subjects are actually in that emotion condition +and ignore the transition period. Clearly, the definition of +different emotions under this external stimulus setting is clear +and subjects are easy to get into a specific emotion. However, +for the WESAD dataset, amusement condition signals are col- +lected when subjects watch funny video clips; stress condition +signals are collected when subjects are asked to provide public +speaking and mental arithmetic tasks; baseline condition sig- +nals are collected when subjects sit/stand at a table and read +magazines. In fact, subjects tend to take some time to transfer +from one emotion condition to the other. However, such a +transition period is also recorded in the WESAD dataset. Thus, +the sample length need to be long enough to make sure not +just the transition period is included. The shorter the time, +the more probable it is that only transitional periods will be +included in the sample. +To sum up, in spite of the fact that the iRealcare dataset has +limited subjects, the actual number of samples is much larger +than the one in the WESAD dataset. Moreover, due to the +exclusive emotion transition period for the WESAD dataset, +the selection of sample length for the iRealcare dataset is more +flexible than the WESAD dataset. We use the widely-used +WESAD dataset as a benchmark for further comparison to +validate our proposed X-GWO-SVM algorithm. +B. Evaluation of exploration-exploitation regulation function +selection +We study the impact of exploration-exploitation regulation +functions on emotion recognition performance. For this study, +we select possible base functions that control the signifi- +cance of each exploration-exploitation regulation function as +listed in Eqs. (15) to (19). Table II and Table III show +the emotion recognition performance for five exploration- +exploitation regulation functions on the iRealcare dataset and +WESAD dataset, respectively. This analysis provides in-depth +insight into the effect of the exploration-exploitation regulation +functions associated with the emotion recognition outcome. +Furthermore, this analysis helps us narrow down the most +suitable exploration-exploitation regulation function in order +to achieve the best performance. +As we mentioned in Section III-B, the declining rate of the +exploration-exploitation regulation function at the beginning +and end with respect to the iteration time represents the +exploration and exploitation ability of the proposed X-GWO- +SVM. From Table II and Table III, we notice that for the +exploration-exploitation regulation function fφ3, where the +function is formed on a basis of the sigmoid function, the +model performance on emotion recognition is poor since it +is under-explored and under-exploited. However, for those +exploration-exploitation regulation functions lying above the +benchmark function fφ1, the model shows significantly bet- +ter performance. Interestingly, the performance drops when +exploration-exploitation regulation functions decline too fast +(fφ2) or too slow (fφ5). The function fφ4 gives the highest +performance for emotion recognition compared to others since +it has the most suitable diverging and converging performance +to the X-GWO-SVM algorithm. Moreover, the fact that fφ4 +outperformed other functions for both datasets is also indica- +tive of its stability. +In summary, the analysis above shows that for all the +exploration-exploitation regulation functions, when the declin- +ing rate of the function at the beginning is too large or too +small, for example, fφ3 or fφ2, emotion recognition accuracy +drops due to the under-exploration or over-exploration. This +results in the X-GWO-SVM more easily falling into local solu- +tions. Similarly, when the declining rate of the function at the +end is too large or too small, fφ2 or fφ3, the performance also +drops due to the under-exploitation or over-exploitation in such +cases becomes too difficult for the algorithm to properly find +the global solution. Hence, we conclude that there is a trade- +off between exploration and exploitation for the exploration- +exploitation regulation functions associated with the proposed +X-GWO-SVM algorithm, for which the proper exploration- +exploitation regulation function fφ4 is applied resulting in +avoiding falling into local solutions. +C. Evaluation of reliability, stability and efficiency of X-GWO- +SVM +This section discusses the performance of X-GWO-SVM +for emotion recognition in terms of reliability, stability, and +efficiency. + +11 +1) Reliability: In our work, when only fusing exploration- +exploitation regulation function fφ4 with GWO-SVM, i.e., +N-GWO-SVM, the classification accuracy and F1-score get +improved (referring to Table IV, Table V and Table VII). This +improvement indicates that involving a nonlinear exploration- +exploitation regulation function can improve the recognition +performance. Similarly, the classification accuracy and F1- +score get further enhanced when the importance of the α wolf +is emphasized, i.e., X-GWO-SVM, which shows the equivalent +importance of the improvement on the X-GWO-SVM. +The classification performance of X-GWO-SVM is superior +to the classification performance of other hyperparameter +optimizer-based systems, such as PSO-SVM and GWO-SVM. +This indicates the high reliability of the proposed algo- +rithm over existing common hyperparameter optimizer-based +schemes. Furthermore, the X-GWO-SVM has the highest ac- +curacy and F1-score among simple machine learning methods, +such as RF, K-NN, decision tree, and linear discriminant +analysis (Table VIII and Table IX). Our analysis indicates +that the X-GWO-SVM is more effective than simple machine +learning approaches at avoiding local solutions. For the deep +learning neural networks, such as CNN, the X-GWO-SVM +can still outperform them except for a more complex single— +self-supervised CNN [29]. Though the accuracy and F1-score +of the X-GWO-SVM are slightly lower than the one from the +self-supervised CNN, considering the efficiency, which will be +discussed in Section V-C3, our algorithm is still competitive. +2) Stability: The variance of the proposed method and +existing works is computed to evaluate the stability of the +methods. All simulation results are applied with 10-fold cross- +validation. Similar to the discussion in Section V-C1, the +X-GWO-SVM is the most stable algorithm among existing +common hyperparameter optimizer-based schemes. Besides, +similar results on both the iRealcare dataset and the WESAD +dataset also indicate the stability of the proposed method. +3) Efficiency: Our works are implemented through both +MATLAB version R2021b and Python 3.7 for feature ex- +traction, model training, and prediction. For MATLAB, the +computation is performed on a laptop with 11th Gen Intel(R) +Core(TM) i7-11800H (2.2GHz and 32GB of RAM). The +computation time for classifying a 200-points (1.56s) iRealcare +sample and a 14000-points (20s) WESAD sample roughly +spends 0.355ms and 0.778ms, respectively, using our proposed +method. For Python, the computation is performed in JETSON +NANO with Quad-core ARM Cortex-A57 MPCore Processor +(1.43GHz and 4GB of RAM). The computation time for +classifying a 200-points (1.56s) iRealcare sample and a 14000- +points (20s) WESAD sample roughly spends 2.659ms and +4.648ms, respectively, using our proposed method. +Compared with the self-supervised CNN, a deep neural net- +work, the proposed X-GWO-SVM is much simpler. The two- +step self-supervised architecture involves deep convolutional +blocks and several fully connected layers in [29], which may +not be realized in lightweight embedded systems. Whereas, +our algorithm has successfully been loaded into JETSON +NANO, an embedded system-on-module and developer kit +with a prediction time of 2.659ms per 200-points iRealcare +sample and 4.648ms per 14000-points WESAD sample. This +provides a way to embed an ECG patch with the proposed +algorithm, achieving edge computing for emotion recognition +on ECG signals. +Moreover, the X-GWO-SVM is the most efficient algo- +rithm among existing common hyperparameter optimizer- +based schemes, which is evaluated by the training time. The +other interesting point that can be found in Table VII is that +all GWO-SVM-based techniques take shorter training time +than the PSO-SVM work, which is compatible with [35]’s +conclusion. +D. Limitations and future directions +The possible limitation of the current study would be that +we only investigate four exploration-exploitation regulation +functions, i.e., fφ2, fφ3, fφ4, and fφ5. Moreover, the dataset +we collected is still insufficient and other existing published +datasets, e.g., AMIGOS [27], Augsburg Biosignal Toolbox +(AuBT) [54], etc., have not been verified by the proposed +X-GWO-SVM algorithm. In future work, we will use other +exploration-exploitation regulation functions for the proposed +algorithm to explore their effectiveness in emotion recognition. +Additionally, more published datasets will be examined by our +method. +Besides, through the results and conclusions reported in +[29], we also observed that deep learning is competitive in +emotion recognition, which may further improve the perfor- +mance of our proposed strategy. Thus in our future work, we +will try to find an effective deep learning method and embed- +ded GWO methods to further improve emotion recognition +performance. +VI. CONCLUSION +In this paper, we presented an X-GWO-SVM technique that +improves the exploration and exploitation abilities of single +channel ECG-based emotion recognition. In order to classify +different emotions, this research used two reliable datasets: +one trustworthy WESAD dataset and one raw self-collected +iRealcare dataset. 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Rahayu, F. Farokhi, and Z. Lin, +“When Machine Learning Meets Privacy: A Survey and Outlook” ACM +Computing Surveys, ACM Computing Surveys (IF10.282), Pub Date : +2021-03-05, DOI: 10.1145/3436755 +[60] S. Shaham, M. Ding, B. Liu, S. Dang Z. Lin, J. Li, “Privacy Preservation +in Location-Based Services: A Novel Metric and Attack Model”, IEEE +Transactions on Mobile Computing, vol. 20, no. 10, pp. 3006-3019, 1 +Oct. 2021. doi: 10.1109/TMC.2020.2993599. +[61] S. Shaham, M. Ding, B. Liu, S. Dang Z. Lin, J. Li, “Privacy-Preserving +Location Data Publishing: A Machine Learning Approach”, IEEE Trans- +actions on Knowledge and Data Engineering, vol. 33, no. 9, pp. 3270- +3283, 1 Sept. 2021. doi: 10.1109/TKDE.2020.2964658. +[62] D. Smith, P. Wang, M. Ding, J. Chan, B. Spak, X. Guan, P.Tyler, +T. Rakotoarivelo, Z. Lin, T. Abbasi,”Privacy-Preserved Optimal En- +ergy Trading, Statistics, and Forecasting for a Neighborhood Area +Network” in Computer, vol. 53, no. 05, pp. 25-34, 2020. doi: +10.1109/MC.2020.2972505 + diff --git a/vtAzT4oBgHgl3EQf7P6B/content/tmp_files/load_file.txt b/vtAzT4oBgHgl3EQf7P6B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc454535e778419809a0a114889fbac47a6a00b3 --- /dev/null +++ b/vtAzT4oBgHgl3EQf7P6B/content/tmp_files/load_file.txt @@ -0,0 +1,1194 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf,len=1193 +page_content='1 A Novel Exploitative and Explorative GWO-SVM Algorithm for Smart Emotion Recognition Xucun Yan, Zihuai Lin, Senior Member, IEEE, Zhiyun Lin, and Branka Vucetic, Life Fellow, IEEE Abstract—Emotion recognition or detection is broadly utilized in patient-doctor interactions for diseases such as schizophrenia and autism and the most typical techniques are speech detection and facial recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, features extracted from these behavior-based emotion recognitions are not reliable since hu- mans can disguise their emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Recording voices or tracking facial expressions for a long term is also not efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, our aim is to find a reliable and efficient emotion recognition scheme, which can be used for non-behavior-based emotion recognition in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This can be solved by implement- ing a single-channel electrocardiogram (ECG) based emotion recognition scheme in a lightweight embedded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, existing schemes have relatively low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For instance, the accuracy is about 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='78% by using a least squares support vector machine (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, we propose a reliable and efficient emotion recognition scheme—exploitative and explorative grey wolf optimizer based SVM (X-GWO-SVM) for ECG-based emo- tion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Two datasets, one raw self-collected iRealcare dataset, and the widely-used benchmark WESAD dataset are used in the X-GWO-SVM algorithm for emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Leave-single-subject-out cross-validation yields a mean accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37% for the iRealcare dataset and a mean accuracy of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93% for the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This work demonstrates that the X-GWO-SVM algorithm can be used for emotion recognition and the algorithm exhibits superior performance in reliability compared to the use of other supervised machine learning methods in earlier works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It can be implemented in a lightweight embedded system, which is much more efficient than existing solutions based on deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Index Terms—Emotion recognition, IoT, Smart health, ECG signals, GWO, SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' INTRODUCTION T HE use of the Internet of Things (IoT) is growing steadily over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It is expected that by 2025, there will be approximately 27 billion connected IoT devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' At present, the IoT is one of the main promoters of technological innovation and one of the areas with greater potential for social and economic transformation [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Through a network of sensors and actuators connected to a wireless network [5]– [14], the operator has the power to remotely gather data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Alternatively, actuators could be programmed to actuate au- tomatically according to values reported by the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Emotion recognition or detection based on IoT wireless sensing and networking has gained lots of attention since it Xucun Yan, Zihuai Lin and Branka Vucetic are with School of Electrical and Information Engineering, University of Sydney, New South Wales 2006, Australia (e-mail: xucun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='yan@sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' zi- huai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='lin@sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' branka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='vucetic@sydney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Zhiyun Lin is with the Department of Electrical and Electronic Engi- neering, Southern University of Science and Technology, Shenzhen 518055, China, and Peng Cheng Laboratory, Shenzhen 518066, China (email: linzy@sustech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Corresponding author: Zhiyun Lin can be broadly utilized in interfaces between humans and computers and patient-doctor interactions for diseases such as schizophrenia and autism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Most emotion detection methods are based on behaviors such as speech detection and face recognition [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, features extracted from the abovementioned behavior-based emotion recognition are not adequate for identifying emotions, because the behavior in- duced by emotion can be disguised by artifacts of human social masking [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For example, emotion recognition based on facial expressions can be easily misled by a poker face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Using physiological signals, such as electroencephalograms (EEGs) [18]–[20], electromyograms (EMGs), and electrocar- diograms (ECGs) [21], is an alternative to identify emotions since physiological signals are one of the most notable means to manifest the central nervous system in which emotions are processed [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Using physiological cues for emotion identification has two advantages over prior approaches to emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The first is that physiological signals generated from automatic reactions are difficult to disguise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The second is that wearable emotion monitoring can continually record physiological in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This differs from the instance of voice recognition where data may only be recorded when individuals are speak- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, using multi-channel biosignals to recognize hu- man emotions is not suitable for practical applications because subjects may be hindered during daily life activities [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It has been proved that ECG signals are a suitable physiological channel with acceptable recognition abilities [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, according to [24]–[27], the accuracy of emotion detection based on a single ECG channel fluctuates a lot for various datasets compared to that of other approaches such as facial emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' On the one hand, recent efforts in emotion recognition using ECG signals have largely relied on relatively simple supervised learning techniques [28], such as random forest (RF), support vector machine (SVM), K- nearest neighbor (K-NN), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, these methods have relatively low accuracy (for instance, the accuracy is about 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='78% [17] by using least squares SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' On the other hand, the current maximum level of single ECG channel- based emotion recognition accuracy reaches 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='9% [29] for Wearable Stress and Affect Detection (WESAD) database and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='2% [29] for a dataset for multi-modal research of affect, personality traits, and mood in individuals and groups (AMIGOS) [27], which utilizes self-supervised convolutional neural network (CNN) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Facial emotion recognition accuracy achieves 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07% [30] for MMI Facial Expression Database and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='91% [30] for the Japanese Female Facial Expression Database, which uses CNN embedded with re- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='01887v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='SP] 5 Jan 2023 2 current neural network (RNN) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Nevertheless, these deep neural network-based techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', CNN, RNN, etc, tend to achieve high accuracy but are complex with low computation efficiency, which cannot be implemented in a lightweight embedded system operating in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, seeking a simple supervised learning scheme to accurately, stably, and efficiently recognize emotions based on a single ECG channel in a lightweight embedded system is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Towards this objective, this paper aims to develop a novel exploitative and explorative GWO-SVM (X-GWO-SVM) for ECG-based emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The goal is to achieve good classification accuracy (as high as utilizing complex neural networks) while simultaneously reducing computation so that it can be implemented in a lightweight embedded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The idea is motivated from the fact that the SVM algorithm can be used to solve single-channel ECG-based emotion recogni- tion issue with lightweight embedded system implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, the existing SVM works do not offer a good classi- fication accuracy performance for ECG-based recognition due to difficulties in finding appropriate hyper-parameters while preventing overfitting of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In general, the selection of hyperparameters is a non-convex optimization issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, many heuristic algorithms such as genetic algorithm (GA), particle swarm optimization (PSO), and grey wolf optimizer (GWO) [31]–[34] are introduced to tackle it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Compared with PSO and a set of search algorithms, GWO provides better performance in computation reduction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', in feature subset selection [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, the GWO approach has been demonstrated to be more stable against initialization than PSO and GA [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, as discussed in [36], conventional GWO-based SVM (GWO-SVM) tech- niques are still easy to fall into local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In this work, an improved method, the X-GWO-SVM method, is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The proposed X-GWO-SVM method is the first to apply GWO-SVM idea to solve ECG-based recognition, and as shown in this paper, this method has higher recognition accuracy than existing SVM and PSO-SVM techniques for ECG emotion recognition use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It can effectively avoid the algorithm falling into a local solution by increasing the exploration ability, and speed up the convergence by increasing the exploitation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In this paper, two datasets, one raw self-collected iRealcare dataset, and the widely used benchmark WESAD dataset are used in the X-GWO-SVM algorithm for emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Leave-single-subject-out cross-validation yields a mean accuracy of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37% and an F1- score of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38% for the iRealcare dataset and a mean accuracy of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93% and an F1-score of 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56% for the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows: 1) We use a self-built wearable IoT ECG patch with only one ECG channel to collect four emotions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', happi- ness, tension, peacefulness and excitement, by playing different videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) We designed a novel X-GWO-SVM algorithm to inter- nally learn hyperparameters on SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It can effectively avoid the algorithm falling into a local solution by increasing the exploration ability, and speed up the convergence by increasing the exploitation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3) This novel X-GWO-SVM algorithm can accurately and efficiently recognize emotions for single-channel ECG- based signals and be implemented in a lightweight embedded system operating in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It improves accuracy compared to existing simple machine learning methods and dramatically reduces complexity compared to some novel deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus, the efficiency is also increased compared to other time-consuming emotion recognition methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The outline of the rest of the paper is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Section II introduces our database and an expanded dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Our model formulation is described in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In Sections IV and Sec- tionV, we present results and discussions, respectively, before concluding with a discussion of potential future directions in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' DATASET ECG signals are composed of the P wave, T wave, and QRS complex, which represent the three phases of an ECG pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In atrial systole, the P wave is the contraction pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The QRS complex signifies ventricular depolarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The T wave represents ventricular re-polarization [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' An ECG device records the electrical changes caused by the activities of the heart, which are collected by electrodes over the skin for a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It has been proved that ECG signals are a suitable physiological channel with acceptable recognition abilities [17] to identify emotions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, single-channel ECG signals are used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In order to verify the general representation ability of X-GWO-SVM, two datasets of ECG signals are used, one raw self-collected iRealcare dataset with 5 subjects and the other widely-used benchmark WESAD dataset with 15 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Description of iRealcare dataset Data collection is one of the most important steps for emotion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The definition of different emotions must be explicit in this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' If the definition is not clear, confusion may occur among different emotions in the classification phase and the classification performance will be influenced negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, emotions normally instantaneously occur and hold for a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The longer the period is, the more irrelevant data is included in ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus, it is hard to properly label the corresponding emotion class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To avoid the aforementioned issue, we self-collect a dataset with high quality and a short period for each emotion, making sure accurate data collection and labeling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The ECG signals are recorded by a low-cost wearable IoT ECG patch, called iRealcare [38]–[42] with 128 Hz sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The data collected by the iRealcare IoT ECG sensor can be transmitted to a smartphone application (APP) via Bluetooth Low Energy (BLE) and then to a cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' From the cloud, we can acquire the raw ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Signals are recorded for four emotions including happiness, tension, peacefulness, and excitement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Except for peacefulness, each emotion is generated based on an external environmental stimulus, which is similar to the published datasets stimulating subjects through audio or video [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The peacefulness describes the normal state, 3 for which the ECG signals are recorded without any external stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Signals for happiness, tension, and excitement are recorded when subjects watch comedies, watch thriller movies and do exercises, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Generally, the record duration should be short as we discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, the record time is in a range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='22-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='16 minutes for each emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It should be noticed that we only record the period that subjects are actually in that emotion condition and ignore the transition period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Clearly, the definition of different emotions under this external stimulus setting is clear and subjects are easy to get into a specific emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Taking into account differences among different subjects, 5 subjects are involved and each subject is recorded with four emotion types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For each subject, there are 192-229 samples for peacefulness, 99- 141 samples for excitement, 156-236 samples for happiness and 166-205 samples for tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' More information on the iRealcare database is shown in Table I and segmentation details are described in Section III-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Description of WESAD dataset The dataset, accessible in [43], is comprised of recordings of 15 subjects (aged 24–35) watching video clips and doing public speaking and mental arithmetic tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The dataset is recorded with a wrist-based device (including the following sensors: photoplethysmography, accelerometer, electrodermal activity, and body temperature) and a chest-based device (including the following sensors: ECG, accelerometer, elec- tromyogram, respiration, and body temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This dataset offers a fusion of physiological parameters to efficiently identify human emotions, as these represent the body’s in- stinctive reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, it is not suitable for practical applications, and it may hinder subjects during daily life activities [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, in this paper, we only study single ECG channel signals for this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The ECG signal is acquired from a RespiBAN Professional using a three-lead configuration with 700Hz sampling rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Three types of emotions (baseline, stress and amusement) are annotated by subjects [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Amusement condition signals are collected when subjects watch funny video clips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Stress condition signals are collected when subjects are asked to provide public speaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Baseline condition signals are collected when subjects sit/stand at a table and read magazines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For each subject, there are 9 samples for amusement, 15-18 samples for stress, and 28-29 samples for baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The segmentation details are described in Section III-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Preprocessing Normally, ECG signals are non-linear with low signal am- plitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The frequency range of ECG signals is from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='05Hz to 100Hz and the dynamic range is below 4mV [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus, the collected ECG signals are susceptible to being disturbed by external factors such as interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To acquire ECG signals with low interference, we conduct the pre-processing of the raw ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' During the data collection and transmission stage, ECG signals are mainly affected by baseline drift, power line interference, and electrode contact noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The baseline drift is caused by body movement and breathing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It can make the entire ECG signal shift down or up at the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The frequency of baseline drift is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5Hz and it will influence the analysis of ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The power- line interference is characterized by 50 or 60Hz, which can be caused by the electromagnetic field of nearby facilities and electromagnetic interference of the power lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Since the iRealcare sensor used BLE instead of cables, the power-line interference will not affect the collected ECG signals from the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The electrode contact noise is caused by the variance of impedance when the skin is stretched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This frequency is typically between 1 and 10Hz [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1) Filtering: The finite impulse response (FIR) filter is used to filter the aforementioned noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It is a reliable and simple filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, the output of a FIR filter is not distorted because it is a linear filter [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' FIR filters are created utilizing window-based techniques, such as the Hamming window, Rectangular window, Hanning window, and the Blackman window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' These different windows are used to design the low pass filter and high pass filter with cut-off frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For our band-pass FIR filter, the cut-off frequencies are set to 3Hz and 100Hz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) Segmentation and splitting: For the iRealcare dataset, the aforementioned 20 groups are denoised, non-overlapping segmented with 200 data points (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56s), and then split into training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Non-overlapping is designated between segments to avoid any potential data leakage between training and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It should be noticed that the selection of the window size (200 data points) is empirical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Prior research employing these datasets utilized a broad variety of window sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For instance, [43] has chosen 5-second windows for WESAD whereas [47] has used 1-second windows for the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Specifically, the training set consists of 16 groups, each of which has four emotions, whereas the test set consists of 4 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Similar to the iRealcare dataset, the WESAD dataset is also filtered by a FIR filter, non-overlapping segmented with 14000 data points (20s), and then 12 subjects are treated as a training set while the rest 3 subjects form a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1 depicts four emotion segments with 200 randomly chosen ECG signal data samples from the iRealcare dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' We can see that for the emotion of peacefulness, the subject’s heart rate is comparatively sluggish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, it is hard to identify the other three emotions based on the original ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' As a result, the design of an efficient feature extraction approach is necessitated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3) Discrete cosine transform (DCT): In this paper, we use the DCT methods to extract the main information of ECG signals in the frequency domain [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It is computed for a compressed version of input ECG signals containing signifi- cant information, and only a small subset of the coefficients is maintained as a feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The main merit of the DCT is its high computational speed which is suitable for data compression [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To improve performance, the Z-score normalization technique is invoked prior to recognition to account for small perturbations in motion artifacts caused by electrodes’ movement on the skin surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The DCT uses a sum of N cosine functions at different 4 TABLE I DATA INFORMATION OF IREALCARE DATABASE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' ID Peacefulness duration/min Excitement duration/min Happiness duration/min Tension duration/min Total duration/min (Segment number) (Segment number) (Segment number) (Segment number) (Segment number) 1 5 ( 192 ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='22 ( 123 ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='16 ( 236 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='33 ( 166 ) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='71 ( 717 ) 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='23 ( 200 ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='63 ( 139 ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='42 ( 208 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='48 ( 172 ) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='76 ( 719 ) 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37 ( 206 ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='69 ( 141 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='91 ( 188 ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='35 ( 205 ) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='32 ( 740 ) 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='98 ( 229 ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='65 ( 140 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='51 ( 173 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='7 ( 180 ) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='84 ( 722 ) 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='06 ( 194 ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='59 ( 99 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07 ( 156 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='66 ( 178 ) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='39 ( 627 ) Sum 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='63 ( 1021 ) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='78 ( 642 ) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='08 ( 961 ) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='52 ( 901 ) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='01 ( 3525 ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 180 200 220 Peacefulness 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 180 200 220 Excitment 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 180 200 220 Happiness 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 Time(s) 180 200 220 Voltage(mV) Tension Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The ECG segments with 200 points (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56s) from four emotions frequencies to express finite data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It converts temporal signals into spectral signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (1) defines the DCT formula for a data sequence x(n), which is a Fourier transform without the conjugate portion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' y(k) = w(k) �N n=1 x(n) cos[ π 2N (2n − 1)(k − 1)], (1) k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', N, where w(k) = � � � 1 √ N , k = 1 � 2 N , 2 ≤ k ≤ N (2) and N is the length of the data sequence [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' During DCT, data samples from each ECG segment are translated into the frequency domain, generating a series of DCT coefficients with length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Then, the generated DCT coefficients are arranged in a decreasing order based on their absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' DCT coefficients with larger absolute values are treated as significant features which will be fed into the proposed X-GWO-SVM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Descending DCT coefficients with dimension u (u ≤ N) can be selected as the extracted features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The determination of a proper dimen- sion u of extracted features will be discussed in Section IV by comparing classification performances at different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2 shows corresponding extracted features with dimension u = 95, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', coefficients with the largest 95 absolute values, from the aforementioned ECG segments (plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It should be noticed that the first coefficient takes the highest energy (highlighted with red color), which stores the most significant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To observe details on the rest coefficients, we zoom in the rest of coefficients (the 2nd to 95th coef- ficients) for each emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Compared to the original ECG signals, the four emotions are clearly differentiated between segments following feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 0 20 40 60 80 0 1000 2000 Peacefulness 1 10 20 30 40 50 60 70 80 90 20 0 20 40 0 20 40 60 80 0 1000 2000 Excitment 1 10 20 30 40 50 60 70 80 90 20 0 20 0 20 40 60 80 0 1000 2000 Happiness 1 10 20 30 40 50 60 70 80 90 20 0 20 40 0 20 40 60 80 Feature dimension u 0 1000 2000 DCT coefficient Tension 1 10 20 30 40 50 60 70 80 90 20 0 20 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The extracted features from ECG segments with dimension of 95 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Exploitative and explorative grey wolf optimizer based support vector machine For the first time, the X-GWO-SVM approach is pro- posed for ECG emotion identification in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The hyperparameter-free property of the proposed method provides a new way for radial basis function-based SVM (RBF-SVM) learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In general, classifying the non-linearly separable data with RBF-SVM requires two hyperparameter which are 5 𝑡 < 𝐿 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Yes Yes Update positions of 𝜉1, 𝜉2, and 𝜉3 based on the fitness values SVM Input training set Trained SVM model Input test set Output Fitness value 𝑖 = 𝑖 + 1 Update 𝜙(𝑡) = 𝟒 𝟏+𝒆𝒕−𝑳 − 𝟐 No Update position of search agents 𝜂𝑗 based on 𝜉1, 𝜉2, and 𝜉3 Update 𝑏𝑗(𝑡) and 𝑐𝑗(𝑡) 𝑗 ≤ 𝑛 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Initialize 1) Maximum iteration L and number of search agents n 2) Positions of search agents {𝜂1, 𝜂2, … , 𝜂𝑖, … 𝜂𝑛}, where d-dimensional 𝜂𝑖 contains {𝐶𝑖, 𝛾𝑖} when 𝑑 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3) Positions of alpha 𝜉1, beta 𝜉2, and delta 𝜉3 No Output 𝜉1 = (𝐶𝑜, 𝛾𝑜) and trained model Yes No 𝑖 ≤ 𝑛 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 𝑡 = 𝑡 + 1 𝑗 = 𝑗 + 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Flow chart of the X-GWO-SVM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Total number of search agents (wolves) is represented by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Co and γo are components stored in the final fittest solution ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' considered—a penalty coefficient C and a spacial parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The objective function of RBF-SVM with C and γ introduced is expressed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (3) and (4): min 1 2||w||2 + C �P i=1 ϵi, (3) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' yi(wT Ψ(xi) + b) − 1 + ϵi ≥ 0 ∀i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', P, K(xi, xj) = e−γ∥xi−xj∥2 = Ψ(xi)T Ψ(xj), (4) where P is the number of training samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' ϵi is a slack variable which is added to relax the constraints of linear SVM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' wT Ψ(xi) + b is the decision function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' yi is the class label;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' xi is the sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' C is the penalty parameter and it controls the trade-off between the size of the margin and the slack variable penalty;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' γ is a spacial parameter which controls data distribution in a new feature space [51], [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Obviously, hyperparameter (C and γ) tuning for RBF-SVM is necessary but complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus, the proposed method can internally learn hyperparameters by emphasizing the importance of the α wolf and non-linearly updating coefficient vectors used in GWO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, this method has higher recognition accuracy than the existing GWO-SVM and PSO-SVM techniques for ECG emotion recognition use, and it can effectively avoid the algorithm falling into a local solution by increasing the exploration ability and speed up the convergence ability by increasing the exploitation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3 demonstrates our X-GWO-SVM method, which is inspired by the activity of grey wolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' There are 4 types of grey wolves, named alpha (α), beta (β), delta (δ), and omega (ω), simulating the leadership hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' These wolves continuously search for prey, the optimal solution in our case, and hunting (optimization) is guided by the fittest solution, second and third best solutions, α, β and δ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The ω wolves follow these three wolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' A total number of search agents (wolves) is represented by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Co and γo are two elements of the searched optimal solution ξ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The X-GWO- SVM method has 10 steps as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1) The X-GWO-SVM related parameters are initialized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', maximum iteration L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' the number of search agents n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' positions of α (ξ1), β (ξ2) and δ (ξ3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' positions of search agents (wolves) η1, η2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', ηi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' ηi ∈ Rd and ξi ∈ Rd are d-dimensional vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In this case, d is equal to 2, representing two optimal hyperparameters (C and γ) required for search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) If the current iteration time t is less than the maximum iteration L, go to the subsequent steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' otherwise, pro- ceed directly to step 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3) For each agent, train RBF-SVM with current position elements ηi = (Ci, γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4) Predict trained RBF-SVM with the test set for each agent and output its loss as a fitness value based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (5): loss(ηi) = 1 M M � i=1 (yi − hi)2, (5) where M is the number of test samples and hi represents the predicted value for the ith test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 5) Sort all fitness values in ascending order and assign positions which have the corresponding top three fitness values as ξ1, ξ2 and ξ3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The mathematical expressions are ξ1(t) = arg max ηi(t),i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=',n loss(ηi(t)), (6) ξ2(t) = arg max ηi(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='ηi(t)̸=ξ1(t) loss(ηi(t)), (7) ξ3(t) = arg max ηi(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='ηi(t)̸=ξ1(t),ξ2(t) loss(ηi(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (8) 6) Update exploration-exploitation regulation function φ(t) based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (9): φ(t) = −2t − L + 1 −t + L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (9) 7) For each search agent, update its position ηi based on following equations: ηi(t + 1) = 1 4ξ1(t) + 1 4 3 � i=1 [ξi(t) (10) − bi(t) ⊙ |ci(t) ⊙ ξi(t) − ηi(t)|], bi(t) = 2φ(t)ri(t) − φ(t)1, ci(t) = 2si(t), (11) where ⊙ and | · | represent Hadamard product operation and element wise absolute value operations, respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' t is the iteration number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1 ∈ R2 and its elements are all ones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' bi ∈ R2 and ci ∈ R2 are coefficient vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 6 The coefficients ri ∈ R2 and si ∈ R2 are random vectors, where elements are in the range 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 8) Accumulate iterative time and go back to step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 9) Output the optimal parameters ξ1 = (Co, γo) and the trained SVM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 10) Calculate the classification accuracy of the model based on the test set and end the X-GWO-SVM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' We demonstrate improvements of the proposed X-GWO- SVM algorithm with respect to its exploration and exploitation ability in following two subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 1) Exploration: Conventionally, components of φ(t) are linearly decreased from 2 to 0 over the course of itera- tions [53], which models wolves approaching the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In our design, we set components of φ(t) non-linearly decrease from 2 to 0 with slower declining rate near 2 and faster declining rate near 0 (referring to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4 (a) demonstrates the components of φ(t) linearly (blue stars) and non-linearly (black circles) decreased from 2 to 0 over the course of iterations when the maximum iteration time L is set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Clearly, for the designed nonlinear decreasing method, we can observe that there is slow declining at the left side of the black dash line (iteration time = 94), aiming to explore a larger range and increase exploration compared to the conventional linear way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' As discussed in [53], bi(t) with random values greater than 1 or less than -1 is used to oblige the search agent to diverge from the prey, which emphasizes exploration and allows the X-GWO-SVM algorithm to search globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It should be noticed that the fluctuation range of bi(t) is also decreased under an effect of φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Components of bi(t) are random values in the interval [−φ(t), φ(t)], where components of φ(t) are non-linearly decreased from 2 to 0 over the course of iterations [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4 (b) shows a variation of bi(t) when linear and nonlinear φ(t) are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The blue and grey shadows indicate variation trends for bi(t) when linear φ(t) and nonlinear φ(t) are applied, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Obviously, the value of bi(t) (black circles) for nonlinear φ(t) applied has a larger range compared with the value of bi(t) (blue stars) for linear φ(t) at the left side of the black dash line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', |bi(t)| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In other words, the next search range for the fittest position in the nonlinear case smoothly attenuates before iteration reaches a threshold—94 in this figure, making sure a large exploration range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) Exploitation (convergence): In [53], when updating the positions, the weights for α, β, and δ wolves are all the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' While for our proposed approach, when updating the positions, we assign more weight to the α wolf (referring to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (10)), which emphasizes the importance of the α wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Consequently, the fittest solution from the previous iteration can be retained and continually influences the subsequent updating step, ensuring a faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' We can observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4 (a) that, for the designed nonlinear decreasing method, there is a much faster decay at the right side of the black dash line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To clearly track the convergence of φ(t), red-filled circles are utilized for the nonlinear case after the 94th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The convergence tends to speed up as the iteration continuously increases, whereas, 0 20 40 60 80 100 Iteration time t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 2 Range of (t) Linear decreasing (t) Nonlinear decreasing (t) (a) 0 20 40 60 80 100 Iteration time t 3 2 1 0 1 2 Range of bi(t) Linear decreasing bi(t) Nonlinear decreasing bi(t) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (a) Components of φ(t) linearly (blue circles) and non-linearly (black circles) decreased from 2 to 0 over the course of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (b) The corresponding variations for components of bi(t) based on the components of φ(t) linearly (blue circles) and non-linearly (black circles) decreased over the course of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The maximum iteration time L is set to 100 for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The black dash line lies at the 94th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The red-filled circles aim to clearly indicate variations of φ(t) and bi(t) for the nonlinear case after the 94th iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' for the conventional linear decreasing method, the components of φ(t) just evenly decrease from 2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' As we discussed before, the value of bi(t) is influenced by the value of φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, a similar phenomenon can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4 (b), where the value of bi(t) converges much faster than the linear case at the right side of the black dash line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', |bi(t)| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In other words, the next search range for the fittest position in the nonlinear case dramatically decreases after iteration time reaches 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To sum up, the proposed X-GWO-SVM algorithm does not require hyperparameter tuning on SVM in order to get good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Additionally, it improves the way of updating position by involving the fittest position α, which emphasizes the importance of the α wolf and keeps the effect of the fittest 7 solution for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' We also enhance the ability of exploitation by nonlinearly decreasing the value of φ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The algorithm improves its global search ability by increasing the exploration ability and speeds up the convergence ability by increasing the exploitation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Measurements The classification performance of various methods can be evaluated by standard statistical measurements: accuracy (ACC) and F1-score (F1), defined as ACC = TP + TN TP + FP + TN + FN, (12) F1 = 2PRE × REC PRE + REC, (13) REC = TP TP + FN, PRE = TP TP + FP, (14) where TP (true positive) is the number of samples correctly predicted as the current class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' TN (true negative) means the number of correctly predicted as other classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' FP (false positive) indicates the number of samples incorrectly detected as the current class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' FN (false negative) denotes the number of samples incorrectly detected as other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Accuracy is the general measurement of the correctly predicted ratio of the total testing samples for each dataset, indicating the method’s capability to classify emotions correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The F1-score, on the other hand, more accurately captures the ideal model for the unbalanced class distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The goal is to maximize these two measures as representations of effective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Feature dimension selection As aforementioned in Section III-A3, determination of a proper number of extracted features is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, we perform feature importance selection in the range of 20 to 135 with step size 5 under the X-GWO-SVM method for iRealcare dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Each simulation result is repeated 10 times for random selection of training and test samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 7 illustrates the accuracy versus the dimension of the feature under the X-GWO-SVM algorithm with 10-fold cross- validation for the iRealcare dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Clearly, the recognition accuracy displays the tendency to rise up at the beginning and decline in late.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The highest mean accuracy is 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37% located at the feature dimension equal to 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, its corresponding box plot (filled with orange color) has relatively low variance, indicating the stability of this feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' After getting the most discriminative result with feature dimension 95, we apply it to other comparison methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Exploration-exploitation regulation function selection As we demonstrated the significance of exploration- exploitation regulation function φ(t) in Section III-B, various exploration-exploitation regulation functions are used in our experiments here to demonstrate that our choice of φ(t) used in the X-GWO-SVM algorithm is the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Expressions on them are shown in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (15) to (19) and these exploration- exploitation regulation functions are plotted in Fig 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It should 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 Dimension of feature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='95 Accuracy 75 80 85 90 95 100 105 110 115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='94 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The accuracy versus the dimension of the feature under the X- GWO-SVM algorithm with 10-cross validation for the iRealcare dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The blue shadow indicates a trend for mean accuracy values of different dimensions of features under the X-GWO-SVM algorithm with 10-cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Box plot is employed with the box top and bottom denoting the 75th and 25th percentiles respectively for the results of 10-cross validation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The red straight line inside the box denotes the median value, while the red dot denotes the mean value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The blue star denotes the outlier value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The most discriminative result is filled with orange color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' be noticed that the conventional linear exploration-exploitation regulation function, a benchmark, is expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Additionally, the one we proposed in the X-GWO-SVM in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (9) is rewritten as fφ4(t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' fφ1(t) = 2 − 2t L , (15) fφ2(t) = 4 1 + et−L − 2, (16) fφ3(t) = −4 1 + e−t + 4, (17) fφ4(t) = −2t − L + 1 −t + L = φ(t), (18) fφ5(t) = 2 cos( π 2tL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (19) Based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 6, we can observe that both fφ2 and fφ3 are deformed from the Sigmoid function, which dramatically decrease from 2 to 0 at the beginning and the end of the iteration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The function fφ4 and function fφ5 successively alleviate this decreasing trend on a basis of function 1 x and cos, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To evaluate the effects of exploration-exploitation regulation function, we apply 10-fold cross-validation to the proposed X- GWO-SVM, varying exploration-exploitation regulation func- tions based on the aforementioned five functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The eval- uated results on exploration-exploitation regulation functions are shown in Table II for iRealcare dataset and Table III for WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Clearly, for iRealcare dataset, X-GWO-SVM combined with fφ4 has the highest accuracy (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37%) and F1- score (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38%) among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, the lowest variance and pretty low training time indicate its stability with low 8 0 20 40 60 80 100 Iteration time t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='5 2 Range of various (t) f 1 f 2 f 3 f 4 f 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Variations of components of φ(t) for different exploration- exploitation regulation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' A similar conclusion can be derived for results on the WESAD dataset, where the highest accuracy (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93%) and F1-score (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56%) are from the combination of X-GWO-SVM with fφ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To this end, we have determined the optimal feature dimension—95, and exploration-exploitation regulation function—fφ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, later evaluations of the proposed X-GWO-SVM are based on these two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' TABLE II RESULTS OF X-GWO-SVM INVOLVED WITH FIVE EXPLORATION-EXPLOITATION REGULATION FUNCTIONS ON IREALCARE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Exploration-exploitation regulation function Mean(ACC) Var(ACC) Mean(F1)/% Training time/s fφ1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='90 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='66E-05 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='91 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='98 fφ2 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='10E-05 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='08 402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='80 fφ3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='40E-02 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='50 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='66 fφ4 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='84E-05 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38 380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='16 fφ5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='31E-05 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='94 368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='14 TABLE III RESULTS OF X-GWO-SVM INVOLVED WITH FIVE EXPLORATION-EXPLOITATION REGULATION FUNCTIONS ON WESAD DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Exploration-exploitation regulation function Mean(ACC) Var(ACC) Mean(F1)/% Training time/s fφ1 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='62E-04 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07 856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='31 fφ2 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='79 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='48E-05 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='44 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='31 fφ3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='62E-04 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07 825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='14 fφ4 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='61E-05 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56 813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='58 fφ5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='79 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='48E-05 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='44 835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='40 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Classification Performance of Proposed Model 1) Classification Performance for of iRealcare dataset: One may suspect that only one of the improvements on X- GWO-SVM can achieve a considerable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus, we investigate the other three methods: 1) using the GWO-SVM method, where none of the improvement on GWO is applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) using the nonlinear φ(t) based grey wolf optimizer (N- GWO-SVM) method, where only the nonlinearly decreasing value of φ(t) is used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3) using PSO-SVM method, where the conventional PSO algorithm is used for searching optimal hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Tables IV to VII show the classification performance for the hyperparameter optimizer-based techniques stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The following metrics are reported: accuracy, F1-score, variation of accuracy, and training duration of the schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' All of them are calculated from 10 repeated classification trials for each scheme (rows in Tabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' IV to VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' TABLE IV THE MEAN ACC OF FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER BASED SCHEMES EVALUATED ON IREALCARE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Scheme Peacefulness/% Excitement/% Happiness/% Tension/% PSO-SVM 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='20 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='87 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='40 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='13 GWO-SVM 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='67 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='73 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='40 N-GWO-SVM 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='60 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='33 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='27 X-GWO-SVM 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='03 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='03 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='33 Table IV shows that GWO-SVM performs significantly better than PSO-SVM for peacefulness (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='20%), excitement (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='67% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='87%), and tension (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='73% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='13%), but N-GWO-SVM only slightly improved perfor- mance on peacefulness (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='60%) and excitement (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Except for a slightly lower performance on happiness com- pared to the PSO-SVM scheme (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07% vs 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='40%), the proposed X-GWO-SVM scheme provides a significant per- formance boost over others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It has the highest accuracy for peacefulness, excitement, and tension of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='03%, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='03%, and 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='33%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Table V presents similar results for the mean F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The proposed X-GWO-SVM scheme provides a significant performance boost over others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' TABLE V THE MEAN F1 OF FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER BASED SCHEMES EVALUATED ON IREALCARE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Scheme Peacefulness/% Excitement/% Happiness/% Tension/% PSO-SVM 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='29 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='46 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='45 GWO-SVM 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='19 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='87 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='77 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='57 N-GWO-SVM 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='59 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='99 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='03 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='36 X-GWO-SVM 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='73 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='31 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='23 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='24 TABLE VI THE MEAN VARIANCE OF ACC FOR FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER BASED SCHEMES EVALUATED ON IREALCARE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Scheme Peacefulness Excitement Happiness Tension PSO-SVM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='17E-02 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 GWO-SVM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='10E-03 N-GWO-SVM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='60E-03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='4E-04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04 X-GWO-SVM 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='01E-04 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='04E-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38E-04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='65E-05 The variance result in Table VI suggests a similar con- clusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Compared to the conventional PSO-SVM scheme or GWO-SVM scheme, except for the variance on peacefulness (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='01E-04 vs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00E-04), our proposed method also shows the lowest variance of accuracy on excitement (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='04E-05), happiness (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38E-04) and tension (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='65E-05), indicating its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Table VII shows the mean accuracy, the mean variance and the mean training time of 10-fold cross-validation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Note that, the proposed X-GWO-SVM scheme has the highest 9 TABLE VII THE MEAN VALUES OF ACC, F1, VARIANCE AND TRAINING TIME FOR FOUR EMOTIONS WITH HYPERPARAMETER OPTIMIZER BASED SCHEMES EVALUATED ON IREALCARE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Scheme Mean(ACC)/% Mean(F1)/% Mean(Var) Training time/s PSO-SVM 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='65 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='82E-02 8824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='80 GWO-SVM 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='97 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='50E-04 446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='24 N-GWO-SVM 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='03 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='25E-04 446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='42 X-GWO-SVM 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='19E-04 380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='16 mean of class accuracy (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37%), the highest mean of class f1-score (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38%), the lowest mean of class variance (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='19E- 04), and the shortest hyperparameter training time (380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='16s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The results demonstrated its high reliability, stability, and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' TABLE VIII THE MEAN ACC, MEAN F1 AND VARIANCE OF NONPARAMETRIC CLASSIFICATION METHODS AND THE PROPOSED METHOD EVALUATED ON IREALCARE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Algorithm Mean(ACC)/% Mean(F1)/% Mean(Var) RF 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='71 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56E-04 K-NN 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='48 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='22 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='35E-04 X-GWO-SVM 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='19E-04 To demonstrate the the significance of our X-GWO-SVM scheme, we also examine other nonparametric classification methods using similar features on the iRealcare dataset, such as RF and K-NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The results are shown in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' All of them are calculated from 10 repeated classification trials for each scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It is apparent from the results that the X- GWO-SVM scheme has the highest accuracy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Actually, RF is more stable than the X-GWO-SVM, while its accuracy and F1-score are much lower than the proposed scheme (81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='71% vs 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37%, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='35% vs 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='38%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In addition, the X-GWO-SVM completely outperforms K- NN in terms of reliability and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Overall, the proposed X-GWO-SVM strategy is more stable and more effective at achieving high mean accuracy on the iRealcare dataset than the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) Classification Performance for WESAD dataset: To demonstrate the reliability and stability of the proposed X- GWO-SVM method, we further examine it on the WESAD dataset in terms of accuracy and F1-score, compared with other existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' By applying the feature dimension selection in the range of 4000 to 10000 with step size 500 under the X-GWO-SVM approach as we described in Section IV-B, the best feature dimension is found to be 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 7 illustrates the accuracy versus the dimension of the feature under the X-GWO-SVM algorithm with 10-cross validation for the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Clearly, the recognition accuracy has a similar pattern to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 6, in which it increases initially and decreases afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The highest mean accuracy is 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93% located at the feature dimension equal to 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The corresponding box plot has the highest accuracy—96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='30%, the lowest accuracy—94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='44%, and the mean F1-score—95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, its corresponding box plot also has a relatively low mean variance (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='49E-05), indicating the stability of this feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The mean training time is 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='07s, in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Table IX presents a comparison between the proposed classification scheme and the state-of-the-art methods pub- lished for single channel ECG-based emotion recognition methods on the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The same testing dataset ensures that the comparison is persuasive and feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It can be observed from Table IX that the proposed classification approach outperforms all simple machine learning methods, such as RF, K-NN, linear discriminant analysis, and decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Though slightly inferior to that of self-supervised CNN, the proposed X-GWO-SVM technique exhibits comparable classification performance among deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' How- ever, considering the computation complexity, the proposed method is much simpler and more efficient than the self- supervised CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Our algorithm has successfully been loaded into a lightweight embedded system with a prediction time of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='659ms per 200-points iRealcare sample and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='648ms per 14000-points WESAD sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Details on these results will be discussed in Section V-C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Overall, the proposed X- GWO-SVM method achieves comparable accuracy and F1- score (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 7 and Table IX ) among neural network-based deep learning classifiers on the WESAD dataset and has an overwhelming performance on other existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 Number of features 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='95 Accuracy Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The accuracy versus the dimension of the feature under the X- GWO-SVM-SVM algorithm with 10-cross validation on WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' TABLE IX COMPARISON OF VARIOUS SINGLE CHANNEL ECG-BASED EMOTION RECOGNITION METHODS ON WESAD DATASET Reference Year Method ACC/% F1/% RF 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='78 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='64 K-NN 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='19 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='39 Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' [43] 2018 Linear discriminant analysis 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='44 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='31 AdaBoost Decision Tree 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='37 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='20 Decision Tree 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='17 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='01 Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' [47] 2019 CNN 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='00 Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' [29] 2020 Self-supervised CNN 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='90 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='30 Fully-supervised CNN 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='20 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='20 Proposed work —— X-GWO-SVM 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='93 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' DISCUSSION The X-GWO-SVM algorithm, for the first time, is proposed and also the first time used in single channel ECG-based emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' By designing a suitable exploration- exploitation regulation function and updating technique, we are able to increase the exploration ability and exploitation ability with the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Two ECG datasets are 10 used: one raw self-collected iRealcare dataset and one credible WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The X-GWO-SVM technique effectively avoids the algorithm from falling into a local solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' hence, it has a greater recognition accuracy than the existing GWO- SVM and PSO-SVM techniques for ECG emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The algorithm enables accurate, stable, and efficient emotion recognition based on single-channel ECG-based signals, which fills a gap for GWO-SVM research on ECG-based emotion recognition and also has the potential for clinical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Evaluation of datasets Despite the restricted number of subjects in the iRealcare dataset, the number of samples for each subject is sufficient since the time of data collection for each emotion is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It is true that the WESAD dataset contains a larger number of subjects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' however, the sample length required for this dataset to achieve high accuracy, which is 14000, drastically reduces the actual number of samples, for example, 9 samples for each subject on amusement, 15-18 samples for each subject on stress, and 28-29 samples for each subject on the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' On the contrary, the sample length required for the iRealcare dataset to achieve high accuracy, which is only 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, the number of samples for the iRealcare dataset is much larger than the one in the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The reason for the caused aforementioned situation might come from the way of giving external stimulus and recording data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For the iRealcare dataset, ECG signals for happiness, tension, and excitement are recorded when subjects watch comedies, watch thriller movies and do exercises, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It should be noticed that emotions normally instantaneously occur and hold for a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Therefore, we only record the period that subjects are actually in that emotion condition and ignore the transition period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Clearly, the definition of different emotions under this external stimulus setting is clear and subjects are easy to get into a specific emotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, for the WESAD dataset, amusement condition signals are col- lected when subjects watch funny video clips;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' stress condition signals are collected when subjects are asked to provide public speaking and mental arithmetic tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' baseline condition sig- nals are collected when subjects sit/stand at a table and read magazines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In fact, subjects tend to take some time to transfer from one emotion condition to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, such a transition period is also recorded in the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus, the sample length need to be long enough to make sure not just the transition period is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The shorter the time, the more probable it is that only transitional periods will be included in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' To sum up, in spite of the fact that the iRealcare dataset has limited subjects, the actual number of samples is much larger than the one in the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, due to the exclusive emotion transition period for the WESAD dataset, the selection of sample length for the iRealcare dataset is more flexible than the WESAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' We use the widely-used WESAD dataset as a benchmark for further comparison to validate our proposed X-GWO-SVM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Evaluation of exploration-exploitation regulation function selection We study the impact of exploration-exploitation regulation functions on emotion recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For this study, we select possible base functions that control the signifi- cance of each exploration-exploitation regulation function as listed in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' (15) to (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Table II and Table III show the emotion recognition performance for five exploration- exploitation regulation functions on the iRealcare dataset and WESAD dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This analysis provides in-depth insight into the effect of the exploration-exploitation regulation functions associated with the emotion recognition outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Furthermore, this analysis helps us narrow down the most suitable exploration-exploitation regulation function in order to achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' As we mentioned in Section III-B, the declining rate of the exploration-exploitation regulation function at the beginning and end with respect to the iteration time represents the exploration and exploitation ability of the proposed X-GWO- SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' From Table II and Table III, we notice that for the exploration-exploitation regulation function fφ3, where the function is formed on a basis of the sigmoid function, the model performance on emotion recognition is poor since it is under-explored and under-exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' However, for those exploration-exploitation regulation functions lying above the benchmark function fφ1, the model shows significantly bet- ter performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Interestingly, the performance drops when exploration-exploitation regulation functions decline too fast (fφ2) or too slow (fφ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The function fφ4 gives the highest performance for emotion recognition compared to others since it has the most suitable diverging and converging performance to the X-GWO-SVM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, the fact that fφ4 outperformed other functions for both datasets is also indica- tive of its stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In summary, the analysis above shows that for all the exploration-exploitation regulation functions, when the declin- ing rate of the function at the beginning is too large or too small, for example, fφ3 or fφ2, emotion recognition accuracy drops due to the under-exploration or over-exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This results in the X-GWO-SVM more easily falling into local solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Similarly, when the declining rate of the function at the end is too large or too small, fφ2 or fφ3, the performance also drops due to the under-exploitation or over-exploitation in such cases becomes too difficult for the algorithm to properly find the global solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Hence, we conclude that there is a trade- off between exploration and exploitation for the exploration- exploitation regulation functions associated with the proposed X-GWO-SVM algorithm, for which the proper exploration- exploitation regulation function fφ4 is applied resulting in avoiding falling into local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Evaluation of reliability, stability and efficiency of X-GWO- SVM This section discusses the performance of X-GWO-SVM for emotion recognition in terms of reliability, stability, and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 11 1) Reliability: In our work, when only fusing exploration- exploitation regulation function fφ4 with GWO-SVM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', N-GWO-SVM, the classification accuracy and F1-score get improved (referring to Table IV, Table V and Table VII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This improvement indicates that involving a nonlinear exploration- exploitation regulation function can improve the recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Similarly, the classification accuracy and F1- score get further enhanced when the importance of the α wolf is emphasized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', X-GWO-SVM, which shows the equivalent importance of the improvement on the X-GWO-SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The classification performance of X-GWO-SVM is superior to the classification performance of other hyperparameter optimizer-based systems, such as PSO-SVM and GWO-SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This indicates the high reliability of the proposed algo- rithm over existing common hyperparameter optimizer-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Furthermore, the X-GWO-SVM has the highest ac- curacy and F1-score among simple machine learning methods, such as RF, K-NN, decision tree, and linear discriminant analysis (Table VIII and Table IX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Our analysis indicates that the X-GWO-SVM is more effective than simple machine learning approaches at avoiding local solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For the deep learning neural networks, such as CNN, the X-GWO-SVM can still outperform them except for a more complex single— self-supervised CNN [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Though the accuracy and F1-score of the X-GWO-SVM are slightly lower than the one from the self-supervised CNN, considering the efficiency, which will be discussed in Section V-C3, our algorithm is still competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 2) Stability: The variance of the proposed method and existing works is computed to evaluate the stability of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' All simulation results are applied with 10-fold cross- validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Similar to the discussion in Section V-C1, the X-GWO-SVM is the most stable algorithm among existing common hyperparameter optimizer-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Besides, similar results on both the iRealcare dataset and the WESAD dataset also indicate the stability of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' 3) Efficiency: Our works are implemented through both MATLAB version R2021b and Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='7 for feature ex- traction, model training, and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For MATLAB, the computation is performed on a laptop with 11th Gen Intel(R) Core(TM) i7-11800H (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='2GHz and 32GB of RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The computation time for classifying a 200-points (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56s) iRealcare sample and a 14000-points (20s) WESAD sample roughly spends 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='355ms and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='778ms, respectively, using our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' For Python, the computation is performed in JETSON NANO with Quad-core ARM Cortex-A57 MPCore Processor (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='43GHz and 4GB of RAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The computation time for classifying a 200-points (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='56s) iRealcare sample and a 14000- points (20s) WESAD sample roughly spends 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='659ms and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='648ms, respectively, using our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Compared with the self-supervised CNN, a deep neural net- work, the proposed X-GWO-SVM is much simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The two- step self-supervised architecture involves deep convolutional blocks and several fully connected layers in [29], which may not be realized in lightweight embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Whereas, our algorithm has successfully been loaded into JETSON NANO, an embedded system-on-module and developer kit with a prediction time of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='659ms per 200-points iRealcare sample and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='648ms per 14000-points WESAD sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' This provides a way to embed an ECG patch with the proposed algorithm, achieving edge computing for emotion recognition on ECG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, the X-GWO-SVM is the most efficient algo- rithm among existing common hyperparameter optimizer- based schemes, which is evaluated by the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The other interesting point that can be found in Table VII is that all GWO-SVM-based techniques take shorter training time than the PSO-SVM work, which is compatible with [35]’s conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Limitations and future directions The possible limitation of the current study would be that we only investigate four exploration-exploitation regulation functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', fφ2, fφ3, fφ4, and fφ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Moreover, the dataset we collected is still insufficient and other existing published datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', AMIGOS [27], Augsburg Biosignal Toolbox (AuBT) [54], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=', have not been verified by the proposed X-GWO-SVM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In future work, we will use other exploration-exploitation regulation functions for the proposed algorithm to explore their effectiveness in emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Additionally, more published datasets will be examined by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Besides, through the results and conclusions reported in [29], we also observed that deep learning is competitive in emotion recognition, which may further improve the perfor- mance of our proposed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' Thus in our future work, we will try to find an effective deep learning method and embed- ded GWO methods to further improve emotion recognition performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' CONCLUSION In this paper, we presented an X-GWO-SVM technique that improves the exploration and exploitation abilities of single channel ECG-based emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In order to classify different emotions, this research used two reliable datasets: one trustworthy WESAD dataset and one raw self-collected iRealcare dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The single channel ECG signals could well be employed in the X-GWO-SVM algorithm for emotion recognition, according to 10-fold cross-validation results from 5 subjects for the iRealcare dataset and 15 subjects for the WE- SAD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The algorithm performed better than past efforts that used various supervised machine learning techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' It also provides a way to implement in the lightweight embedded system, which is much more efficient than existing solutions of using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' The method has the potential to be used in clinical settings and also fills a gap in GWO-SVM research on ECG-based emotion identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' In our future work, we will apply radio sensing techniques, such as [42], [55]–[58], instead of wearable devices for emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQf7P6B/content/2301.01887v1.pdf'} +page_content=' We will also develop privacy preservation algorithms 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mode 100644 index 0000000000000000000000000000000000000000..a6617209d3494558e5fa7cb9c8805628f74b15f7 --- /dev/null +++ b/wNE2T4oBgHgl3EQf3AhK/content/tmp_files/2301.04166v1.pdf.txt @@ -0,0 +1,3502 @@ +MNRAS 000, 1–20 (2023) +Preprint 12 January 2023 +Compiled using MNRAS LATEX style file v3.0 +A Wide-Field View on Multiple Stellar Populations in 28 Milky Way +Globular Clusters +E. Leitinger,1,2★ H. Baumgardt,1 I. Cabrera-Ziri3 M. Hilker2 and E. Pancino4,5 +1School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia +2European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany +3Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12-14, D-69120 Heidelberg, Germany +4INAF – Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, I-50125 Firenze, Italy +5Space Science Data Center, ASI, via del Politecnico snc, I-00133 Roma, Italy +Accepted 19 December 2022 +ABSTRACT +The majority of Galactic globular clusters (GCs) contain multiple stellar populations displaying +specific chemical abundance variations. In particular, GCs generally contain a ‘primordial’ +population with abundances similar to field stars, along with an ‘enriched’ population ex- +hibiting light element anomalies. In this paper we present a homogeneous and wide-view +analysis of multiple stellar populations in 28 Galactic GCs. By using a combination of HST +photometry together with wide-field, ground-based photometry we are able to analyse be- +tween 84% and 99% of all stars in each cluster. For each GC, we classify stars into separate +sub-populations using the well-established 𝐶UBI colour index, and investigate the spatial dis- +tributions of these populations. Our results show that dynamically young GCs can contain +either centrally concentrated enriched or primordial populations, or no centrally concentrated +population. Dynamically old GCs show fully mixed populations as expected. The existence of +clusters born with centrally concentrated primordial (and homogeneously mixed) populations +exacerbates the mass-budget problem facing many cluster formation scenarios. The diversity +in these results also highlights the need for additional theories that can account for the wide +variety of initial conditions that we find. We finally investigate the enriched star fraction as +a function of different global parameters in our GC sample, using also data for young and +low-mass clusters from the Small- and Large Magellanic Clouds and confirm earlier results +that the enriched star fraction strongly correlates with the initial mass of a cluster. +Key words: +(Galaxy:) globular clusters: general – Stars: abundances – (stars:) +Hertzsprung–Russell and colour–magnitude diagrams – stars: kinematics and dynamics – +Galaxy: evolution +1 +INTRODUCTION +Most Galactic globular clusters (GCs) contain multiple stellar pop- +ulations (MPs), distinguished by star-to-star variations in light ele- +ment abundances that are not explained by simple stellar evolution. +Stars are determined as ‘primordial’ (usually as P1) if their ele- +mental abundances are similar to the surrounding field stars of the +cluster, and ‘enriched’ (P2) if they demonstrate an enhancement in +some light elements (e.g. He, N, Na and Al), but a depletion in +others (e.g. C, O and sometimes Mg) in comparison to P1 (Gratton +et al. 2012; Charbonnel 2016; Bastian & Lardo 2018). However, +heavier element variations such as Fe only are present in a minority +of clusters (Carretta et al. 2009; Willman & Strader 2012; Bastian & +Pfeffer 2022). The formation history of GCs necessary to produce +MPs is a matter of ongoing debate (Forbes et al. 2018; Gratton et al. +★ E-mail: ellen.leitinger@uq.net.au +2019; Cassisi & Salaris 2020). We know that the observed abun- +dance patterns are compatible with the chemistry of the CNO-cycle +(and hot subcycles) and that this happens mostly in massive stars +or in the H-burning shells of red giants, which leads to the theory +that stellar formation of the enriched populations is fuelled by GC +internal processes. +An important piece of information regarding the formation +history of MPs in GCs is the spatial distribution of the stars in each +population. If a cluster has not undergone significant dynamical +mixing during its lifetime, we can assume it still maintains its initial +spatial configurations. If we then observe that one stellar population +is located primarily within the centre of such a cluster, we can +assume this was the initial configuration of the stars during cluster +formation. The analysis presented in this paper focuses in part on +the spatial distribution of the MPs, which serves as a way to test +the validity of the current processes theorised to describe cluster +formation. +© 2023 The Authors +arXiv:2301.04166v1 [astro-ph.GA] 10 Jan 2023 + +2 +E. I. Leitinger et al. +One such process is the AGB scenario, first proposed by Cot- +trell & Da Costa (1981), in which first generation (P1) AGB stars +expel enriched material by stellar winds, which accumulates in the +center of the cluster and mixes with primordial material to spark +a second event of star formation - creating P2 stars. However, for +clusters in which the P2 population is equal to, or more massive +than, the P1 population, the AGB scenario encounters a ‘mass bud- +get’ problem since, assuming a standard stellar mass function, the +enriched material created from P1 stars is not sufficient to create the +P2 stars we observe in some clusters (e.g. Prantzos & Charbonnel +2006; Cabrera-Ziri et al. 2015). An implication of the AGB scenario +is that an enriched star formation event occurring in the center of +the cluster will lead to centrally concentrated P2 stars. +Another formation process involves enrichment due to Super +Massive Stars (SMS) (Denissenkov & Hartwick 2014; Gieles et al. +2018) that form due to runaway collisions in the early stages of +cluster formation and reside in the center of a cluster, providing a +‘conveyer belt’ of enriched material with different He fractions. This +theory can overcome the mass budget problem as the continuous +stellar collisions provide additional Hydrogen, which constantly +rejuvinates the SMS. In this theory, P2 star formation occurs in the +regions surrounding the SMSs. +Fast rotating massive stars were proposed by Decressin et al. +(2007a,b) to account for the observed chemical inhomogeneities, as +massive stars create the required enriched material for additional +star formation events, while the fast rotation brings the material to +the surface of the star and ejects it. In this scenario, secondary star +formation events occur in the region surrounding the fast rotating +massive stars after the enriched material is diluted by left over +primordial gas. +Finally, massive interacting binaries have been suggested as +a probable cause for the chemical enrichment found in MPs of +GCs by de Mink et al. (2009) and Renzini et al. (2022). Renzini +et al. (2022) theorised that above a certain critical mass threshold, +massive stars skip the supernova stage and instead implode into +black holes, therefore ensuring the remaining stars in the cluster do +not contain an abundance spread in Fe. As the centers of GCs are +much denser than the outer regions, binary stars are expected to be +destroyed or ejected at a higher rate in the center than they do in the +outer regions due to increased collisions. Lucatello et al. (2015) +discovered a higher fraction of binaries within the P1 population, +as opposed to the P2 population in 10 Galactic GCs, which seems +to support theories that assume P2 stars are centrally concentrated. +Dalessandro et al. (2019) studied the radial distribution of +20 Galactic GCs as a function of the age/relaxation time fraction +(hereby referred to as ‘dynamical age’) using HST photometry +and N-body model simulations. They found that clusters with +low dynamical ages preferentially contain centrally concentrated +P2 populations. It is expected that clusters with lower dynamical +ages have not undergone much dynamical mixing in their lifetime +and are therefore still exhibiting properties close to their initial +conditions. Clusters with higher dynamical ages were found to have +spatially blended multiple stellar populations, in agreement with +the idea that these clusters have undergone significant dynamical +mixing. The results found by Dalessandro et al. (2019) provide +observational evidence for formation theories in which enriched +populations are formed within the center of the cluster. In their +review, Bastian & Lardo (2018) concluded that GCs might not +have homogeneous histories, suggesting instead that MPs can be +formed through a variety of individual scenarios. In this case we +could assume that the scenarios mentioned above are responsible +for clusters with centrally concentrated P2 stars. However, if a +cluster contains centrally concentrated P1 stars, there are no current +theories to explain this. +In this work, we study a diverse sample of 28 Galactic GCs +in order to provide a comprehensive insight into the various pos- +sibilities of cluster properties. Large scale photometric analyses +have been performed on Galactic GCs by Monelli et al. (2013); +Milone et al. (2017); Stetson et al. (2019), revealing intriguing scal- +ing relations that may help us understand the origin of MPs. So +far, combined space- and ground-based photometry for the purpose +of obtaining a thorough spatial analysis of MPs and their charac- +teristics exists only for a small number of clusters. We used both +space-based and ground-based photometry to perform a homoge- +neous analysis of the wide-field spatial extent of a large sample of +GCs, using the well-established color combination CUBI and chro- +mosome map methods in order to separate the MPs. In this paper we +categorise MPs in space- and ground-based photometry separately, +before combining the results to investigate correlations in terms of +spatial distributions, enriched star fractions and global properties. +We also compare our results with theoretical data and combine the +Galactic GCs with Local Group GCs to further investigate trends. +2 +OBSERVATIONAL DATA +The photometric catalogues used in this work include the wide- +field ground-based Johnson-Cousins UBVRI photometric data pro- +vided by Stetson et al. (2019), along with the space-based HST +UV Globular Cluster Survey data (‘HUGS’) (Piotto et al. 2015a; +Nardiello et al. 2018) with photometry obtained through UV/blue +and WFC3/UVIS filters. For this first project we will focus our anal- +ysis of multiple stellar populations only on the RGB stars of these +catalogues, combining both the HST and ground-based photometry +in order to observe a wide-field view of each cluster, covering at +least 84% of the stars. The Stetson et al. (2019) photometric cat- +alogue includes 48 GCs and the HUGS survey includes 57 GCs, +but only 32 of these clusters overlap and exist in both catalogues. +Of these 32 clusters, we successfully classified distinct MPs in 28 +of them. We excluded clusters from our sample if they contained +too few RGB stars after removing non-members and performing +photometric cleaning, or if the classification of cluster stars into +different sub-populations was inconclusive. The ground-based cat- +alogues cover almost the full extent of each cluster, but cluster +centers have much higher stellar densities than the outer regions, +causing blending to affect the photometry of stars close to the center. +This is where using HST photometry for the inner regions of clusters +has an advantage, as crowding is less of an issue with space-based +photometry. +In this section we detail the steps taken to remove non- +members, non-RGB stars and bad photometry from each photo- +metric catalogue before separating the multiple stellar populations +in Section 3 and characterising the cluster properties in Section 4. +Both the ground-based and HST catalogues encountered issues +with different types of incompletenesses. In areas of the observed +fields where either no stars were measured in a relevant filter, or the +photometry was too poor to be usable, we could not reliably make +assumptions about the properties of stars in that area. We calculated +completeness fractions for the remaining stars so that we account for +the stars that were missed. We describe the spatial incompleteness +in Section 2.1, the photometric incompleteness in Section 2.6 and +the surface density incompleteness in Section 2.7. +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +3 +Figure 1. Spatial completeness of the HST photometry for the globular +cluster NGC 5024. Artificial test stars are shown in red (green), if our +test indicated they fall outside (inside) the area covered by photometry. +Stars shown in black are real stars located in regions that fall below 50% +completeness. +2.1 +Spatial Completeness Correction +Our first step in processing both the ground-based and HST cata- +logues was determining the spatial completeness fraction 𝑓𝑆 of each +catalogue independently. Using the original catalogues for both the +ground-based and HST photometry, the spatial position of each star +in right ascension and declination were calculated as an offset from +the cluster center. The data was not cleaned for stars without mea- +sured photometry, defined as mag < 0 for HST and mag > 99 for +the ground-based photometry, since these entries in the catalogues +still indicated the presence of a star. +We distributed a series of concentric rings spaced by 1.0′′ +in distance around the cluster centers and distributed 360 artificial +points evenly spaced by 1 degree along each ring. For each of +these artificial points, we determined the distance between the point +and its nearest star, from the surrounding stars in our photometry. +A point was considered to be covered by the photometry if the +minimum distance was less than a tolerance distance - usually close +to 1 arcsec, but otherwise dependent on the cluster. This method has +the flexibility to be able to account for arbitrary field geometries, +including large gaps within the field. +The spatial completeness 𝑓𝑆 of each annulus was set equal to +the fraction of points that were covered by photometry in the field: +𝑓𝑆 = +𝑁in +𝑁total , where 𝑁in is the number of points inside the observed +field and 𝑁total = 360, the total number of points for that annulus. +We discarded photometry outside the radius in which the spatial +completeness drops below 50%, shown as black points in Figure +1, using NGC 5024 as an example. Surviving stars were assigned +a spatial completeness fraction (0.5 ≤ 𝑓𝑆 ≤ 1.0), based on the +completeness of the annulus they were located within. The HST +and ground-based data was combined without allowing spatial gaps +in the field by ensuring the ground-based data begins at the same +radius at which the HST data ends for all clusters. +1000 +500 +0 +500 +RA offset [arcsec] +1000 +500 +0 +500 +1000 +Dec offset [arcsec] +1000 +500 +0 +500 +RA offset [arcsec] +1000 +500 +0 +500 +1000 +Dec offset [arcsec] +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +dEBV +0.06 +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +Interpolated dEBV +Figure 2. Top panel: Differential reddening map of NGC 6121. Bottom +panel: Interpolation of the reddening map onto the ground-based photometry +after spatial completeness correction in order to assign individual values of +dEBV based on the nearest-neighbour in the top panel. +2.2 +Differential Reddening Correction +To compute differential reddening maps, we used a method simi- +lar to other methods employed in the literature (e.g., Milone et al. +2012), which will be described in detail in a forthcoming publication +(Pancino et al., in preparation). We used the ground-based photom- +etry by Stetson et al. (2019), selecting stars with photometric errors +lower than 0.3 mag in 𝐵𝑉𝐼, 𝜒 < 3, and |sharp| < 0.5. We computed +a fiducial line as the median ridge line of the main sequence of each +cluster, down to about 2–4 magnitudes below the turnoff point. We +selected stars not further than the 5 and 95% percentiles from the +fiducial line in the three color planes𝑉,𝐵–𝑉;𝑉,𝑉–𝐼; and𝑉,𝐵–𝐼. This +allowed us to remove a large fraction of contaminating field stars. +The color difference of each selected star from the reference line +was computed in the three planes along the reddening line, assum- +ing R𝑉 = 3.1 and using Dean et al. (1978) to compute the reddening +line direction in each plane. We then rescaled these raw color differ- +ences and combined them into one single estimate of ΔE(B–V) for +each star. To disentangle photometric errors and other effects from +the actual differential reddening signal, we smoothed these maps in +right ascension and declination. by replacing the ΔE(B–V) of each +star with the median of its 𝑘 neighbors, with 𝑘 ranging from 50 +to 300 (typically in the range 150-200) depending on the cluster. +MNRAS 000, 1–20 (2023) + +Covered by photometry +Not covered by photometry +Discarded stars'since +100 +completeness<50% +DEC offset [arcsec] +50 +0 +-50 +-100 +-100 +-50 +0 +50 +100 +RA offset [arcsec]4 +E. I. Leitinger et al. +8 +10 +12 +14 +16 +18 +20 +22 +24 +I +1.00 +0.75 +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +Sharp +Figure 3. Sharp parameter cuts for the ground-based photometry of NGC +5024. Black points represent stars that survived the cut, red points were +removed. The two red vertical lines represent rough limits in magnitudes +to isolate the RGB. An ‘envelope’ function in red encloses stars with large +enough photometric quality, as defined by Equation 1. +10 +12 +14 +16 +18 +20 +I +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Figure 4. 𝜒 parameter cuts performed only on the ground-based photometry. +Black points represent the stars of NGC 5024 that survived the sharp cuts +of Figure 3, while red points were removed. All stars beneath the red line, +defined by Equation 2 are kept. +This also allowed us to compute an uncertainty for each differential +reddening estimate as the median absolute deviation of the values +for the 𝑘 neighbours. +To correct the ground-based photometry, the reddening map +was interpolated for each star in both the HST and ground-based +catalogues, as shown in the bottom panel of Figure 2. We used +the standard ratio of absolute to selective extinction of 𝑅𝑉 = 3.1, +with the exception of NGC 6121, for which the value of 𝑅𝑉 = +3.76 ± 0.07 was used as suggested by Hendricks et al. (2012). +Magnitude corrections for the ground-based 𝑈 and 𝐵 bands were +applied using extinction ratios according to Cardelli et al. (1989), +while the 𝑅 and 𝐼 bands were corrected according to Dean et al. +(1978). Similarly for the HST photometry, differential reddening +was corrected for the 𝐹275𝑊, 𝐹336𝑊, 𝐹438𝑊 and 𝐹814𝑊 bands +using extinction ratios from the SVO Filter Profile Service (Rodrigo +et al. 2012; Rodrigo & Solano 2020). +2.3 +Photometric Quality Indicators +We removed stars with less reliable photometry by using different +quality indicators based on the available parameters provided by +the HST and ground-based catalogues. For the ground-based pho- +tometry, we implemented quality cuts based on magnitude errors +and the 𝜒 and sharp parameters described in the work of Stetson +& Harris (1988). For the HST photometry we implemented cuts in +sharp while also using the membership probability and quality-fit +parameters (QFIT) for each star provided by Nardiello et al. (2018). +For the ground-based photometry, the U,B,V and I bands with as- +sociated errors > 9 mag were cut. For the HST photometry, using +the same constraints as Dalessandro et al. (2019), stars belonging to +the cluster were selected using membership probability > 75% and +𝑄𝐹𝐼𝑇 > 0.9 in each of the 𝐹336𝑊, 𝐹438𝑊, 𝐹606𝑊 and 𝐹814𝑊 +bands. +For both photometry sets, cuts were made based on the sharp +values following a method similar to Stetson et al. (2003), but re- +placing the −1 ≥ sharp ≥ 1 criterion with an ‘envelope’ function. +We defined an exponential function above and below the bulk of the +values to remove stars with sharp values too far from the mean: +|sharp| < 0.15 + 𝑒𝑥𝑝 +� 𝑚𝑎𝑔 − 22 +1.5 +� +, +(1) +where 𝑚𝑎𝑔 += +𝐼 for the ground-based photometry and +𝑚𝑎𝑔 = 𝐹814𝑊 for HST. Figure 3 shows the cut for ground-based +photometry in which stars enclosed within the envelope are kept. +For the ground-based photometry we also used the 𝜒 parame- +ter, which determines the observed vs. expected pixel-to-pixel scat- +ter. By adapting the method from Stetson et al. (2003), a function +was applied to remove outliers: +𝜒 < 1.2 + 2 × 10(−0.2(𝐼−12)). +(2) +Stars which met the criterium are shown in black in Figure 4, while +stars in red were rejected. +2.4 +Proper Motion Cleaning +The HST photometry includes a membership probability parameter +(see Nardiello et al. 2018) to help discard stars that do not belong to +the cluster, as discussed in Section 2.3. Determining the true mem- +bers of a cluster for the ground-based photometry was done using +proper motions of the stars after cross-matching with the Gaia DR3 +catalogue (Gaia Collaboration et al. 2016, 2021). This catalogue +is comprehensive in scale, but has difficulties with incompleteness +in the center of clusters and lower accuracy due to the high stellar +crowding (Vasiliev & Baumgardt 2021). +In order to enforce an equivalent MS turn-off limit between all +catalogues, we first located the MS turn-off in the Gaia G band and +applied a cut exactly at this magnitude to isolate the RGB stars. This +was a precaution against matching faint stars from one catalogue +to bright stars in another catalogue (for stars in close proximity to +each other). We then cross-matched between the Gaia, HST and +ground-based catalogues within a 0.5′′ tolerance and determined +the equivalent MS turn-off in the HST and ground-based catalogues. +We isolated the RGB stars in each catalogue using the equivalent +MS turn-off limits found from this process. +Proper motion cleaning was only performed on the ground- +based photometry outside the HST footprint due to the aforemen- +tioned high stellar densities in the center of the clusters. The stars +matched with the Gaia catalogue were then proper motion cleaned +using a 𝜒2 test, defined in Equation 3, using both the right ascen- +sion 𝜇𝛼∗ and declination 𝜇𝛿 proper motion components and cor- +responding errors. The cluster proper motion values (𝜇𝛼∗,𝑐𝑙𝑢𝑠𝑡𝑒𝑟 +and 𝜇𝛿,𝑐𝑙𝑢𝑠𝑡𝑒𝑟) were taken from Vasiliev & Baumgardt (2021). We +include a proper motion error of 0.2 mas/yr to account for both the +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +5 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +V - I +12 +13 +14 +15 +16 +17 +18 +19 +20 +V +Non-members +Members +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 +* [mas/yr] +10.0 +7.5 +5.0 +2.5 +0.0 +2.5 +5.0 +7.5 + [mas/yr] +Figure 5. Ground-based photometry for NGC 5024, demonstrating the effect +of proper motion cleaning. Upper panel: CMD of stars above the approx- +imate MS turn-off, with accepted stars in black and rejected stars in red. +Lower panel: The proper motion distributions of stars matched with Gaia +EDR3, divided into members (black) and non-members (red). +internal velocity dispersion of the cluster and any proper motion +errors that may be underestimated. +𝜒2 = +(𝜇𝛼∗,𝑐𝑙𝑢𝑠𝑡𝑒𝑟 − 𝜇𝛼∗)2 +(𝜇𝛼∗,𝑒𝑟𝑟)2 + 0.2[mas/yr]2 + +(𝜇𝛿,𝑐𝑙𝑢𝑠𝑡𝑒𝑟 − 𝜇𝛿)2 +(𝜇𝛿,𝑒𝑟𝑟)2 + 0.2[mas/yr]2 (3) +The cut-off limit for the 𝜒2 value was slightly varied for each cluster, +depending on the background stellar density and how clearly the +cluster motion was distinguishable from the background. In order +to limit the effect of large errors allowing non-members to pass, we +implemented an error tolerance relative to the proper motion of the +cluster. The resulting cluster member stars are shown in black in +both panels of Figure 5, while rejected stars are shown in red. The +ground-based stars within 100′′ of the cluster center were added to +the confirmed cluster member stars for the photometric cleaning in +Section 2.5. We did this as these inner ground-based stars assisted +with photometric cleaning and were removed anyway once the HST +and ground-based photometry were combined. +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +U - V +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +3.00 +B - I +Fitted Data +Clipped Data +Polynomial Fit +Figure 6. Polynomial fitting of RGB stars in colour-colour combinations +𝑈 − 𝑉 vs 𝐵 − 𝐼 for the ground-based photometry of NGC 5024. The line +of best fit for the RGB stars is in green, cluster members are in black and +non-members removed via the 𝑁 − 𝜎 clipping method are in red. +2.5 +Photometric Cleaning +The purpose of the photometric cleaning process was to remove +non-members and non-RGB stars so that the resulting distribution +of RGB stars could be separated into multiple populations. We +identified the Horizontal Branch (HB) and AGB stars in CMDs +created from both the HST and ground-based photometry, as well as +red and blue outlier stars that stray too far from the RGB. These stars +were then manually removed from both sides of the RGB, allowing +us to easily approximate and fit polynomials to the location of the +RGB in the cluster CMD. +We applied a polynomial fit to the RGB in colour-colour and +colour-magnitude diagrams using the Astropy LinearLSQFitter (As- +tropy Collaboration et al. 2018), so that outliers could be removed +using an 𝑁 − 𝜎 clipping method. The colour-colour combination +of 𝑈 − 𝑉 vs 𝐵 − 𝐼 ground-based bands shown in Figure 6 was +used for the polynomial fit, where the median (𝑚fit) was required +(as opposed to the mean) as outliers surrounding the RGB stars +can heavily affect mean values. Depending on the contamination of +non-members and AGB stars in each cluster, the number of standard +deviations to be cut from the median was adjusted within the range +2 ≤ 𝑁 ≤ 3. Highly contaminated clusters required a closer cut +and therefore a smaller value of 𝑁. Non-members were identified +according to (𝑈 −𝑉)obs − (𝑈 −𝑉)fit > 𝑚fit ± (𝑁𝜎(U−V)), meaning +all stars with a colour difference greater than 𝑁 standard deviations +from the median of the polynomial fit were clipped. The process was +iterated a maximum of three times. We also used this process for +the HST photometry by using the closest equivalent colour-colour +combination in the available HST bands. +We then applied the same 1D polynomial fitting and 𝑁 − 𝜎 +clipping method to the following colour-index combinations in the +ground-based photometry: (𝑉 − 𝐼), (𝐵− 𝐼) and (𝑈 − 𝐵), and the HST +photometry: (𝐹606𝑊 − 𝐹814𝑊), (𝐹438𝑊 − 𝐹814𝑊), (𝐹336𝑊 − +MNRAS 000, 1–20 (2023) + +6 +E. I. Leitinger et al. +0.8 +1.0 +1.2 +1.4 +V - I +13 +14 +15 +16 +17 +18 +I +Fitted Data +Clipped Data +1.5 +2.0 +2.5 +B - I +0.0 +0.5 +1.0 +U - B +1.70 +1.65 +1.60 +1.55 +1.50 +CUBI +Figure 7. 𝑁 − 𝜎 clipping through various colour-index combinations for the ground-based photometry of NGC 5024. Outliers are shown in red, while stars +that closely fit the polynomial applied to each distribution are shown in black. Right panel: The same method was used on the 𝐶UBI distribution. +𝐹438𝑊) and (𝐹336𝑊 − 𝐹814𝑊). Finally, a special photometric +index 𝐶UBI was used, which separates stars based on their chemical +properties, namely N and He abundances. 𝐶UBI was first introduced +by Monelli et al. (2013) for ground-based photometry using Johnson +filters with a focus on the RGB. It can also be adapted to the HST +filters, as demonstrated by Milone et al. (2013). For each star in +the ground-based photometry: 𝐶UBI = (𝑈 − 𝐵) − (𝐵 − 𝐼), while +𝐶UBI = (𝐹336𝑊 − 𝐹438𝑊) − (𝐹438𝑊 − 𝐹814𝑊) in the HST +photometry. We applied the same 𝑁 − 𝜎 clipping method on the +resulting 𝐶UBI distributions. The full sequence of polynomial fitting +with 𝑁−𝜎 clipping is illustrated in Figure 7, where red outliers were +removed for each colour-index combination before finally removing +outliers from the 𝐶UBI distribution. +2.6 +Photometric Completeness Correction +While the spatial completeness analysis of Section 2.1 compensates +for cluster regions without observed stars caused by the limitations +of the field, the photometric completeness compensates for a lack +of stars due to poor or missing photometry. The aim is to assign a +weighting to the surviving stars, such that they account for the frac- +tion of stars that are lost during photometric cleaning. We assumed +that both the HST and ground-based catalogues were complete at +the magnitudes of the RGB, as Anderson et al. (2008) derives the +completeness for the HST data as 100% for stars brighter than the +SBG for most clusters and Stetson et al. (2019) reports the ground- +based data is complete across all radii for stars between 𝑉 = 19 +and 𝑉 = 12. To determine the photometric completeness factor +(0 ≤ 𝑓𝑃 ≤ 1.0), we compared the number of RGB stars before +and after the photometric cleaning processes. We divided the orig- +inal spatial distribution of RGB stars radially into annuli and the +number of stars before (𝑁1) vs. the number of stars after (𝑁2) deter- +mined the photometric completeness factor for stars in each annulus: +𝑓𝑃 = 𝑁2/𝑁1. As we expect that the original HST and ground-based +0 +200 +400 +600 +800 +1000 +1200 +1400 +Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Completeness fraction +fP +fS +fS × fP +Completeness cut-off +Figure 8. The individual spatial (blue) and photometric (green) complete- +ness fractions, as well as the product of both completeness fractions ( 𝑓𝑆 𝑓𝑃 +in black) as a function of radius for the ground-based stars in NGC 5024. The +dotted red line indicates the cut-off at 15%, which is the minimum accepted +completeness fraction. +catalogues contain the vast majority of stars, this completeness fac- +tor accounts for the stars we remove in our cleaning, not stars missed +by the catalogues. +The combined completeness fraction for each RGB star in both +the HST and ground-based photometry was calculated as the product +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +7 +100 +101 +102 +103 +Radius [arcsec] +12 +11 +10 +9 +8 +7 +Log (N/arcsec2) +Trager (1995) +Ground-based RGB +HST RGB +Figure 9. A comparison of the number density profiles as a function of radius +for NGC 5024. The surface density profile from Trager et al. (1995) is shown +in black. The HST cleaned and weighted RGB stars (cyan) transition into the +ground-based cleaned and weighted RGB stars (magenta) at approximately +100′′ and matches well against the Trager et al. (1995). +of the spatial and photometric completeness 𝑓𝑇 = 𝑓𝑆 𝑓𝑃, which can +be seen as a function of radius in Figure 8 for only the ground-based +photometry of NGC 5024 as an example. The dense cluster center +suffers a drop in completenesses due to the blending of stars in the +ground-based catalogue, which were removed mainly through the +quality cuts of Section 2.3. Additionally, the outer regions 𝑅 > 800′′ +begin to drop in completenesses mainly due to the photometric +cleaning of Section 2.5. We stopped at the radius at which the +combined completeness fraction dropped below 𝑓𝑇 < 0.15 for the +ground-based and HST photometry. +2.7 +Number Density Completeness +In order to check the validity of our completeness corrections, we +calculated the surface density based on completeness corrected stel- +lar number counts and compared this against the surface brightness +profiles of Trager et al. (1995). The number density profile of the +cleaned RGB stars in our sample was weighted by the spatial and +photometric completenesses 𝑓𝑇 . After correction for the combined +completenesses, we applied the same shift factor to both the HST +and ground-based data to convert between number density and sur- +face density. The Trager et al. (1995) data was used as a reference +profile and is shown in black in Figure 9. We then compared the +number density profiles of the HST (cyan) and ground-based data +(magenta) to the reference profile for each cluster in order to con- +firm the viability of the total incompleteness factors as a weighting +to compensate for missing photometry. We found a good match be- +tween the HST and ground-based photometry and the Trager et al. +(1995) profile for all 28 GCs in our sample. +3 +IDENTIFICATION OF MULTIPLE POPULATIONS +We now move on to separate the multiple stellar populations using +both the 𝐶UBI distribution method (Section 3.1) and the chromo- +1.8 +1.7 +1.6 +1.5 +CUBI +14 +15 +16 +17 +18 +F814W +4th percentile +96th percentile +RGB +2 +1 +0 +1 +CUBI +Figure 10. Left panel: The 𝐶UBI distribution of NGC 5024 stars in black +using HST photometry, with the 4𝑡ℎ percentile ridgeline in blue and the +96𝑡ℎ percentile ridgeline in red. Grey horizontal lines indicate the photo- +metric error in the 𝐶UBI distribution at different magnitudes. Right panel: +The resulting distribution Δ𝐶UBI of the same stars after normalisation as +described by Equation 4. +some map method (Section 3.2), before finally analysing their radial +distributions (Section 3.3). +3.1 +Gaussian Mixture Models applied to 𝐶UBI Distributions +The multiple stellar populations of each cluster were identified using +the photometric index 𝐶UBI described in Section 2.5. The general +method for categorizing stars into multiple populations throughout +this paper involved applying Gaussians to the Δ𝐶UBI distribution +of stars, which is a normalised version of the 𝐶UBI distribution +as shown in Figure 10. To normalise the distribution, the 4𝑡ℎ and +96𝑡ℎ percentiles of the combined 𝐶UBI values for all stars were +determined and fitted with a 1D polynomial, as per the method +detailed in Milone et al. (2017). We used Equation 4 to calculate +the normalised distribution Δ𝐶UBI from the distributions of 𝐶UBI +in both the HST and ground-based photometry. +Δ𝐶UBI = +𝐶UBI − 𝑋𝑏𝑙𝑢𝑒[𝐼] +𝑋𝑟𝑒𝑑 [𝐼] − 𝑋𝑏𝑙𝑢𝑒[𝐼] − 1 +(4) +The red (𝑋𝑟𝑒𝑑) and blue (𝑋𝑏𝑙𝑢𝑒) fiducial ridgelines in the left +panel of Figure 10 were created at equally sized increments of +𝐹814𝑊 and 𝐼 magnitude bins for the HST and ground-based +photometry, respectively. An example of the resulting Δ𝐶UBI +distribution is shown in the right panel of Figure 10. We note that +for all clusters in our sample, the photometric error in the 𝐶UBI +distribution is much smaller than the colour spread in 𝐶UBI due to +the presence of multiple stellar populations. Due to this, we are +confident that the separation between multiple populations in the +𝐶UBI distribution is not influenced by photometric errors in the +bands. +With this normalised distribution of stars, Gaussian Mixture +MNRAS 000, 1–20 (2023) + +8 +E. I. Leitinger et al. +2.0 +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +CUBI +14 +15 +16 +17 +18 +F814W +P1 +P2 +p +0.8 +2 +4 +6 +8 +10 +Number of Components +550 +575 +600 +625 +650 +675 +700 +725 +Information Criterion +AIC +BIC +1.25 +1.50 +1.75 +2.00 +2.25 +2.50 +2.75 +F336W - F814W +14 +15 +16 +17 +18 +F814W +75 +50 +25 +0 +25 +50 +75 + RA [arcsec] +100 +75 +50 +25 +0 +25 +50 +75 +100 + DEC [arcsec] +Figure 11. Population separation of NGC 5024 using HST photometry. Top left: The best-fit GMM (solid black line) with the corresponding individual +Gaussians (dashed), together with the Δ𝐶UBI distribution of stars separated into their respective P1 and P2 populations. In grey we show stars with ambiguous +classification, i.e. membership probability to either population of 𝑝 ≤ 0.8. Top right: The AIC and BIC both show a minimum at 𝑛 = 2, indicating a clear +identification of two populations. Bottom left: The CMD of the two populations from the MS turn-off to the tip of the RGB. Bottom right: The spatial distribution +of the two populations showing isotropic behaviour. +Models (GMMs) from the scikit-learn package (Pedregosa et al. +2011) were applied in order to find the most probable distribu- +tion of the mutliple populations. The method uses an expectation- +maximization approach in order to determine the best mixture of one +or more Gaussians to fit the Δ𝐶UBI distribution. Both the Akaike in- +formation criterion (AIC) and Bayesian information criterion (BIC) +were used to determine the most probable number of populations +when provided with the Δ𝐶UBI distribution of a cluster. The min- +ima of both the AIC - which estimates the relative quality of the +statistical models based on in-sample prediction error, and the BIC +- which selects the most probable model based on likelihood func- +tions, indicated the most probable number of populations within a +cluster from a range of 1 ≤ 𝑛 ≤ 10 different components. For most +clusters the AIC and BIC found 𝑛 = 2 components. The top right +panel of Figure 11 shows the range of possible components when +applying GMMs to NGC 5024, with both AIC and BIC providing +minima at 𝑛 = 2. Clusters with minima at 𝑛 = 1 were discarded. +From the most probable GMM samples, the final separation +of the populations was created in terms of two or more Gaussians +encompassing the full sample of stars. The top left panel of Figure +11 shows the combination of two Gaussians on the Δ𝐶UBI distribu- +tion of stars. Each star was assigned to a population based on the +probability that it belonged to a particular Gaussian. This member- +ship probability was also used to divide the multiple populations +for clusters with three populations, as discussed further in Section +3.2. We required stars to have membership probability 𝑝 ≥ 0.8 +between the P1 and P2 populations. This resulted in a small gap +between each of the Gaussians, shown as gray points in Figure 11, +ensuring that the stars belong to the population they were assigned +to with high confidence. We experimented with this threshold using +0.5 ≤ 𝑝 ≤ 1.0 in increments of 0.05 and found the overall results +and conclusions of this work were not affected by the exact value of +the threshold. Similarly, we tested the effect of changing the limit of +the primordial and enriched classifications for clusters with inter- +esting radial distributions1. Briefly, we randomly sampled arbitrary +limits in the Δ𝐶UBI colour (i.e. the point where the Gaussians over- +lap) and classified stars left of the limit as primordial and stars to the +right as enriched. The limit was drawn from a uniform distribution +covering the inner 2𝜎 of the Δ𝐶UBI colour to avoid a cut too close +to either colour end. We did this to prevent having almost all stars +classified into one population with only a few left to be classified in +another. For the purpose of these tests, we continued the remainder +of the analysis with these arbitrary classifications in order to sta- +tistically determine the significance of our resulting radial profiles. +We sampled the arbitrary limits 200 times per cluster and each time +we sampled anywhere from 90 to 100% of the stars on either side of +1 NGC 3201, NGC 6101 and NGC 7078 – see Section 4.3.2 +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +9 +the limit to also observe the effect of randomly removing individual +stars from each population. +3.2 +Chromosome Maps +In addition to the 𝐶UBI colour distribution classification, for the +HST photometry it is also possible to separate the populations us- +ing chromosome maps. Introduced by Milone et al. (2017), a chro- +mosome map is a colour-colour plot that has been normalised in +a way which allows efficient separation of sub-populations of dif- +ferent abundances. It uses the RGB width in a 𝐹275𝑊 − 𝐹814𝑊 +vs. 𝐹814𝑊 CMD, along with the RGB width of the pseudo-colour +combination 𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 vs. 𝐹814𝑊. Following the +method in Milone et al. (2017), we defined a dividing line between +populations in the Δ𝐹275𝑊 ,𝐹814𝑊 vs. Δ𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 +distribution. We found that clusters such as NGC 2808 contained +several distinct populations which can be split using a chromosome +map. In these instances, the multiple populations tend to be easier +to distinguish using a chromosome map, as they can become some- +what blended together when using the Δ𝐶UBI distribution alone. +Therefore, by creating chromosome maps and then using the GMM +method in two dimensions, as shown in Figure 12, we were able +to directly compare the populations separated using a Δ𝐶UBI plot, +against the populations separated by a chromosome map. The aim +was to implement the same membership probability defined in Sec- +tion 3.1 of 𝑝 ≥ 0.8 to cut out the ambiguous stars, shown in grey in +Figure 12, before checking how the remaining stars were assigned +to populations according to the two methods. +The HST photometry includes the UV filter F275W which +has no ground-based equivalent. We therefore relied on the 𝐶UBI +distribution of the HST and ground-based photometry for a consis- +tent analysis. The HST F275W photometry was only used to confirm +whether the𝐶UBI classification was consistent with the chromosome +map method. To do this, the RGB stars of the 𝐶UBI distribution were +separated into multiple populations with both methods. In Figure +13 we show the chromosome map of NGC 5024, where we colour +code the stars classified as P1 and P2 with the Δ𝐶UBI distribution +method in orange and blue, respectively. This figure shows that for +the majority of the stars, the classification of different populations +using Δ𝐶UBI was consistent with the classification based on the +chromosome map. +In all clusters, there was a small percentage of stars where +the P1/P2 classification obtained using the chromosome map and +Δ𝐶UBI disagree. We see that there are Δ𝐶UBI P1 stars in Figure +13 (blue) that inhabit the region in which the bulk of the P2 stars +(orange) are located, and vice versa. We found the average fraction +of stars that were classified differently by each method was ∼ 10% +for the 28 GCs in our final sample, with a minimum of 4% and +a maximum of 20% after the probability cut. Clusters with high +contamination percentages had heavily blended populations in the +chromosome map, meaning the distribution of stars followed a more +continuous distribution as opposed to distinct clumps. This caused +difficulties in accurately determining the classification of popula- +tions in one or both separation methods and therefore these clusters +were excluded from our analysis. To further check the consistency +of the population classification, we used overlapping stars that were +covered by both (ground based and HST) photometric catalogues +and had been independently classified into the different sub- +populations using each data set. We found consistent classifications +of populations for stars common to both data sets, as demon- +strated with large bold blue (P1) and orange (P2) points in Figure 13. +0.3 +0.2 +0.1 +0.0 +F275W, F814W +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +CF275W, F336W, F438W +P1 +P2 +Probability cut +0 +50 +0 +50 +Figure 12. Chromosome map using the HST photometry for NGC 5024, +with Gaussian Mixture Models (GMMs) applied in two dimensions. The +lower left plot shows the chromosome map with populations P1 (blue) and +P2 (orange) as defined by the two Gaussians in the top and right panels. In +grey are stars which lie in-between the two populations, with membership +probabilites 𝑝 ≤ 0.8 for either population. +0.30 +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +0.05 +F275W, F814W +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +CF275W, F336W, F438W +P2 +P1 +Figure 13. Chromosome map using the HST photometry for NGC 5024, as +shown in Figure 12; however, we now use the Δ𝐶UBI separation to assign +the stars into P1 (blue) and P2 (orange) populations. Stars with membership +probabilities 𝑝 ≤ 0.8 are also removed. There is still a very good separation, +as also shown in Figure 12, so we can see ‘contaminant’ stars by eye as blue +points located in the orange clump and vice versa. Bold circles indicate stars +that overlap in both the HST and ground-based photometry, colour-coded +to show the agreement between their independent classifications in each +photometric catalogue. +MNRAS 000, 1–20 (2023) + +10 +E. I. Leitinger et al. +After ensuring consistent results between the different clas- +sification methods/catalogues, we combined the HST and ground- +based photometry by removing stars in the ground-based data which +overlap with the HST field. By doing this, we ensure the ground- +based data begins at the same radius where the HST data ends, +ensuring there are no gaps between the fields. We then use this +combined data set to study the behaviour of MPs across the full +extent of these clusters. +3.3 +Radial Distributions of Different Populations +A useful tool in understanding the behaviour of MPs as a function +of radius is calculating the cumulative radial distribution of the stars +in each population. If one population is more centrally concentrated +within the cluster, we see a comparatively steeper slope in its cu- +mulative radial distribution than we do for the other population. +However, if the populations are homogeneously mixed throughout +the cluster, we see similar slopes for both distributions. The 𝐴+ +parameter introduced by Alessandrini et al. (2016) is a way to quan- +tify differing radial profiles, as it is an integration of the ‘area’ +between the two distributions. The cumulative radial distributions +in this work provide a spatially complete view of each cluster by +combining the innermost region using HST photometry with the +outer region using ground-based photometry. To calculate the cu- +mulative radial distributions we used the method introduced and +detailed by Alessandrini et al. (2016) and Dalessandro et al. (2019). +The 𝐴+ parameter considers the area between the cumulative ra- +dial distributions of two populations, so for clusters exhibiting three +distinct stellar populations such as NGC 1851, NGC 2808, NGC +6101 and NGC 7078, we combined the P2 and P3 stars into a single +‘enriched’ population, referred to as P2 for simplicity. This clas- +sification follows the logic of Milone et al. (2017), in which the +primordial stars (P1) are identified as the group of stars aligning +with Δ𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 = Δ𝐹275𝑊, 𝐹814𝑊 = 0 in a chro- +mosome map, while P2 stars are any stellar populations located +above the primordial stars. +We calculated a modified version of the 𝐴+ parameter using +Equation 5 in order to characterize the weighted cumulative radial +distributions of stars in each population using the total completeness +fractions 𝑓𝑇 calculated in Section 2.6. +𝐴+(𝑅) = +∫ 𝑅 +𝑅𝑚𝑖𝑛 +�𝜙𝑃1(𝑅′) − 𝜙𝑃2(𝑅′)� 𝑑𝑅′ +(5) +Here, 𝜙 is the normalised, cumulative sum of the weights, 𝑤 = 1/ 𝑓𝑇 , +of the stars in either the P1 or P2 population. Our 𝐴+ parameter indi- +cates whether a cluster has a P1 concentration in the center (𝐴+ > 0), +a P2 concentration in the center (𝐴+ < 0), or a homogeneous mix +of populations (𝐴+ ∼ 0) throughout the cluster. The uncertainty in +𝐴+ was determined via bootstrapping. Briefly, the P1 and P2 stars +of each cluster were sampled randomly for a total of 500 iterations +using a sample size of 1000, with an 𝐴+ value calculated each +time. The final uncertainty for each cluster was calculated from the +standard deviation of the 500 iterations. +Figure 14 shows the weighted and normalised cumulative ra- +dial distributions of the two stellar populations found in NGC 5024 +along the top panels, with the bottom panels showing the corre- +sponding number ratio of enriched to total stars (P2/Ptotal) as a +function of radius. NGC 5024 is an example of why the full ex- +tent of the cluster should be analysed when considering the radial +distributions of populations within a cluster. The left panels show +the behaviour of the cluster for only the HST field (1293 stars). +We already see by eye that both cumulative profiles are almost +identical, which is also supported numerically by the parameter +𝐴+ = −0.03 ± 0.02. The cumulative radial distribution of the HST +photometry alone would suggest that the populations of this cluster +are fully mixed and spatially indistinguishable. However, the middle +panels show the result of this same analysis on the ground-based +photometry (438 stars). Here P2 is more centrally concentrat‹ed +(𝐴+ = −0.57 ± 0.26), with the outer regions dominated by P1 stars. +Finally, in the right panel, the full extent of the cluster is analysed +by combining both the HST and ground-based stars, producing a +value of 𝐴+ = −0.84±0.11 and supporting the result that P2 is cen- +trally concentrated. This information is lost when only observing +the cluster center and using the resulting 𝐴+ parameter to describe +the behaviour of the cluster as a whole. It is especially important +to consider the outer regions of clusters, since dynamical mixing of +the populations will affect the center of the cluster within shorter +timescales than it does for the outer stars (Dalessandro et al. 2019). +To show the consistency of behaviour between the two photometric +data sets, we plot the enriched star fraction P2/Ptotal as a function +of radius in the lower panels of Figure 14. Here, the inner region +also shows a mostly constant P2 concentration and the outer region +shows a strong decline in P2 stars, supporting the result of the cu- +mulative radial distributions while also showing agreement in the +transition region between data sets. +4 +RESULTS +For the 28 Galactic GCs in our sample we now investigate the +trends associated with the 𝐴+ parameter and the enriched star +fraction P2/Ptotal. In Section 4.1 we explore the global trends using +the cumulative radial distributions, in Section 4.2 we explore the +global trends using the enriched star fractions P2/Ptotal, and finally +in Section 4.3 we discuss individual notable clusters that have low +dynamical ages. +Throughout this section we use cluster parameters provided +by the Galactic Globular Cluster Database by Baumgardt et al. +(2019), updated to the Gaia DR3 data as described by Vasiliev & +Baumgardt (2021) and Baumgardt & Vasiliev (2021). We take the +initial cluster mass and current cluster mass values, the former being +calculated from the current cluster masses and cluster orbits using +Equation 3 from Baumgardt & Makino (2003). The relaxation time +(𝑇𝑅𝐻 ) of each cluster was also used, giving the time scale in which +each cluster will become dynamically mixed, which was derived by +Baumgardt & Hilker (2018). We define the dynamical age as the +ratio of the age of a star cluster to its relaxation time and estimate +the mass loss ratio (Mc/Mi) as the ratio of the current (Mc) and +initial (Mi) mass of the cluster. We also take the projected half-light +radius (𝑅hlp), half-mass radius and orbital parameter values for each +cluster from this database. The cluster ages are taken from the work +of Kruijssen et al. (2019), while metallicity values are taken from +Harris (2010). +Previous work has found a clear correlation of the width of +the RGB in clusters with MPs as a function of cluster metallic- +ity [Fe/H], absolute visual magnitude 𝑀𝑉 and initial mass of the +cluster (Monelli et al. 2013; Milone et al. 2017). Since we have com- +bined two independent photometric catalogues to get an extended +spatial view, it was important that we replicated the well-established +trends observed by others who used the same catalogues. In par- +ticular, we followed the method set out by Monelli et al. (2013) +for the ground-based catalogue and determined the RGB widths +(WRGB) in the same manner for the 28 Galactic GCs in our sample. +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +11 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.03 ± 0.02 +2 +4 +6 +8 +10 +Projected Radius [HLR] +200 +400 +600 +A + = -0.57 ± 0.26 +0 +2 +4 +6 +8 +10 +Projected Radius [HLR] +A + +total = -0.84 ± 0.11 + A + +4 = -0.42 ± 0.05 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.60 +200 +400 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.35 +0 +200 +400 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.52 +Figure 14. Cumulative radial distributions of different populations and enriched star fractions in NGC 5024 for the full radial range of the cluster. The scale +shown for reference in the top panels is the distance from the centre of the cluster in units of projected half light radius [HLR]. Upper left: The weighted, +normalised, cumulative radial distribution of P1 (blue) and P2 (red) stars in the HST photometry within 𝑟 < 100′′ of the cluster center. Upper middle: The +ground-based photometry from 100′′ < 𝑟 < 740′′, analysed in the same way as the HST data in the upper left panel. Upper right: The 𝐴+ parameter for the +combined data set, covering 0 < 𝑟 < 740′′. The radius at which the HST photometry meets the ground-based photometry is shown by a black, dashed line. +We quote both 𝐴+ +4 for the calculated 𝐴+ value at a radial limit of 4.27𝑅hlp and 𝐴+ +total for the full radial range. Lower left: The fraction of P2 stars as a function +of radius for the HST photometry (black). Each bin has an equal number of stars, with the radial range of the bins illustrated at the bottom of the plot (green). +Lower middle: The ground-based photometry analysed in the same way as the HST data in the lower left panel. Lower right: The P2 fraction as a function of +radius for the combined data set. The total P2/Ptotal fractions are indicated in each panel for each corresponding radius range. +We found a strong correlation between WRGB and [Fe/H], with +a Spearman correlation coefficient 𝑟𝑠 = 0.693 and associated p- +value = 4 × 10−5, as well as an anti-correlation between WRGB and +𝑀𝑉 , with 𝑟𝑠 = −0.331 and a p-value = 0.08. For the HST data, +we followed the method of Milone et al. (2017) and reproduced +the correlation between WF275W,F814W and [Fe/H] for clusters +with 𝑀𝑉 > −7.3, providing a Spearman correlation coefficient of +𝑟𝑠 = 0.704 and a p-value = 4 × 10−5. We also reproduced the trend +between WF275W,F814W and 𝑀𝑉 , with 𝑟𝑠 = −0.104 and p-value += 0.6. We conclude that our data exhibits the same well-established +trends as previous work. +4.1 +Global Trends using Cumulative Radial Distributions +(𝐴+) +We analysed large regions of the targets in our sample of 28 Galac- +tic GCs and calculated the cumulative radial distribution parameters +𝐴+. We then identified clusters in which the 𝐴+ values indicated a +high central concentration of either primordial or enriched stars at +a significance larger than 3𝜎. These significantly segregated clus- +ters will be discussed in detail in Sections 4.3.1 and 4.3.2. The +maximum radii for the outermost stars in the ground-based fields +differed greatly for each cluster, so in order to make the results in +different clusters comparable to each other, we analysed the spatial +distribution of stars only out to 4.27𝑅hlp in all clusters. We chose +this limit as it was the minimum radius for our final sample of stars +in NGC 6101, with most clusters extending beyond this radial limit. +The only clusters that did not reach this limit were NGC 3201, NGC +5053, NGC 6121 and NGC 6838 where the maximum radii for the +ground-based photometry were in the range of 2.5𝑅ℎ𝑙𝑝 (NGC 6838) +< 𝑟𝑚𝑎𝑥 < 4.0𝑅ℎ𝑙𝑝 (NGC 3201). Limiting all clusters to this lower +range would remove important information on the cluster proper- +ties in the outermost regions. Therefore, for these four clusters we +assumed that the relative fraction of primordial and enriched stars +is constant from the outermost radius covered by our photometry +to 4.27𝑅hlp. Since we extrapolate out to 4.27𝑅hlp by sampling real +stars in the outer radial bins, we do not expect that this will add +significant uncertainty to the 𝐴+ parameters as we also propagate +the uncertainties of these sampled stars. +Figure 15 shows the resulting 𝐴+ +4 parameters (calculated at a +maximum radius of 4.27𝑅hlp) as a function of dynamical age. We +found that dynamically old clusters all have 𝐴+ ∼ 0, in agreement +with the idea that due to relaxation, populations become mixed. +This also agrees with the findings of Dalessandro et al. (2019). +In dynamically young clusters, we found a larger range of +𝐴+ values. Surprisingly, we not only found centrally concentrated +MNRAS 000, 1–20 (2023) + +12 +E. I. Leitinger et al. +2 +4 +6 +8 +10 +12 +14 +16 +Age / Relaxation Time +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +A + +4 +6101 +5024 +5272 +3201 +6809 +5053 +4590 +288 +5904 +7089 +2808 +6205 +7078 +6752 +6341 +1261 +7099 +6254 +5286 +4833 +6934 +5986 +1851 +6981 +6366 +6218 +6838 +6121 +Figure 15. The total cumulative radial distributions in terms of the 𝐴+ +4 parameter for the 28 Galactic GCs as a function of their dynamical age. All clusters +are limited to a radius equivalent to 4.27𝑅hlp for direct comparison. Using this radius limit, clusters with an 𝐴+ value greater than 3-𝜎 significance from zero +are displayed as labelled black points. An 𝐴+ value close to zero indicates the MPs are spatially mixed throughout the analysed spatial extent of the cluster. +Significantly positive 𝐴+ values indicate that the primordial (P1) population is more centrally concentrated, while negative values indicate the enriched (P2) +population is more centrally concentrated. Refer to Section 4.3.3 for a special discussion on NGC 7078. +P2 populations (𝐴+ < 0, e.g. NGC 2808, NGC 5024, NGC 5272 +and NGC 6809) consistent with the findings of Dalessandro et al. +(2019), but also clusters with centrally concentrated P1 populations +(𝐴+ > 0, e.g. NGC 3201 and NGC 6101), and clusters with full +spatially mixed populations (𝐴+ ∼ 0, e.g. NGC 288, NGC 4590, +NGC 5053, NGC 5904, NGC 70782 and NGC 7089) in the same +small dynamical age range (age/relaxation time < 4.5). The central +concentration of a primordial population seems to be in tension with +the prediction of globular cluster formation models where P2 stars +are preferentially concentrated towards the centre. +We also investigated the relationship between 𝐴+ +4 and the mass +loss fraction (Mc/Mi). Clusters that have lost >70% of their ini- +tial masses due to dynamical evolution should be entirely mixed +according to Vesperini et al. (2013). However, given that their sim- +ulations do not include the effects of stellar evolution, our present +day masses cannot be directly compared with Vesperini et al. (2013). +To do this we need to take into account that star clusters lose ∼ 50% +of their mass during a Hubble time due to stellar evolution (e.g. +high mass stars dying first), so the Vesperini et al. (2013) clusters +that have lost >70% of their initial mass correspond to the clus- +ters with Mc/Mi ≳ 0.15 in Figure 16. Therefore, clusters with +Mc/Mi ≳ 0.15 are giving us a peek into the diversity of configu- +rations the P1 and P2 populations of stars in globular clusters can +display at the time of birth. As expected, in Figure 16 we found that +the clusters with significant central concentrations in either P1 or P2 +have undergone the least amount of mass loss, with the exception of +2 See Section 4.3.3 for a detailed discussion on NGC 7078 +NGC 6809. Generally, as more mass is lost by a cluster, the initial +concentrations of the multiple populations are also lost, as the stars +become spatially mixed. We therefore concentrate our analysis on +the clusters that should have retained the largest amount of their +initial conditions in terms of dynamical age and mass loss. +While 20 Galactic GCs were investigated by Dalessandro et al. +(2019), our study overlaps with only 8 of these clusters. We tested +for consistency with their results by matching the constraints of +their analysis and found all 8 overlapping clusters produce the same +cumulative radial distributions as Dalessandro et al. (2019). These +constraints included limiting the HST field to 2 𝑅hlp in order to +match the radial range covered by their analysis and only including +the ground-based data for the analysis of NGC 288 within this +same radial range. For our independent analysis, we included the +ground-based photometry without restricting the radial range to 2 +𝑅hlp and still found agreement with Dalessandro et al. (2019) for 7 +out of the 8 overlapping clusters, since both the HST and ground- +based photometry show 𝐴+ ∼ 0. The one cluster that did not agree +with their results is NGC 6101, where we found P1 to be centrally +concentrated. When we considered only the HST photometry for +NGC 6101, we found 𝐴+ ∼ 0 in agreement with Dalessandro et al. +(2019), but with the inclusion of the ground-based photometry and +therefore the outer region of the cluster, we found a P1 central +concentration. This suggests that conclusions arrived at by studying +only the inner regions of a cluster may be misleading, especially +in dynamically young clusters. A more extensive coverage of such +clusters is required to obtain a full picture. +Our results for the dynamically young clusters suggest that +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +13 +0.1 +0.2 +0.3 +0.4 +0.5 +Mcurrent/Minitial +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +A + +4 +288 +2808 +3201 +4590 +5024 +5272 +5904 +6101 +6205 +6809 +7078 +7089 +5053 +2 +4 +6 +8 +10 +12 +14 +Age / Relaxation Time +Figure 16. The total cumulative radial distributions in terms of the 𝐴+ +4 parameter (calculated at a maximum radius of 4.27𝑅hlp) for the 28 Galactic GCs as a +function of their mass loss ratio. Each cluster is also colour-coded by its dynamical age. Clusters categorised as ‘dynamically young’ (age/relaxation time < +4.5) are displayed as labelled points. Refer to Section 4.3.3 for a special discussion on NGC 7078. +clusters are able to form with either enriched stars in the center, +primordial stars in the center, or enriched and primordial stars dis- +tributed in the same way. This is an intriguing result, considering +that the majority of globular cluster formation models will naturally +produce clusters in which the P2 stars are centrally concentrated. +Our results therefore argue for the need of additional theories that +can explain how clusters form with mixed stellar populations or +centrally concentrated primordial stars. +4.2 +Global Trends using Enriched Star Fractions (P2 / Ptotal) +For each of the 28 Galactic GCs in our sample we calculated the en- +riched star fraction P2/Ptotal with associated standard errors, where +enriched stars included both the P2 and P3 stellar populations. Un- +like the cumulative radial distribution analysis, we did not imple- +ment a radial limit of 4.27𝑅hlp for each cluster, but instead calculated +the P2/Ptotal fraction for the full possible extent of each cluster, tak- +ing into account the total completeness fraction (see Section 2). The +top panel of Figure 17 shows the P2/Ptotal fraction as a function of +the initial cluster mass. We obtain a strong correlation between +these two parameters with 𝑟𝑠 = 0.8 and p-value = 1 × 10−9, similar +to the correlation found by Milone et al. (2017) and Milone et al. +(2020) using the P1/Ptotal fraction against log(𝑀[𝑀⊙]). We found +no significant correlations for the global fraction P2/Ptotal as a func- +tion of either metallicity or age (see Figure 17). After removing the +mass trend from our data, we similarly found that the residuals are +uncorrelated with age or metallicity. We neither found significant +correlations with orbital parameters such as peri- and apogalactic +distances and eccentricity, nor with the slope of the mass function. +In order to test how young and low mass clusters fit into the +global trends, we included an additional 7 Local Group clusters: +NGC 121, NGC 336, NGC 416, NGC 1783, NGC 1978, Lindsay 1 +and Fornax 3. The Local Group clusters were separated into mul- +tiple stellar populations using only HST photometry, but with the +same method as outlined in Section 3. There is no need to combine +HST and ground-based photometry for these clusters due to the +fact that the half-light radius for each cluster is well within a single +HST field, meaning the majority - if not all - stars are covered ny a +single field. We only calculated the enriched star fractions P2/Ptotal +for these additional clusters. In order to separate the populations, +we used the narrow-band filter 𝐹343𝑁, which contains the NH ab- +sorption line and can be used in the colour combination 𝐶UBUn = +(𝑈 − 𝐵) − (𝐵 − 𝑈𝑛) = (𝐹336𝑊 − 𝐹438𝑊) − (𝐹438𝑊 − 𝐹343𝑁) +as introduced by Niederhofer et al. (2017). In the same way we +confirmed consistency between the Δ𝐶UBI distribution and chro- +mosome maps, we also produced consistent results between the +𝐶UBUn and Δ𝐶UBI distributions. +We also included an additional 4 low-mass Milky Way clusters +to our sample (Ruprecht 106, Palomar 12, Terzan 7 and E3), using +previous work which performed spectroscopic analysis of stars and +found no evidence of multiple populations. E3 and Ruprecht 106 +do not contain enriched populations according to the analysis of +MNRAS 000, 1–20 (2023) + +14 +E. I. Leitinger et al. +4.25 +4.50 +4.75 +5.00 +5.25 +5.50 +5.75 +6.00 +6.25 +log(Initial Mass) [M ] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +P2 / Ptotal +rs = 0.8 p-value: 1e-09 +SMC +LMC +Fornax +MW +2.0 +1.5 +1.0 +0.5 +[Fe/H] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +P2 / Ptotal +rs = -0.39 p-value: 0.01 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Age [Gyr] +1e10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +P2 / Ptotal +rs = 0.32 p-value: 0.04 +Figure 17. The enriched stellar population fraction as a function of global +parameters for Galactic GCs (black circles). Added are SMC GCs (green +triangles), LMC GCs (blue squares) and Fornax GCs (orange crosses). The +large error bars plotted in grey for clusters with P2/Ptotal = 0 are due to +the low number of stars with spectroscopic abundance measurements. Top +panel: The fraction of P2 stars as a function of the initial mass of each +cluster shows a clear correlation between the two parameters. Middle panel: +There is no significant relationship between the enriched star fraction and +metallicity of the cluster. Bottom panel: There is no significant relationship +between the enriched star fraction and the age of the cluster. +Monaco et al. (2018); Salinas & Strader (2015) and Dotter et al. +(2018); Frelijj et al. (2021), respectively, and we therefore set them to +P2/Ptotal = 0 with standard errors of 1/ +√ +𝑁, where 𝑁 is the number +of stars analysed. Similarly, the current consensus is that Terzan +7 and Palomar 12 do not contain multiple populations, based on +the spectroscopic analysis of ≤ 5 RGB stars (Sbordone et al. 2005; +Cohen 2004), and we therefore set P2/Ptotal = 0. The standard errors +for the enriched star fraction associated with Terzan 7 and Palomar +12 were comparatively much larger than for other clusters, in order +to reflect the uncertainty of declaring a non-detection of MPs with +a sample of only 5 RGB stars. The age and metallicity of Lindsay +1 were taken from Glatt et al. (2009), while those of E3 were taken +from Forbes & Bridges (2010), and of Ruprecht 106 from Kruijssen +et al. (2019), who averaged the values determined by Forbes & +Bridges (2010) and Dotter et al. (2010, 2011). All other additional +cluster ages and metallicities were taken from Usher et al. (2019). +The addition of these 11 young and low-mass GCs to the sam- +ple did not significantly influence the trends found for the P2/Ptotal +fractions against global parameters. The initial mass correlation +in Figure 17 is supported by the addition of these clusters, which +continue the trend into the lower initial mass range. The relation- +ship between P2/Ptotal and metallicity [Fe/H] previously showed a +Spearman rank order coefficient of 𝑟𝑠 = 0.11 and p-value = 0.58 +for the original 28 Galactic GCs. After the addition of the 11 young +and low-mass GCs, this coefficient changed to 𝑟𝑠 = −0.39 with a +p-value = 0.01, showing a slight but ultimately inconclusive anti- +correlation. For P2/Ptotal against age, the Spearman correlation only +changes from 𝑟𝑠 = −0.21 with a p-value = 0.29 for the original 28 +Galactic GCs, to 𝑟𝑠 = 0.32 with a p-value = 0.04 for the full sample, +again showing an inconclusive (weak) correlation. There appears to +be no significant trend between enriched star fractions and metallic- +ity or age, but the addition of a larger sample of young and low-mass +clusters may alter this result. +4.3 +Dynamically Young Clusters +By ‘dynamically young’ we refer to the clusters in our sample with +dynamical ages < 4.5. Vesperini et al. (2013) found that dynamical +age is a good indicator for the degree of dynamical mixing, with +small dynamical ages corresponding to clusters which have retained +the initial conditions of their formation. Following this criterion, +the clusters described in detail throughout this section are assumed +to have preserved their initial conditions. We have divided this +section into three parts, focusing on dynamically young clusters +with: enriched (P2) populations concentrated in the center in Section +4.3.1, the primordial (P1) population in the center in Section 4.3.2 +and spatially mixed populations in Section 4.3.3. The cumulative +radial distribution plots for the covered extent of all dynamically +young clusters can be found in Appendix A. +4.3.1 +Clusters with centrally concentrated P2 stars +In this section we discuss the individual results of the clus- +ters NGC 2808, NGC 5024, NGC 5272 and NGC 6809, which +contain a significant central concentration of the enriched (P2) stars. +NGC 2808 was separated into multiple stellar populations +by Milone et al. (2015) using a chromosome map with HST +photometry. We find that the inner region covered by the HST +field indicated that primordial and enriched stars are spatially +mixed with 𝐴+ = 0.02 ± 0.02, whereas Dalessandro et al. (2019) +found 𝐴+ = −0.029 ± 0.001, in agreement with our results +over the same approximate spatial range, i.e. 2 𝑅hlp. However, +the inclusion of stars in the ground-based photometry shows +a significant P2 central concentration for the full range of the +cluster, with 𝐴+ +total = −0.49 ± 0.08. Limiting the spatial range +to 4.27𝑅hlp resulted in a value of 𝐴+ +4 = −0.08 ± 0.03, further +strengthening the idea that omitting the outer stars from radially +dependent analyses can hide the true properties of clusters. NGC +2808 contains the largest sample of stars from all 28 analysed +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +15 +Table 1. Parameters for the 28 Galactic GCs studied in this paper. The individual columns give the final number of stars in our sample after the analysis of +Sections 2 and 3, split between the HST (NHST) and the ground-based (NGB) catalogues. The total cumulative radial distribution parameter 𝐴+ +4 was calculated +for all clusters at a maximum radius of 4.27𝑅hlp in units of projected half-light radius, except for clusters specified in Section 4.1. We also include the 𝐴+ +total +values calculated for the largest extent of each cluster covered by our datasets. The enriched star fractions P2/Ptot,4 were also calculated for a radius of 4.27𝑅hlp, +and the full range (P2/Ptot). The next column gives the maximum analysed radius in each cluster in units of projected half-light radius (𝑟max [HLR]). The final +columns give the dynamical ages (Age/Trh), mass loss fractions (Mc/Mi) and projected half-light radii (𝑅hlp) (see Section 4 for details). +Cluster +NHST +NGB +𝐴+ +4 +𝐴+ +total +P2/Ptot,4 +P2/Ptot +𝑟max [HLR] +Age / Trh +Mc/Mi +𝑅hlp [pc] +Fe/H +NGC 288 +190 +356 +0.21 ± 0.08 +0.25 ± 0.09 +0.37 ± 0.04 +0.37 ± 0.04 +4.61 +3.18 ± 0.23 +0.244 ± 0.007 +5.83 +-1.32 +NGC 1261 +903 +153 +0.02 ± 0.06 +0.40 ± 0.15 +0.58 ± 0.03 +0.58 ± 0.03 +25.74 +5.68 ± 0.32 +0.316 ± 0.005 +3.25 +-1.27 +NGC 1851 +1241 +256 +-0.08 ± 0.07 +-0.94 ± 0.25 +0.70 ± 0.03 +0.67 ± 0.03 +24.06 +7.60 ± 0.43 +0.283 ± 0.004 +1.74 +-1.18 +NGC 2808 +3356 +1401 +-0.08 ± 0.03 +-0.49 ± 0.08 +0.75 ± 0.02 +0.74 ± 0.01 +20.86 +3.61 ± 0.10 +0.412 ± 0.003 +2.45 +-1.14 +NGC 3201 +187 +363 +0.46 ± 0.12 +0.46 ± 0.12 +0.49 ± 0.04 +0.51 ± 0.04 +4.00 +3.48 ± 0.24 +0.462 ± 0.009 +3.80 +-1.59 +NGC 4590 +217 +144 +-0.13 ± 0.10 +0.04 ± 0.15 +0.53 ± 0.05 +0.52 ± 0.05 +5.86 +2.73 ± 0.18 +0.448 ± 0.036 +4.44 +-2.23 +NGC 4833 +535 +336 +-0.07 ± 0.07 +0.02 ± 0.08 +0.51 ± 0.03 +0.52 ± 0.03 +4.57 +6.36 ± 0.28 +0.199 ± 0.010 +3.26 +-1.85 +NGC 5024 +1293 +438 +-0.42 ± 0.05 +-0.84 ± 0.11 +0.56 ± 0.03 +0.52 ± 0.02 +10.31 +1.33 ± 0.07 +0.505 ± 0.036 +6.43 +-2.10 +NGC 5053 +0 +181 +0.07 ± 0.16 +0.07 ± 0.16 +0.46 ± 0.08 +0.45 ± 0.08 +3.50 +1.33 ± 0.16 +0.410 ± 0.089 +12.37 +-2.27 +NGC 5272 +1259 +619 +-0.36 ± 0.05 +-0.17 ± 0.10 +0.64 ± 0.02 +0.64 ± 0.02 +14.36 +2.92 ± 0.14 +0.476 ± 0.020 +3.39 +-1.50 +NGC 5286 +1990 +242 +0.09 ± 0.05 +-0.23 ± 0.12 +0.61 ± 0.02 +0.59 ± 0.02 +9.56 +6.21 ± 0.25 +0.283 ± 0.010 +2.37 +-1.69 +NGC 5904 +970 +657 +-0.10 ± 0.06 +-0.44 ± 0.15 +0.70 ± 0.03 +0.68 ± 0.02 +14.28 +3.23 ± 0.20 +0.451 ± 0.008 +3.51 +-1.29 +NGC 5986 +1278 +329 +-0.08 ± 0.04 +-0.12 ± 0.05 +0.59 ± 0.03 +0.59 ± 0.02 +5.99 +7.06 ± 0.43 +0.181 ± 0.013 +2.77 +-1.59 +NGC 6101 +252 +229 +0.66 ± 0.13 +0.70 ± 0.13 +0.52 ± 0.05 +0.53 ± 0.05 +4.27 +1.15 ± 0.11 +0.479 ± 0.087 +9.56 +-1.98 +NGC 6121 +197 +208 +-0.08 ± 0.10 +-0.08 ± 0.10 +0.64 ± 0.05 +0.62 ± 0.05 +2.89 +15.69 ± 0.76 +0.089 ± 0.001 +2.49 +-1.16 +NGC 6205 +1093 +421 +-0.04 ± 0.04 +-0.02 ± 0.07 +0.68 ± 0.03 +0.68 ± 0.03 +10.19 +3.86 ± 0.22 +0.390 ± 0.015 +3.46 +-1.53 +NGC 6218 +247 +333 +0.20 ± 0.07 +0.12 ± 0.09 +0.57 ± 0.04 +0.55 ± 0.04 +5.38 +12.68 ± 0.43 +0.220 ± 0.007 +2.83 +-1.37 +NGC 6254 +649 +524 +0.02 ± 0.05 +-0.04 ± 0.09 +0.62 ± 0.03 +0.61 ± 0.03 +6.82 +5.90 ± 0.39 +0.297 ± 0.006 +2.96 +-1.56 +NGC 6341 +740 +238 +-0.16 ± 0.07 +-0.52 ± 0.23 +0.57 ± 0.03 +0.55 ± 0.03 +20.29 +5.32 ± 0.15 +0.316 ± 0.004 +2.39 +-2.31 +NGC 6366 +88 +371 +0.19 ± 0.14 +0.09 ± 0.16 +0.45 ± 0.05 +0.47 ± 0.05 +4.38 +12.39 ± 1.11 +0.144 ± 0.007 +3.77 +-0.59 +NGC 6752 +437 +376 +0.08 ± 0.08 +-0.05 ± 0.14 +0.71 ± 0.04 +0.72 ± 0.04 +9.59 +4.88 ± 0.18 +0.382 ± 0.005 +2.87 +-1.54 +NGC 6809 +216 +364 +-0.51 ± 0.07 +-0.49 ± 0.07 +0.56 ± 0.04 +0.56 ± 0.04 +4.51 +3.64 ± 0.18 +0.261 ± 0.011 +4.58 +-1.94 +NGC 6838 +135 +213 +0.05 ± 0.13 +0.01 ± 0.12 +0.34 ± 0.06 +0.34 ± 0.06 +2.53 +12.98 ± 1.27 +0.253 ± 0.013 +3.35 +-0.78 +NGC 6934 +499 +119 +0.21 ± 0.07 +0.05 ± 0.14 +0.61 ± 0.04 +0.61 ± 0.04 +8.87 +6.39 ± 0.52 +0.371 ± 0.052 +2.95 +-1.47 +NGC 6981 +329 +123 +0.00 ± 0.09 +0.19 ± 0.15 +0.40 ± 0.05 +0.40 ± 0.05 +7.69 +8.48 ± 0.98 +0.082 ± 0.015 +4.14 +-1.42 +NGC 7078 +1352 +272 +-0.03 ± 0.06 +0.37 ± 0.10 +0.62 ± 0.03 +0.64 ± 0.03 +15.00 +4.10 ± 0.12 +0.471 ± 0.005 +2.03 +-2.37 +NGC 7089 +1815 +422 +-0.06 ± 0.05 +-0.04 ± 0.16 +0.64 ± 0.02 +0.64 ± 0.02 +24.35 +3.54 ± 0.14 +0.346 ± 0.006 +3.04 +-1.65 +NGC 7099 +295 +110 +0.14 ± 0.09 +-0.11 ± 0.16 +0.55 ± 0.05 +0.54 ± 0.05 +8.63 +5.83 ± 0.20 +0.231 ± 0.010 +2.54 +-2.27 +clusters, with 4757 stars in total. It presents a good opportunity +for obtaining substantial amounts of individual spectra for further +analysis. The final sample of the cluster contained 1323 P1 +stars and 3433 P2 stars, which exacerbates the mass budget +problem, especially considering that NGC 2808 has a young dy- +namical age and should still retain Mc/Mi ∼ 0.41 of its initial mass. +A spectroscopic analysis of NGC 5024 was performed by +Boberg et al. (2016) for 53 RGB stars within 500 arcseconds of +the cluster center, discovering a centrally concentrated enriched +population. This agrees with our cumulative radial distribution of +𝐴+ +4 = −0.42 ± 0.05, which includes stars from the cluster center to +739 arcseconds. However, for the two different methods used by +Boberg et al. (2016), they find P2/Ptotal ∼ 0.3, while our results +for the total enriched fraction shows P2/Ptotal = 0.52 ± 0.02. +Since Boberg et al. (2016) only used RGB stars with magnitudes +𝑉 < 15.5, while our analysis includes the full RGB of stars with +magnitudes 𝑉 < 19.3, we argue that our enriched star fraction +includes a larger and more complete sample and is therefore more +indicative of the enriched star fraction. We found NGC 5024 has +the highest amount of remaining initial mass with Mc/Mi ∼ 0.51, +along with one of the lowest dynamical ages, meaning its initial +conditions should not have changed significantly over time. From +Figure 14 we see that the photometry from the inner region alone +provides a different picture than the combination of HST and +ground-based photometry, supporting the idea that dynamical +mixing of the populations affects the center of the cluster before the +outer regions. Although the HST region contained 1293 stars and +the ground-based photometry contained 438 stars, these outermost +stars prove to be crucial in arriving at the full picture. +NGC 5272 was previously analysed by Dalessandro et al. +(2019), who used a combination of HST photometry and Ström- +gren photometry from Massari et al. (2016). Additionally, Lardo +et al. (2011) used SDSS photometry for RGB stars beyond 100 +arcseconds from the cluster center. Both discovered a centrally con- +centrated enriched population, consistent with our cumulative radial +distribution of 𝐴+ +4 = −0.36 ± 0.05 for stars within 4.27 𝑅hlp. How- +ever, we found that extending to the full possible extent of the cluster +returned a value of 𝐴+ +total = −0.17 ± 0.10, showing a less signif- +icant P2 central concentration overall. We found NGC 5272 has +retained a high fraction of its initial mass, estimated to be close to +Mc/Mi ∼ 0.48, so we consider NGC 5272 to also largely preserve +its initial conditions. Our cumulative radial distribution for the HST +photometry alone shows no dynamical mixing between the popula- +tions with 𝐴+ ∼ 0, but the ground-based photometry indicates the +outer regions are not yet mixed. +Rain et al. (2019) identified two populations in NGC 6809 +based on 11 RGB stars using high resolution FLAMES/UVES spec- +tra. Their spectroscopic identification of two populations is consis- +tent with our photometric identification of two populations in both +photometric data sets. We found a centrally concentrated enriched +population in both the HST and ground-based photometry, which +indicates a lack of dynamical mixing within the center of the clus- +ter when compared with NGC 5024 and NGC 5272. Interestingly, +NGC 6809 is dynamically young but has lost a significant amount +of its initial mass, with Mc/Mi ∼ 0.26. NGC 6809 has the smallest +galactocentric distance in our sample, with 𝑅𝐺𝐶 = 4.01 ± 0.03 kpc +MNRAS 000, 1–20 (2023) + +16 +E. I. Leitinger et al. +and an escape velocity of 𝑣𝑒𝑠𝑐 = 17.3 km/s. Tidal disruption affects +clusters with smaller galactocentric distances more strongly (Baum- +gardt et al. 2019) and the size of an accreted cluster in particular will +respond to the tidal field of the MW upon accretion (Miholics et al. +2014). As NGC 6809 is both suggested to be an accreted cluster +(Massari et al. 2019) and has a small galactocentric distance and +relatively low escape velocity, we expect that although the cluster is +dynamically young, tidal disruption after its accretion has affected +its initial conditions. It therefore becomes somewhat difficult to con- +fidently conclude whether our discovery of a centrally concentrated +P2 population is representative of its initial spatial distribution. +4.3.2 +Clusters with centrally concentrated P1 stars +One of the most interesting results of this work is the centrally +concentrated primordial populations found in NGC 3201 and NGC +6101. In order to test the validity of these findings, we present a +more thorough analysis of the two clusters in this section. +NGC 3201 is considered dynamically young, but previous stud- +ies of the cluster have produced complicated results that cause un- +certainty around whether we can assume it maintains its initial +configuration. NGC 3201 is proposed to be an accreted cluster pre- +viously belonging to Sequoia/Gaia-Enceladus (Massari et al. 2019). +Lucatello et al. (2015) found that the P1 population in NGC 3201 +hosts a higher fraction of binary stars than the P2 population, which +they suggested to be due to the dense conditions of the central region +that enhance the destruction and ejection of binaries. This result as- +sumes that only P2 stars can be centrally concentrated. Kamann +et al. (2020) used HST photometry and MUSE spectroscopy and +also found that NGC 3201 contains a higher binary fraction in the +P1 population than it does for P2. They compare this result to sim- +ulations suggesting P1 binaries are only overabundant outside the +half-light radius (Hong et al. 2015, 2016). These simulations also +assume a P2 central concentration, as they use this configuration +for the initial conditions of their simulation. Our discovery of a P1 +concentration (𝐴+ +4 = 0.46 ± 0.12) therefore does not support the +previous hypothesis proposed to describe the relative binary frac- +tions between different sub-populations, but our result is not unique +in that Hartmann et al. (2022) also discovered a P1 central con- +centration by combining HST photometry with photometry from +the S-PLUS survey. Bianchini et al. (2019) and Wan et al. (2021) +investigated the peculiar kinematics in the outskirts of NGC 3201, +which contains tidal tails and exhibits flattened velocity dispersions +in the outskirts. +When analysing NGC 3201, we found that it suffered from sig- +nificant differential reddening. However, after correcting for its ef- +fect (see Section 2.2), the final spatial distribution of the populations +showed no indication of problems due to differential reddening. In +NGC 3201, we found that P1 stars had the highest concentration +at intermediate radii around 150”, with P2 stars being dominant in +the outer parts and also towards the center of the cluster. A KS test +showed that the central concentration of P2 was significant at a ∼ 2𝜎 +level and significant at the 8𝜎 level towards the outer parts, leading +to a U-shaped distribution in the relative fraction of P2 stars. +In order to properly test the validity of the primordial central +concentration discovery in the 𝐴+ parameter, we performed the +probability cut and population limit tests outlined at the end of +Section 3.1. By testing the effect of different limits in Δ𝐶UBI to +separate the populations, we found 𝐴+ = 0.28 ± 0.25. Similarly, +by testing different probability thresholds for the membership +of stars belonging to P1 and P2, we found 𝐴+ = 0.35 ± 0.03. +These tests confirm that the presence of a centrally concentrated +primordial population is a consistent/robust result regardless of the +method chosen to classify P1/P2 stars. Our discovery of a centrally +concentrated primordial population could indicate that the peculiar +kinematics found by Bianchini et al. (2019) and Wan et al. (2021) +is driven by the enriched population of stars in the outskirts. NGC +3201 has intriguing characteristics and our discovery of a P1 central +concentration further adds to these previous results. However, it +is difficult to describe the complexity of NGC 3201 using only +the 𝐴+ parameter and future work would benefit from a parameter +which incorporates both the radial spatial distributions between +populations and the enriched star fraction for such clusters. +Dalessandro et al. (2019) analysed NGC 6101 and found +𝐴+ = −0.003 ± 0.001, indicating the populations are homoge- +neously mixed. In our analysis we found 𝐴+ = −0.07 ± 0.02 for +252 stars in the HST photometry, whereas 𝐴+ = 0.57 ± 0.19 +was found using 229 stars in the ground-based photometry alone. +Our combined cumulative radial distributions indicate a centrally +concentrated primordial population. NGC 6101 is the only case +in our sample for which the HST and ground-based separations +using the Δ𝐶UBI distributions returned a different number of +populations. The chromosome map returned two populations, as +did the Δ𝐶UBI distribution for the HST photometry. However, in the +ground-based Δ𝐶UBI distribution, three populations were returned. +The blending of the populations was also somewhat present in the +chromosome map, but two populations are nonetheless distinct +enough for separation, as is also shown in Figure 7 of Milone et al. +(2017) where the primordial population contains more stars than +the enriched population. We found NGC 6101 has retained almost +half of its initial mass (Mc/Mi ∼ 0.48) and has gone through the +least amount of dynamical mixing of all 28 clusters. With a low +metallicity of [Fe/H]= −1.98 dex (Harris 2010), the populations +in a Δ𝐶UBI distribution are closer together than in more metal +rich targets, since 𝐶UBI is most sensitive to molecular bands, +which are weaker at low metallicities. This leads to difficulties in +separating the populations. Due to that, we thoroughly tested how +the separation of populations affected the final cumulative radial +distributions. The result of trying different probability thresholds +for the memberships of stars belonging to P1 and P2 returned a +value of 𝐴+ = 0.59 ± 0.06, while the test of sampling arbitrary +limits in the Δ𝐶UBI colour distributions returned 𝐴+ = 0.39 ± 0.19, +showing a robust signal that P1 is concentrated in all cases. +Some simulations have studied the concept of an initially cen- +trally concentrated population evolving over time. For example, the +simulations of Vesperini et al. (2013) show that for a dynamically +young cluster with an initial P2 central concentration, the P2 frac- +tion as a function of radius will decrease significantly in the outer +regions of the cluster, due to the slowing of two-body relaxation +at larger distances from the cluster centre. However, we note that +the same could be concluded if P1 were to have been formed more +centrally concentrated, as there is no physical distinction between +stars labeled P1 or P2 in these simulations, other than their ini- +tial configurations. Therefore, the behaviour we observe from the +dynamically young clusters in our sample is indicative of the ini- +tial conditions, where the P1 population was born initially more +centrally concentrated. +When viewing only the inner HST region (𝑟 < 1𝑅hlp) of NGC +3201 (Figure A3), we found that the P2/Ptotal fraction decreases +with increasing radius. We performed a K-S test on the P1 and P2 +distributions within this range to quantify this, based on the standard +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +17 +two-sample test described in Section 12.4 of Monahan (2001), but +modified to also include the weights (𝑤 = 1/ 𝑓𝑇 ) of each star, fol- +lowing the method described in Equations 3-5 of Baumgardt et al. +(2022). The weighted K-S test showed the P1 and P2 distributions +have a 2% probability of following the same distribution, meaning +there is likely a P2 central concentration for the inner region. How- +ever, if we consider stars beyond 1𝑅hlp the enriched star fraction +increases for the outer regions. Figure 7 of Vesperini et al. (2013) +shows a simulated scenario in which the enriched star fraction as a +function of radius could demonstrate similar U-shaped behaviour, +however, it is not immediately clear that this represents the same +phenomenon observed in NGC 3201. +For example, the radius at which Vesperini et al. (2013) expects +this increase (𝑟 > 5𝑅hlp) is much larger than the radius at which +we observe the increase (𝑟 ∼ 1𝑅hlp). Moreover, the dynamical +ages (Age/Trh) at which the U-shaped behaviour occurs in the +simulations is expected to be Age/Trh ≥ 5, whereas NGC 3201 +has a dynamical age of Age/Trh = 3.48 ± 0.24. Finally, Vesperini +et al. (2013) describes this increase as a "weak final rise" on the +order of ∼ 10%, whereas in NGC 3201 we observe an ∼ 300% +increase at an ∼ 8𝜎 significance between the minimum at ∼ 1𝑅hlp +and the maximum at ∼ 4𝑅hlp of the enriched star fraction. Detailed +simulations will be necessary to test how the initial conditions of +NGC 3201 looked. +4.3.3 +Spatially mixed populations +We focus in this section on the dynamically young clusters that +have retained most of their initial conditions but are nevertheless +spatially mixed and do not contain one centrally concentrated +population. These clusters include NGC 288, NGC 4590, NGC +5053, NGC 5904, NGC 6205, NGC 7078 and NGC 7089. +NGC 288 was analysed by Dalessandro et al. (2019) using +HST photometry, in which they found that it contains spatially +mixed populations with 𝐴+ = −0.045 ± 0.002. Similarly, we +found two spatially mixed populations, with 𝐴+ +4 = 0.21 ± 0.08 (P1 +centrally concentrated only at < 3𝜎 level). Additionally, Hartmann +et al. (2022) used both, HST photometry and photometry from the +S-PLUS survey, calculating cumulative radial distributions that +show mixed populations in the central HST regions, but with a +P2 central concentration in the outer regions. The discrepancies +between our results in the outer regions - aside from the use of +different photometric bands - appears to be due to differences in +our analysis methods. More specifically, our sample of stars are +corrected for photometric incompleteness, we exclude stars from +our analysis in which the P1/P2 classifications are ambiguous +(𝑝 > 80%), our limiting radius is 4.27 𝑅ℎ𝑙𝑝 compared to their 5.5 +𝑅ℎ𝑙𝑝 and our sample includes an extra 116 stars in comparison. We +found that NGC 288 has retained only a fraction Mc/Mi ∼ 0.24 of +its initial mass, with an enriched fraction of P2/Ptotal = 0.37±0.04. +At a glance, it seems plausible that mass loss is responsible for +ejecting either primordial or enriched stars from the outer regions, +resulting in spatially mixed populations. However, it is also possible +that NGC 288 formed with spatially mixed populations, as the +initial configuration is difficult to determine due to the significant +amount of mass loss. +We found that NGC 4590 contains spatially mixed pop- +ulations for both the HST and ground-based photometry, but +based on a comparatively small sample size of 361 stars. Baum- +gardt et al. (2019) found that NGC 4590 has large perigalactic +(8.95 ± 0.06 kpc) and apogalactic (29.51 ± 0.42 kpc) distances, +and Massari et al. (2019) suggests one of the Helmi streams +is the progenitor of this cluster. We found NGC 4590 retains +approximately Mc/Mi ∼ 0.45 of its initial mass and is one of +the dynamically youngest clusters in our sample, but nonetheless +contains fully spatially mixed populations. The large peri- and +apogalactic distances suggest tidal stripping is unlikely to have +removed a significant fraction of stars, but the accretion of NGC +4590 to the MW may have led to a stronger than predicted mass loss. +Previous work has found NGC 5053 to be dynamically +complicated: it contains significant tidal tails (Jordi & Grebel +2010; Lauchner et al. 2006) and a possible tidal bridge to NGC +5024 (Chun et al. 2010). Although NGC 5053 has one of the +lowest dynamical ages and is predicted to retain a significant +fraction of its initial mass with Mc/Mi ∼ 0.41, we found its stellar +populations are spatially mixed. NGC 5053 was the only cluster +for which we relied solely on the ground-based photometry. Due to +the insufficient number of RGB stars in the HST photometry, the +full extent of the ground-based photometry - including the cluster +center - was used instead. The core of NGC 5053 has the lowest +density of any cluster in our sample, and it has a large half-light +radius, greatly reducing the blending effect in the cluster center +that usually plagues ground-based photometry. As it is possible +that NGC 5053 and NGC 5024 were accreted together within the +same dwarf galaxy, we note that this event may have affected the +mass loss of both clusters. +The work of Lee (2019) using Strömgren photometry and the +CUBI index found two populations in NGC 5904 with spatially +mixed populations. In a follow-up paper, Lee (2021) stated that +this previously determined bimodal distribution could actually +be separated further into three populations using Strömgren and +Ca-CN-CH-NH photometry. With this difference in classifications, +their cumulative radial distributions changed from showing spatially +mixed populations throughout the extent of the cluster - consistent +with our results - to instead showing the most carbon-poor and +nitrogen-rich population as centrally concentrated. Lardo et al. +(2011) also separated NGC 5904 into two populations using +SDSS photometry, which they refer to as UV-blue and UV-red. +The resulting cumulative radial distributions from Lardo et al. +(2011) show the UV-red stars are more centrally concentrated. Our +final sample of NGC 5904 contains a large sample size of 1627 +RGB stars and was consistent between the HST and ground-based +photometry in identifying two stellar populations, exhibiting +complete spatial mixing between populations and a consistent +enriched fraction of P2/Ptotal = 0.68 ± 0.02 throughout the cluster. +We found that our results are consistent with only the initial +findings of Lee (2019), as we did not find three populations within +NGC 5904 using the combined HST and ground-based photometry. +The introduction of spectroscopy to classify the populations based +on chemical abundances such as carbon and nitrogen may help to +check the validity of our photometrically separated populations. +NGC 6205 was found to have a mass loss ratio close to +Mc/Mi ∼ 0.39 and is spatially mixed to its outermost regions at +10.19𝑅ℎ𝑙𝑝. Similarly, we found NGC 7089 has a mass loss ratio +of Mc/Mi ∼ 0.35 with spatially mixed populations extending +out to 24𝑅ℎ𝑙𝑝. Both clusters have large masses and are close to +the upper limit of our definition of ‘dynamically young’. NGC +6205 has previously been analysed by Savino et al. (2018) using +both HST and Strömgren photometry, in which they estimate +MNRAS 000, 1–20 (2023) + +18 +E. I. Leitinger et al. +an enriched fraction of approximately 80%, compared to our +fraction of P2/Ptotal += 0.68 ± 0.03. In terms of cumulative +radial distributions, they also found no evidence for a centrally +concentrated population in both the inner and outer regions of +NGC 6205 (extending to approximately 700 arcseconds). NGC +7089 was analysed by Hartmann et al. (2022) using HST and +S-PLUS survey photometry, discovering a P2 central concentration +in both the HST field and outer region, which is at odds with +our results of spatially mixed populations throughout the cluster. +We also find our results at odds with Lardo et al. (2011), who +identified a centrally concentrated population using cumulative +radial distributions from SDSS photometry for both NGC 6205 +and NGC 7089. If the dynamical age of NGC 6205 is long enough +for dynamical mixing to occur throughout the entire cluster, we +would expect this to occur for NGC 7078 and NGC 6809 as +well, as per Figure 15. However, we found clusters with similar +dynamical ages have strongly varying spatial concentrations instead. +Previous photometric analysis of NGC 7078 has found +contradictory results: Larsen et al. (2015) combined HST and +SDSS photometry of RGB stars and discovered three stellar +populations, which yielded a centrally concentrated primordial +population; however, Lardo et al. (2011) found only two popula- +tions using SDSS photometry and consequentially discovered a +centrally concentrated enriched population instead. The 𝑚𝐹336𝑊 +vs. 𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 plot of Piotto et al. (2015b) (Figure +22) shows at least two populations within NGC 7078 using HST +photometry, while Milone et al. (2017) distinctly separated the +HST photometry into three populations using a chromosome map. +We found NGC 7078 contained one of the largest discrepancies for +the P1 and P2 populations between the chromosome map and the +Δ𝐶UBI distribution, with a contamination of approximately 20%. +The low metallicity of NGC 7078 makes it difficult to separate +the populations in the Δ𝐶UBI distribution, as the molecular bands +responsible for the colour variations in Δ𝐶UBI become weaker, +translating into smaller colour differences (see discussion in e.g. +Balbinot et al. 2022, and references therein). Additionally, this +cluster suffers severely from differential reddening, which adds +noise to the signal of the multiple populations. Taking these caveats +into account, we advise the reader to take the following results for +NGC 7078 with caution. We checked other low metallicity clusters +in our sample ([Fe/H] < −1.8) and found they did not suffer from +this same confusion, and NGC 7078 is the only cluster in our +sample affected by this. +Because of the significant overlap between the Gaussians fitted +by GMM to separate the populations in the Δ𝐶UBI distribution +for NGC 7078, we took a different approach. In the HST region, +we rely on the chromosome map classification of populations, +finding a resulting 𝐴+ value close to zero, which indicates the +centre of the cluster is spatially mixed. Guided by the HST data, +we establish colour cuts for the ground-based data which gave us +relatively pure P1 and P2 stars. More specifically, we selected only +the extremes of the P1 (Δ𝐶UBI < −0.7) and P2 (Δ𝐶UBI > −0.3) +populations. For reference we cross-matched our RGB stars +with APOGEE DR17 (Majewski et al. 2017; Abdurro’uf et al. +2022), where the stars in our final sample correspond to [Al/Fe] +abundances of [Al/Fe]< 0.05 for P1 stars and [Al/Fe]> 0.4 for +P2 stars. The final 𝐴+ +4 and 𝐴+ +𝑡𝑜𝑡𝑎𝑙 values quoted are therefore the +combination of chromosome map classifications for the HST stars +and our sample of extreme P1 and P2 stars selected as described +above for the ground-based stars. We find 𝐴+ +4 = −0.03 ± 0.06, +indicative of spatially mixed populations out to 4.27𝑅hlp, +but with an overall signal indicating a P1 central concentration +for the full extent (out to 15𝑅hlp) of the cluster (𝐴+ +total = 0.37±0.10). +4.4 +Constraints on the loss of P1 stars +Previous mass-loss scenarios involving internal enrichment aim to +solve the mass budget problem by suggesting P1 stars are primarily +located in the outskirts of GCs during formation, e.g. Krause et al. +(2020). With P2 stars concentrated in the centre, mass-loss in the +outskirts would then be responsible for the removal of P1 stars from +the clusters (e.g. D’Ercole et al. 2008; Vesperini et al. 2010). Bastian +& Lardo (2015) explored this concept by analysing the correlations +between enriched star fractions and cluster properties such as mass, +metallicity and Galactocentric distance using literature data from +33 GCs. For scenarios in which self-enrichment is responsible for +the MP phenomenon, the enriched star fraction is expected to vary +from the initial birth of the cluster to the present day, but was instead +found to be constant throughout time, within errors. They concluded +that the mass budget problem cannot be solved by assuming mass- +loss in the outskirts of clusters, claiming that alternative theories +are needed instead. Gratton et al. (2019) suggested a combination of +polluting and diluting scenarios may explain the resulting chemical +abundance spreads observed in GCs, with an emphasis that the inter- +acting binaries theory (Vanbeveren et al. 2012) may be responsible +for the ejection of stars in clusters. According to their relative spa- +tial distribution (i.e. 𝐴+), we have found varying behaviours for the +initial spatial configurations of MPs in our sample of Galactic GCs, +where dynamically young clusters in our sample show P1 centrally +concentrated stars, as well as a homogeneous mix of populations. +However, this by itself does not necessarily translate to the exacer- +bation of the mass budget problem as one also needs to account for +the relative number of P1 stars in the outskirts of the clusters (i.e. +where stars more likely to escape from the cluster reside). +Our analysis has revealed that in the outer regions P1 stars +do not constitute the majority of the stars, with the exception of +NGC 5024 and NGC 6809 (see bottom right panels of Figure 14 +and Figures in Appendix A). This suggests that contrary to what +is required by different models, during the dynamical evolution of +these clusters P2 stars would be lost to the field population at a +similar or higher rate than P1 stars. This would have important +implications on the interpretations of the number of P2 stars found +in the field, and their use to anchor the contribution of dissolved +GCs to their host galaxy mass. +5 +CONCLUSIONS +We have performed a spatially complete analysis of a large and +diverse sample of 28 Galactic GCs, showing that GCs which still +maintain their initial conditions can contain a central concentration +of enriched or primordial stars, as well as a homogeneous mix of +both. We found centrally concentrated enriched populations in NGC +2808, NGC 5024, NGC 5272 and NGC 6809. They can be explained +with existing formation theories that involve internal polluters, such +as SMS, FRMS or AGB stars. We can also rely on the notion of +dynamical mixing to explain why GCs with large dynamical ages +tend to have spatially homogeneous stellar populations over time. +However, dynamically young GCs with a centrally concentrated pri- +mordial population (NGC 3201 and NGC 6101) cannot be explained +with current formation theories. These models cannot account ei- +ther for dynamically young GCs that already contain fully spatially +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +19 +mixed stellar populations such as NGC 288, NGC 4590, NGC 5053, +NGC 5904, NGC 6205, NGC 7078 and NGC 7089. Furthermore, +the existence of dynamically young clusters with fully mixed pop- +ulations or a centrally concentrated P1, pose more challenges if +P1 stars are required to be preferentially lost during the long term +dynamical evolution of the cluster. +Interpolations or simulations based off an incomplete view of +clusters have previously been used to constrain the possible fractions +of primordial or enriched stars. In our analysis, we used a spatially +complete view of each cluster to calculate the enriched star fractions +(P2 / Ptotal), which showed a clear correlation with the initial mass, +but no clear correlations against other global parameters such as +age and metallicity. Our sample of 28 Galactic GCs, 4 low-mass +Galactic GCs and 7 Local Group GCs provided a range of 0 ≤ P2 +/ Ptotal < 0.75 for the total enriched star fractions. We found that +in some clusters, the enriched star fraction as a function of radius +was constant across the extent of the cluster, while others exhibited +either increasing or decreasing enriched star fractions. +Current theories of GC formation and theoretical simulations +have assumed the possibility of only a P2 central concentration, due +in part to an analysis which limits itself to only the central regions of +clusters and assumes conclusions on the properties of the full clus- +ter. We argue the need for future theories and simulations to also +consider alternative configurations of initial conditions. The next +stage of this research will explore the spectroscopic data available +for our sample of 28 GCs in the same manner: combining data for +the inner and outer regions of each cluster for a spatially complete +view. We aim to check the validity of our photometric separations +by spectroscopically separating the stellar populations based on +chemical abundances. We will also use our current classifications +of populations to explore the kinematic differences, along with dif- +ferences in chemical abundances and binary fractions in order to +provide further observational information relating to the possible +initial conditions and the final, dynamically mixed conditions of the +clusters in our sample. +ACKNOWLEDGEMENTS +We thank the referee for their insightful feedback on the manuscript, +which improved the quality of our work. +The authors are very grateful to Florian Niederhofer for providing +us an independent, HST based reddening map of NGC 7078. +This study was supported by the Klaus Tschira Foundation. +DATA AVAILABILITY +The Hubble Space Telescope UV Globular Cluster Survey +(“HUGS”) photometric catalogue: https://archive.stsci. +edu/prepds/hugs/ +The wide-field, ground-based Johnson-Cousins UBVRI photomet- +ric catalogue, courtesy of Stetson et al. 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G., 2009, A&A, 507, L1 +APPENDIX A: CUMULATIVE RADIAL DISTRIBUTIONS +OF DYNAMICALLY YOUNG CLUSTERS +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +21 +0.5 +1.0 +1.5 +2.0 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = 0.02 ± 0.02 +5 +10 +15 +20 +Projected Radius [HLR] +200 +400 +600 +800 +1000 +A + = -0.58 ± 0.24 +0 +5 +10 +15 +20 +Projected Radius [HLR] +A + +total = -0.49 ± 0.08 + A + +4 = -0.08 ± 0.03 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.76 +200 +400 +600 +800 +1000 +Projected Radius [arcsec] +P2 / Ptotal: 0.66 +0 +200 +400 +600 +800 +1000 +Projected Radius [arcsec] +P2 / Ptotal: 0.74 +Figure A1. As in Figure 14, but for NGC 2808. +0.2 +0.4 +0.6 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = 0.00 ± 0.03 +1 +2 +3 +4 +Projected Radius [HLR] +100 +200 +300 +400 +500 +600 +A + = 0.27 ± 0.10 +0 +1 +2 +3 +4 +Projected Radius [HLR] +A + +total = 0.25 ± 0.09 + A + +4 = 0.21 ± 0.08 +20 +40 +60 +80 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.33 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.38 +0 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.37 +Figure A2. As in Figure 14, but for NGC 288. +MNRAS 000, 1–20 (2023) + +22 +E. I. Leitinger et al. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.03 ± 0.02 +1 +2 +3 +4 +Projected Radius [HLR] +100 +200 +300 +400 +500 +600 +700 +A + = 0.78 ± 0.13 +0 +1 +2 +3 +4 +Projected Radius [HLR] +A + +total = 0.46 ± 0.12 + A + +4 = 0.46 ± 0.12 +0 +20 +40 +60 +80 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.53 +100 +200 +300 +400 +500 +600 +700 +Projected Radius [arcsec] +P2 / Ptotal: 0.50 +0 +200 +400 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.51 +Figure A3. As in Figure 14, but for NGC 3201. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = 0.02 ± 0.04 +1 +2 +3 +4 +5 +6 +Projected Radius [HLR] +100 +200 +300 +400 +500 +A + = 0.39 ± 0.26 +0 +1 +2 +3 +4 +5 +6 +Projected Radius [HLR] +A + +total = 0.04 ± 0.15 + A + +4 = -0.13 ± 0.10 +0 +20 +40 +60 +80 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.56 +100 +200 +300 +400 +500 +Projected Radius [arcsec] +P2 / Ptotal: 0.48 +0 +100 +200 +300 +400 +500 +Projected Radius [arcsec] +P2 / Ptotal: 0.52 +Figure A4. As in Figure 14, but for NGC 4590. +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +23 +0 +1 +2 +3 +4 +Projected Radius [HLR] +0 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + +total = 0.07 ± 0.16 + A + +4 = 0.07 ± 0.16 +0 +1 +2 +3 +4 +Projected Radius [HLR] +0 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.45 +Figure A5. As in Figure 14, but for NGC 5053. No HST photometry was used, so only the ground-based photometry was included in the final sample. +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.02 ± 0.02 +2 +4 +6 +8 +10 +12 +14 +Projected Radius [HLR] +200 +400 +600 +800 +1000 +A + = 1.14 ± 0.23 +0 +2 +4 +6 +8 +10 +12 +14 +Projected Radius [HLR] +A + +total = -0.17 ± 0.10 + A + +4 = -0.36 ± 0.05 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.70 +200 +400 +600 +800 +1000 +Projected Radius [arcsec] +P2 / Ptotal: 0.52 +0 +200 +400 +600 +800 +1000 +Projected Radius [arcsec] +P2 / Ptotal: 0.64 +Figure A6. As in Figure 14, but for NGC 5272. +MNRAS 000, 1–20 (2023) + +24 +E. I. Leitinger et al. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.01 ± 0.03 +2.5 +5.0 +7.5 +10.0 +12.5 +Projected Radius [HLR] +250 +500 +750 +1000 +1250 +A + = -0.49 ± 0.25 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +Projected Radius [HLR] +A + +total = -0.44 ± 0.15 + A + +4 = -0.10 ± 0.06 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.71 +250 +500 +750 +1000 +1250 +Projected Radius [arcsec] +P2 / Ptotal: 0.65 +0 +250 +500 +750 +1000 +1250 +Projected Radius [arcsec] +P2 / Ptotal: 0.68 +Figure A7. As in Figure 14, but for NGC 5904. +0.2 +0.4 +0.6 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.07 ± 0.02 +1 +2 +3 +4 +Projected Radius [HLR] +100 +200 +300 +400 +500 +600 +A + = 0.57 ± 0.19 +0 +1 +2 +3 +4 +Projected Radius [HLR] +A + +total = 0.70 ± 0.13 + A + +4 = 0.66 ± 0.13 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.33 +100 +200 +300 +400 +500 +Projected Radius [arcsec] +P2 / Ptotal: 0.65 +0 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.53 +Figure A8. As in Figure 14, but for NGC 6101. +MNRAS 000, 1–20 (2023) + +A Wide-Field View on Multiple Populations +25 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.07 ± 0.02 +2 +4 +6 +8 +10 +Projected Radius [HLR] +200 +400 +600 +800 +1000 +A + = 0.22 ± 0.19 +0 +2 +4 +6 +8 +10 +Projected Radius [HLR] +A + +total = -0.02 ± 0.07 + A + +4 = -0.04 ± 0.04 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.68 +200 +400 +600 +800 +1000 +Projected Radius [arcsec] +P2 / Ptotal: 0.66 +0 +200 +400 +600 +800 +1000 +Projected Radius [arcsec] +P2 / Ptotal: 0.68 +Figure A9. As in Figure 14, but for NGC 6205. +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = -0.05 ± 0.02 +1 +2 +3 +4 +Projected Radius [HLR] +200 +400 +600 +800 +A + = -0.41 ± 0.09 +0 +1 +2 +3 +4 +Projected Radius [HLR] +A + +total = -0.49 ± 0.07 + A + +4 = -0.51 ± 0.07 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.70 +200 +400 +600 +800 +Projected Radius [arcsec] +P2 / Ptotal: 0.49 +0 +200 +400 +600 +800 +Projected Radius [arcsec] +P2 / Ptotal: 0.56 +Figure A10. As in Figure 14, but for NGC 6809. +MNRAS 000, 1–20 (2023) + +26 +E. I. Leitinger et al. +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = 0.01 ± 0.05 +4 +6 +8 +10 +12 +14 +Projected Radius [HLR] +100 +200 +300 +400 +500 +600 +A + = 1.76 ± 0.33 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +Projected Radius [HLR] +A + +total = 0.37 ± 0.10 + A + +4 = -0.03 ± 0.06 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.63 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.67 +0 +100 +200 +300 +400 +500 +600 +Projected Radius [arcsec] +P2 / Ptotal: 0.64 +Figure A11. As in Figure 14, but for NGC 7078. Refer to Section 4.3.3 for a special discussion on NGC 7078. +0.0 +0.5 +1.0 +1.5 +Projected Radius [HLR] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Cumulative Radial Distribution +P1 +P2 +A + = 0.08 ± 0.02 +5 +10 +15 +20 +25 +Projected Radius [HLR] +200 +400 +600 +800 +1000 1200 +A + = 0.07 ± 0.44 +0 +5 +10 +15 +20 +25 +Projected Radius [HLR] +A + +total = -0.04 ± 0.16 + A + +4 = -0.06 ± 0.05 +0 +20 +40 +60 +80 +100 +Projected Radius [arcsec] +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +P2 / Ptotal +P2 / Ptotal: 0.65 +200 +400 +600 +800 +1000 1200 +Projected Radius [arcsec] +P2 / Ptotal: 0.61 +0 +250 +500 +750 +1000 +1250 +Projected Radius [arcsec] +P2 / Ptotal: 0.64 +Figure A12. As in Figure 14, but for NGC 7089. +MNRAS 000, 1–20 (2023) + diff --git a/wNE2T4oBgHgl3EQf3AhK/content/tmp_files/load_file.txt b/wNE2T4oBgHgl3EQf3AhK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efe07e38c64aea62ec209ccb6cd411cc5e1dddbf --- /dev/null +++ b/wNE2T4oBgHgl3EQf3AhK/content/tmp_files/load_file.txt @@ -0,0 +1,2428 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf,len=2427 +page_content='MNRAS 000, 1–20 (2023) Preprint 12 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 A Wide-Field View on Multiple Stellar Populations in 28 Milky Way Globular Clusters E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger,1,2★ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Baumgardt,1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Cabrera-Ziri3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Hilker2 and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Pancino4,5 1School of Mathematics and Physics, The University of Queensland, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lucia, QLD, 4072, Australia 2European Southern Observatory, Karl-Schwarzschild-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Germany 3Astronomisches Rechen-Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Zentrum für Astronomie der Universität Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Mönchhofstraße 12-14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' D-69120 Heidelberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Germany 4INAF – Osservatorio Astrofisico di Arcetri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Largo Enrico Fermi 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I-50125 Firenze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Italy 5Space Science Data Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' ASI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' via del Politecnico snc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I-00133 Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Italy Accepted 19 December 2022 ABSTRACT The majority of Galactic globular clusters (GCs) contain multiple stellar populations displaying specific chemical abundance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In particular, GCs generally contain a ‘primordial’ population with abundances similar to field stars, along with an ‘enriched’ population ex- hibiting light element anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this paper we present a homogeneous and wide-view analysis of multiple stellar populations in 28 Galactic GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' By using a combination of HST photometry together with wide-field, ground-based photometry we are able to analyse be- tween 84% and 99% of all stars in each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For each GC, we classify stars into separate sub-populations using the well-established 𝐶UBI colour index, and investigate the spatial dis- tributions of these populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our results show that dynamically young GCs can contain either centrally concentrated enriched or primordial populations, or no centrally concentrated population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Dynamically old GCs show fully mixed populations as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The existence of clusters born with centrally concentrated primordial (and homogeneously mixed) populations exacerbates the mass-budget problem facing many cluster formation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The diversity in these results also highlights the need for additional theories that can account for the wide variety of initial conditions that we find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We finally investigate the enriched star fraction as a function of different global parameters in our GC sample, using also data for young and low-mass clusters from the Small- and Large Magellanic Clouds and confirm earlier results that the enriched star fraction strongly correlates with the initial mass of a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Key words: (Galaxy:) globular clusters: general – Stars: abundances – (stars:) Hertzsprung–Russell and colour–magnitude diagrams – stars: kinematics and dynamics – Galaxy: evolution 1 INTRODUCTION Most Galactic globular clusters (GCs) contain multiple stellar pop- ulations (MPs), distinguished by star-to-star variations in light ele- ment abundances that are not explained by simple stellar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Stars are determined as ‘primordial’ (usually as P1) if their ele- mental abundances are similar to the surrounding field stars of the cluster, and ‘enriched’ (P2) if they demonstrate an enhancement in some light elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' He, N, Na and Al), but a depletion in others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' C, O and sometimes Mg) in comparison to P1 (Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Charbonnel 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bastian & Lardo 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, heavier element variations such as Fe only are present in a minority of clusters (Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Willman & Strader 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bastian & Pfeffer 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The formation history of GCs necessary to produce MPs is a matter of ongoing debate (Forbes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' ★ E-mail: ellen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='leitinger@uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='au 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Cassisi & Salaris 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We know that the observed abun- dance patterns are compatible with the chemistry of the CNO-cycle (and hot subcycles) and that this happens mostly in massive stars or in the H-burning shells of red giants, which leads to the theory that stellar formation of the enriched populations is fuelled by GC internal processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' An important piece of information regarding the formation history of MPs in GCs is the spatial distribution of the stars in each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' If a cluster has not undergone significant dynamical mixing during its lifetime, we can assume it still maintains its initial spatial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' If we then observe that one stellar population is located primarily within the centre of such a cluster, we can assume this was the initial configuration of the stars during cluster formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The analysis presented in this paper focuses in part on the spatial distribution of the MPs, which serves as a way to test the validity of the current processes theorised to describe cluster formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04166v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='GA] 10 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' One such process is the AGB scenario, first proposed by Cot- trell & Da Costa (1981), in which first generation (P1) AGB stars expel enriched material by stellar winds, which accumulates in the center of the cluster and mixes with primordial material to spark a second event of star formation - creating P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, for clusters in which the P2 population is equal to, or more massive than, the P1 population, the AGB scenario encounters a ‘mass bud- get’ problem since, assuming a standard stellar mass function, the enriched material created from P1 stars is not sufficient to create the P2 stars we observe in some clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Prantzos & Charbonnel 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Cabrera-Ziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' An implication of the AGB scenario is that an enriched star formation event occurring in the center of the cluster will lead to centrally concentrated P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Another formation process involves enrichment due to Super Massive Stars (SMS) (Denissenkov & Hartwick 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Gieles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2018) that form due to runaway collisions in the early stages of cluster formation and reside in the center of a cluster, providing a ‘conveyer belt’ of enriched material with different He fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This theory can overcome the mass budget problem as the continuous stellar collisions provide additional Hydrogen, which constantly rejuvinates the SMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this theory, P2 star formation occurs in the regions surrounding the SMSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Fast rotating massive stars were proposed by Decressin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2007a,b) to account for the observed chemical inhomogeneities, as massive stars create the required enriched material for additional star formation events, while the fast rotation brings the material to the surface of the star and ejects it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this scenario, secondary star formation events occur in the region surrounding the fast rotating massive stars after the enriched material is diluted by left over primordial gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Finally, massive interacting binaries have been suggested as a probable cause for the chemical enrichment found in MPs of GCs by de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2009) and Renzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Renzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2022) theorised that above a certain critical mass threshold, massive stars skip the supernova stage and instead implode into black holes, therefore ensuring the remaining stars in the cluster do not contain an abundance spread in Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As the centers of GCs are much denser than the outer regions, binary stars are expected to be destroyed or ejected at a higher rate in the center than they do in the outer regions due to increased collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lucatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2015) discovered a higher fraction of binaries within the P1 population, as opposed to the P2 population in 10 Galactic GCs, which seems to support theories that assume P2 stars are centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) studied the radial distribution of 20 Galactic GCs as a function of the age/relaxation time fraction (hereby referred to as ‘dynamical age’) using HST photometry and N-body model simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' They found that clusters with low dynamical ages preferentially contain centrally concentrated P2 populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' It is expected that clusters with lower dynamical ages have not undergone much dynamical mixing in their lifetime and are therefore still exhibiting properties close to their initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Clusters with higher dynamical ages were found to have spatially blended multiple stellar populations, in agreement with the idea that these clusters have undergone significant dynamical mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The results found by Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) provide observational evidence for formation theories in which enriched populations are formed within the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In their review, Bastian & Lardo (2018) concluded that GCs might not have homogeneous histories, suggesting instead that MPs can be formed through a variety of individual scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this case we could assume that the scenarios mentioned above are responsible for clusters with centrally concentrated P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, if a cluster contains centrally concentrated P1 stars, there are no current theories to explain this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this work, we study a diverse sample of 28 Galactic GCs in order to provide a comprehensive insight into the various pos- sibilities of cluster properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Large scale photometric analyses have been performed on Galactic GCs by Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), revealing intriguing scal- ing relations that may help us understand the origin of MPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' So far, combined space- and ground-based photometry for the purpose of obtaining a thorough spatial analysis of MPs and their charac- teristics exists only for a small number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We used both space-based and ground-based photometry to perform a homoge- neous analysis of the wide-field spatial extent of a large sample of GCs, using the well-established color combination CUBI and chro- mosome map methods in order to separate the MPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this paper we categorise MPs in space- and ground-based photometry separately, before combining the results to investigate correlations in terms of spatial distributions, enriched star fractions and global properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also compare our results with theoretical data and combine the Galactic GCs with Local Group GCs to further investigate trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2 OBSERVATIONAL DATA The photometric catalogues used in this work include the wide- field ground-based Johnson-Cousins UBVRI photometric data pro- vided by Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), along with the space-based HST UV Globular Cluster Survey data (‘HUGS’) (Piotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2018) with photometry obtained through UV/blue and WFC3/UVIS filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For this first project we will focus our anal- ysis of multiple stellar populations only on the RGB stars of these catalogues, combining both the HST and ground-based photometry in order to observe a wide-field view of each cluster, covering at least 84% of the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) photometric cat- alogue includes 48 GCs and the HUGS survey includes 57 GCs, but only 32 of these clusters overlap and exist in both catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Of these 32 clusters, we successfully classified distinct MPs in 28 of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We excluded clusters from our sample if they contained too few RGB stars after removing non-members and performing photometric cleaning, or if the classification of cluster stars into different sub-populations was inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The ground-based cat- alogues cover almost the full extent of each cluster, but cluster centers have much higher stellar densities than the outer regions, causing blending to affect the photometry of stars close to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This is where using HST photometry for the inner regions of clusters has an advantage, as crowding is less of an issue with space-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In this section we detail the steps taken to remove non- members, non-RGB stars and bad photometry from each photo- metric catalogue before separating the multiple stellar populations in Section 3 and characterising the cluster properties in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Both the ground-based and HST catalogues encountered issues with different types of incompletenesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In areas of the observed fields where either no stars were measured in a relevant filter, or the photometry was too poor to be usable, we could not reliably make assumptions about the properties of stars in that area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We calculated completeness fractions for the remaining stars so that we account for the stars that were missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We describe the spatial incompleteness in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1, the photometric incompleteness in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 and the surface density incompleteness in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Spatial completeness of the HST photometry for the globular cluster NGC 5024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Artificial test stars are shown in red (green), if our test indicated they fall outside (inside) the area covered by photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Stars shown in black are real stars located in regions that fall below 50% completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 Spatial Completeness Correction Our first step in processing both the ground-based and HST cata- logues was determining the spatial completeness fraction 𝑓𝑆 of each catalogue independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Using the original catalogues for both the ground-based and HST photometry, the spatial position of each star in right ascension and declination were calculated as an offset from the cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The data was not cleaned for stars without mea- sured photometry, defined as mag < 0 for HST and mag > 99 for the ground-based photometry, since these entries in the catalogues still indicated the presence of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We distributed a series of concentric rings spaced by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0′′ in distance around the cluster centers and distributed 360 artificial points evenly spaced by 1 degree along each ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For each of these artificial points, we determined the distance between the point and its nearest star, from the surrounding stars in our photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' A point was considered to be covered by the photometry if the minimum distance was less than a tolerance distance - usually close to 1 arcsec, but otherwise dependent on the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This method has the flexibility to be able to account for arbitrary field geometries, including large gaps within the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The spatial completeness 𝑓𝑆 of each annulus was set equal to the fraction of points that were covered by photometry in the field: 𝑓𝑆 = 𝑁in 𝑁total , where 𝑁in is the number of points inside the observed field and 𝑁total = 360, the total number of points for that annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We discarded photometry outside the radius in which the spatial completeness drops below 50%, shown as black points in Figure 1, using NGC 5024 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Surviving stars were assigned a spatial completeness fraction (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 ≤ 𝑓𝑆 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0), based on the completeness of the annulus they were located within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The HST and ground-based data was combined without allowing spatial gaps in the field by ensuring the ground-based data begins at the same radius at which the HST data ends for all clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 1000 500 0 500 RA offset [arcsec] 1000 500 0 500 1000 Dec offset [arcsec] 1000 500 0 500 RA offset [arcsec] 1000 500 0 500 1000 Dec offset [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 dEBV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 Interpolated dEBV Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Top panel: Differential reddening map of NGC 6121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bottom panel: Interpolation of the reddening map onto the ground-based photometry after spatial completeness correction in order to assign individual values of dEBV based on the nearest-neighbour in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 Differential Reddening Correction To compute differential reddening maps, we used a method simi- lar to other methods employed in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=', Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2012), which will be described in detail in a forthcoming publication (Pancino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We used the ground-based photom- etry by Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), selecting stars with photometric errors lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 mag in 𝐵𝑉𝐼, 𝜒 < 3, and |sharp| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We computed a fiducial line as the median ridge line of the main sequence of each cluster, down to about 2–4 magnitudes below the turnoff point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We selected stars not further than the 5 and 95% percentiles from the fiducial line in the three color planes𝑉,𝐵–𝑉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='𝑉,𝑉–𝐼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' and𝑉,𝐵–𝐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This allowed us to remove a large fraction of contaminating field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The color difference of each selected star from the reference line was computed in the three planes along the reddening line, assum- ing R𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 and using Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1978) to compute the reddening line direction in each plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We then rescaled these raw color differ- ences and combined them into one single estimate of ΔE(B–V) for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To disentangle photometric errors and other effects from the actual differential reddening signal, we smoothed these maps in right ascension and declination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' by replacing the ΔE(B–V) of each star with the median of its 𝑘 neighbors, with 𝑘 ranging from 50 to 300 (typically in the range 150-200) depending on the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=" MNRAS 000, 1–20 (2023) Covered by photometry Not covered by photometry Discarded stars'since 100 completeness<50% DEC offset [arcsec] 50 0 50 100 100 50 0 50 100 RA offset [arcsec]4 E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 8 10 12 14 16 18 20 22 24 I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 Sharp Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Sharp parameter cuts for the ground-based photometry of NGC 5024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Black points represent stars that survived the cut, red points were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The two red vertical lines represent rough limits in magnitudes to isolate the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' An ‘envelope’ function in red encloses stars with large enough photometric quality, as defined by Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 10 12 14 16 18 20 I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝜒 parameter cuts performed only on the ground-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Black points represent the stars of NGC 5024 that survived the sharp cuts of Figure 3, while red points were removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' All stars beneath the red line, defined by Equation 2 are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This also allowed us to compute an uncertainty for each differential reddening estimate as the median absolute deviation of the values for the 𝑘 neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To correct the ground-based photometry, the reddening map was interpolated for each star in both the HST and ground-based catalogues, as shown in the bottom panel of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We used the standard ratio of absolute to selective extinction of 𝑅𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1, with the exception of NGC 6121, for which the value of 𝑅𝑉 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 was used as suggested by Hendricks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Magnitude corrections for the ground-based 𝑈 and 𝐵 bands were applied using extinction ratios according to Cardelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1989), while the 𝑅 and 𝐼 bands were corrected according to Dean et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Similarly for the HST photometry, differential reddening was corrected for the 𝐹275𝑊, 𝐹336𝑊, 𝐹438𝑊 and 𝐹814𝑊 bands using extinction ratios from the SVO Filter Profile Service (Rodrigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Rodrigo & Solano 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 Photometric Quality Indicators We removed stars with less reliable photometry by using different quality indicators based on the available parameters provided by the HST and ground-based catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the ground-based pho- tometry, we implemented quality cuts based on magnitude errors and the 𝜒 and sharp parameters described in the work of Stetson & Harris (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the HST photometry we implemented cuts in sharp while also using the membership probability and quality-fit parameters (QFIT) for each star provided by Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the ground-based photometry, the U,B,V and I bands with as- sociated errors > 9 mag were cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the HST photometry, using the same constraints as Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), stars belonging to the cluster were selected using membership probability > 75% and 𝑄𝐹𝐼𝑇 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='9 in each of the 𝐹336𝑊, 𝐹438𝑊, 𝐹606𝑊 and 𝐹814𝑊 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For both photometry sets, cuts were made based on the sharp values following a method similar to Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2003), but re- placing the −1 ≥ sharp ≥ 1 criterion with an ‘envelope’ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We defined an exponential function above and below the bulk of the values to remove stars with sharp values too far from the mean: |sharp| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 + 𝑒𝑥𝑝 � 𝑚𝑎𝑔 − 22 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 � , (1) where 𝑚𝑎𝑔 = 𝐼 for the ground-based photometry and 𝑚𝑎𝑔 = 𝐹814𝑊 for HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Figure 3 shows the cut for ground-based photometry in which stars enclosed within the envelope are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the ground-based photometry we also used the 𝜒 parame- ter, which determines the observed vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' expected pixel-to-pixel scat- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' By adapting the method from Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2003), a function was applied to remove outliers: 𝜒 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 + 2 × 10(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2(𝐼−12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2) Stars which met the criterium are shown in black in Figure 4, while stars in red were rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 Proper Motion Cleaning The HST photometry includes a membership probability parameter (see Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2018) to help discard stars that do not belong to the cluster, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Determining the true mem- bers of a cluster for the ground-based photometry was done using proper motions of the stars after cross-matching with the Gaia DR3 catalogue (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2016, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This catalogue is comprehensive in scale, but has difficulties with incompleteness in the center of clusters and lower accuracy due to the high stellar crowding (Vasiliev & Baumgardt 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In order to enforce an equivalent MS turn-off limit between all catalogues, we first located the MS turn-off in the Gaia G band and applied a cut exactly at this magnitude to isolate the RGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This was a precaution against matching faint stars from one catalogue to bright stars in another catalogue (for stars in close proximity to each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We then cross-matched between the Gaia, HST and ground-based catalogues within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5′′ tolerance and determined the equivalent MS turn-off in the HST and ground-based catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We isolated the RGB stars in each catalogue using the equivalent MS turn-off limits found from this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Proper motion cleaning was only performed on the ground- based photometry outside the HST footprint due to the aforemen- tioned high stellar densities in the center of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The stars matched with the Gaia catalogue were then proper motion cleaned using a 𝜒2 test, defined in Equation 3, using both the right ascen- sion 𝜇𝛼∗ and declination 𝜇𝛿 proper motion components and cor- responding errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The cluster proper motion values (𝜇𝛼∗,𝑐𝑙𝑢𝑠𝑡𝑒𝑟 and 𝜇𝛿,𝑐𝑙𝑢𝑠𝑡𝑒𝑟) were taken from Vasiliev & Baumgardt (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We include a proper motion error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 mas/yr to account for both the MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 V - I 12 13 14 15 16 17 18 19 20 V Non-members Members 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 [mas/yr] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 [mas/yr] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Ground-based photometry for NGC 5024, demonstrating the effect of proper motion cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Upper panel: CMD of stars above the approx- imate MS turn-off, with accepted stars in black and rejected stars in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lower panel: The proper motion distributions of stars matched with Gaia EDR3, divided into members (black) and non-members (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' internal velocity dispersion of the cluster and any proper motion errors that may be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝜒2 = (𝜇𝛼∗,𝑐𝑙𝑢𝑠𝑡𝑒𝑟 − 𝜇𝛼∗)2 (𝜇𝛼∗,𝑒𝑟𝑟)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2[mas/yr]2 + (𝜇𝛿,𝑐𝑙𝑢𝑠𝑡𝑒𝑟 − 𝜇𝛿)2 (𝜇𝛿,𝑒𝑟𝑟)2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2[mas/yr]2 (3) The cut-off limit for the 𝜒2 value was slightly varied for each cluster, depending on the background stellar density and how clearly the cluster motion was distinguishable from the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In order to limit the effect of large errors allowing non-members to pass, we implemented an error tolerance relative to the proper motion of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The resulting cluster member stars are shown in black in both panels of Figure 5, while rejected stars are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The ground-based stars within 100′′ of the cluster center were added to the confirmed cluster member stars for the photometric cleaning in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We did this as these inner ground-based stars assisted with photometric cleaning and were removed anyway once the HST and ground-based photometry were combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 U - V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 B - I Fitted Data Clipped Data Polynomial Fit Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Polynomial fitting of RGB stars in colour-colour combinations 𝑈 − 𝑉 vs 𝐵 − 𝐼 for the ground-based photometry of NGC 5024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The line of best fit for the RGB stars is in green, cluster members are in black and non-members removed via the 𝑁 − 𝜎 clipping method are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Photometric Cleaning The purpose of the photometric cleaning process was to remove non-members and non-RGB stars so that the resulting distribution of RGB stars could be separated into multiple populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We identified the Horizontal Branch (HB) and AGB stars in CMDs created from both the HST and ground-based photometry, as well as red and blue outlier stars that stray too far from the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These stars were then manually removed from both sides of the RGB, allowing us to easily approximate and fit polynomials to the location of the RGB in the cluster CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We applied a polynomial fit to the RGB in colour-colour and colour-magnitude diagrams using the Astropy LinearLSQFitter (As- tropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2018), so that outliers could be removed using an 𝑁 − 𝜎 clipping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The colour-colour combination of 𝑈 − 𝑉 vs 𝐵 − 𝐼 ground-based bands shown in Figure 6 was used for the polynomial fit, where the median (𝑚fit) was required (as opposed to the mean) as outliers surrounding the RGB stars can heavily affect mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Depending on the contamination of non-members and AGB stars in each cluster, the number of standard deviations to be cut from the median was adjusted within the range 2 ≤ 𝑁 ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Highly contaminated clusters required a closer cut and therefore a smaller value of 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Non-members were identified according to (𝑈 −𝑉)obs − (𝑈 −𝑉)fit > 𝑚fit ± (𝑁𝜎(U−V)), meaning all stars with a colour difference greater than 𝑁 standard deviations from the median of the polynomial fit were clipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The process was iterated a maximum of three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also used this process for the HST photometry by using the closest equivalent colour-colour combination in the available HST bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We then applied the same 1D polynomial fitting and 𝑁 − 𝜎 clipping method to the following colour-index combinations in the ground-based photometry: (𝑉 − 𝐼), (𝐵− 𝐼) and (𝑈 − 𝐵), and the HST photometry: (𝐹606𝑊 − 𝐹814𝑊), (𝐹438𝑊 − 𝐹814𝑊), (𝐹336𝑊 − MNRAS 000, 1–20 (2023) 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 V - I 13 14 15 16 17 18 I Fitted Data Clipped Data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 B - I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 U - B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 CUBI Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝑁 − 𝜎 clipping through various colour-index combinations for the ground-based photometry of NGC 5024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Outliers are shown in red, while stars that closely fit the polynomial applied to each distribution are shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Right panel: The same method was used on the 𝐶UBI distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐹438𝑊) and (𝐹336𝑊 − 𝐹814𝑊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Finally, a special photometric index 𝐶UBI was used, which separates stars based on their chemical properties, namely N and He abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐶UBI was first introduced by Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) for ground-based photometry using Johnson filters with a focus on the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' It can also be adapted to the HST filters, as demonstrated by Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For each star in the ground-based photometry: 𝐶UBI = (𝑈 − 𝐵) − (𝐵 − 𝐼), while 𝐶UBI = (𝐹336𝑊 − 𝐹438𝑊) − (𝐹438𝑊 − 𝐹814𝑊) in the HST photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We applied the same 𝑁 − 𝜎 clipping method on the resulting 𝐶UBI distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The full sequence of polynomial fitting with 𝑁−𝜎 clipping is illustrated in Figure 7, where red outliers were removed for each colour-index combination before finally removing outliers from the 𝐶UBI distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 Photometric Completeness Correction While the spatial completeness analysis of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 compensates for cluster regions without observed stars caused by the limitations of the field, the photometric completeness compensates for a lack of stars due to poor or missing photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The aim is to assign a weighting to the surviving stars, such that they account for the frac- tion of stars that are lost during photometric cleaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We assumed that both the HST and ground-based catalogues were complete at the magnitudes of the RGB, as Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2008) derives the completeness for the HST data as 100% for stars brighter than the SBG for most clusters and Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) reports the ground- based data is complete across all radii for stars between 𝑉 = 19 and 𝑉 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To determine the photometric completeness factor (0 ≤ 𝑓𝑃 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0), we compared the number of RGB stars before and after the photometric cleaning processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We divided the orig- inal spatial distribution of RGB stars radially into annuli and the number of stars before (𝑁1) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' the number of stars after (𝑁2) deter- mined the photometric completeness factor for stars in each annulus: 𝑓𝑃 = 𝑁2/𝑁1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As we expect that the original HST and ground-based 0 200 400 600 800 1000 1200 1400 Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Completeness fraction fP fS fS × fP Completeness cut-off Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The individual spatial (blue) and photometric (green) complete- ness fractions, as well as the product of both completeness fractions ( 𝑓𝑆 𝑓𝑃 in black) as a function of radius for the ground-based stars in NGC 5024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The dotted red line indicates the cut-off at 15%, which is the minimum accepted completeness fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' catalogues contain the vast majority of stars, this completeness fac- tor accounts for the stars we remove in our cleaning, not stars missed by the catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The combined completeness fraction for each RGB star in both the HST and ground-based photometry was calculated as the product MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 7 100 101 102 103 Radius [arcsec] 12 11 10 9 8 7 Log (N/arcsec2) Trager (1995) Ground-based RGB HST RGB Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' A comparison of the number density profiles as a function of radius for NGC 5024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The surface density profile from Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1995) is shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The HST cleaned and weighted RGB stars (cyan) transition into the ground-based cleaned and weighted RGB stars (magenta) at approximately 100′′ and matches well against the Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' of the spatial and photometric completeness 𝑓𝑇 = 𝑓𝑆 𝑓𝑃, which can be seen as a function of radius in Figure 8 for only the ground-based photometry of NGC 5024 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The dense cluster center suffers a drop in completenesses due to the blending of stars in the ground-based catalogue, which were removed mainly through the quality cuts of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Additionally, the outer regions 𝑅 > 800′′ begin to drop in completenesses mainly due to the photometric cleaning of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We stopped at the radius at which the combined completeness fraction dropped below 𝑓𝑇 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 for the ground-based and HST photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7 Number Density Completeness In order to check the validity of our completeness corrections, we calculated the surface density based on completeness corrected stel- lar number counts and compared this against the surface brightness profiles of Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The number density profile of the cleaned RGB stars in our sample was weighted by the spatial and photometric completenesses 𝑓𝑇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' After correction for the combined completenesses, we applied the same shift factor to both the HST and ground-based data to convert between number density and sur- face density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1995) data was used as a reference profile and is shown in black in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We then compared the number density profiles of the HST (cyan) and ground-based data (magenta) to the reference profile for each cluster in order to con- firm the viability of the total incompleteness factors as a weighting to compensate for missing photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found a good match be- tween the HST and ground-based photometry and the Trager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (1995) profile for all 28 GCs in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 3 IDENTIFICATION OF MULTIPLE POPULATIONS We now move on to separate the multiple stellar populations using both the 𝐶UBI distribution method (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1) and the chromo- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 CUBI 14 15 16 17 18 F814W 4th percentile 96th percentile RGB 2 1 0 1 CUBI Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Left panel: The 𝐶UBI distribution of NGC 5024 stars in black using HST photometry, with the 4𝑡ℎ percentile ridgeline in blue and the 96𝑡ℎ percentile ridgeline in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Grey horizontal lines indicate the photo- metric error in the 𝐶UBI distribution at different magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Right panel: The resulting distribution Δ𝐶UBI of the same stars after normalisation as described by Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' some map method (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2), before finally analysing their radial distributions (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 Gaussian Mixture Models applied to 𝐶UBI Distributions The multiple stellar populations of each cluster were identified using the photometric index 𝐶UBI described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The general method for categorizing stars into multiple populations throughout this paper involved applying Gaussians to the Δ𝐶UBI distribution of stars, which is a normalised version of the 𝐶UBI distribution as shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To normalise the distribution, the 4𝑡ℎ and 96𝑡ℎ percentiles of the combined 𝐶UBI values for all stars were determined and fitted with a 1D polynomial, as per the method detailed in Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We used Equation 4 to calculate the normalised distribution Δ𝐶UBI from the distributions of 𝐶UBI in both the HST and ground-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Δ𝐶UBI = 𝐶UBI − 𝑋𝑏𝑙𝑢𝑒[𝐼] 𝑋𝑟𝑒𝑑 [𝐼] − 𝑋𝑏𝑙𝑢𝑒[𝐼] − 1 (4) The red (𝑋𝑟𝑒𝑑) and blue (𝑋𝑏𝑙𝑢𝑒) fiducial ridgelines in the left panel of Figure 10 were created at equally sized increments of 𝐹814𝑊 and 𝐼 magnitude bins for the HST and ground-based photometry, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' An example of the resulting Δ𝐶UBI distribution is shown in the right panel of Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We note that for all clusters in our sample, the photometric error in the 𝐶UBI distribution is much smaller than the colour spread in 𝐶UBI due to the presence of multiple stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Due to this, we are confident that the separation between multiple populations in the 𝐶UBI distribution is not influenced by photometric errors in the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' With this normalised distribution of stars, Gaussian Mixture MNRAS 000, 1–20 (2023) 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 CUBI 14 15 16 17 18 F814W P1 P2 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 2 4 6 8 10 Number of Components 550 575 600 625 650 675 700 725 Information Criterion AIC BIC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 F336W - F814W 14 15 16 17 18 F814W 75 50 25 0 25 50 75 RA [arcsec] 100 75 50 25 0 25 50 75 100 DEC [arcsec] Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Population separation of NGC 5024 using HST photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Top left: The best-fit GMM (solid black line) with the corresponding individual Gaussians (dashed), together with the Δ𝐶UBI distribution of stars separated into their respective P1 and P2 populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In grey we show stars with ambiguous classification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' membership probability to either population of 𝑝 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Top right: The AIC and BIC both show a minimum at 𝑛 = 2, indicating a clear identification of two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bottom left: The CMD of the two populations from the MS turn-off to the tip of the RGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bottom right: The spatial distribution of the two populations showing isotropic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Models (GMMs) from the scikit-learn package (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2011) were applied in order to find the most probable distribu- tion of the mutliple populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The method uses an expectation- maximization approach in order to determine the best mixture of one or more Gaussians to fit the Δ𝐶UBI distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Both the Akaike in- formation criterion (AIC) and Bayesian information criterion (BIC) were used to determine the most probable number of populations when provided with the Δ𝐶UBI distribution of a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The min- ima of both the AIC - which estimates the relative quality of the statistical models based on in-sample prediction error, and the BIC which selects the most probable model based on likelihood func- tions, indicated the most probable number of populations within a cluster from a range of 1 ≤ 𝑛 ≤ 10 different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For most clusters the AIC and BIC found 𝑛 = 2 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The top right panel of Figure 11 shows the range of possible components when applying GMMs to NGC 5024, with both AIC and BIC providing minima at 𝑛 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Clusters with minima at 𝑛 = 1 were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' From the most probable GMM samples, the final separation of the populations was created in terms of two or more Gaussians encompassing the full sample of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The top left panel of Figure 11 shows the combination of two Gaussians on the Δ𝐶UBI distribu- tion of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Each star was assigned to a population based on the probability that it belonged to a particular Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This member- ship probability was also used to divide the multiple populations for clusters with three populations, as discussed further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We required stars to have membership probability 𝑝 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 between the P1 and P2 populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This resulted in a small gap between each of the Gaussians, shown as gray points in Figure 11, ensuring that the stars belong to the population they were assigned to with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We experimented with this threshold using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 ≤ 𝑝 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 in increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 and found the overall results and conclusions of this work were not affected by the exact value of the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Similarly, we tested the effect of changing the limit of the primordial and enriched classifications for clusters with inter- esting radial distributions1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Briefly, we randomly sampled arbitrary limits in the Δ𝐶UBI colour (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' the point where the Gaussians over- lap) and classified stars left of the limit as primordial and stars to the right as enriched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The limit was drawn from a uniform distribution covering the inner 2𝜎 of the Δ𝐶UBI colour to avoid a cut too close to either colour end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We did this to prevent having almost all stars classified into one population with only a few left to be classified in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the purpose of these tests, we continued the remainder of the analysis with these arbitrary classifications in order to sta- tistically determine the significance of our resulting radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We sampled the arbitrary limits 200 times per cluster and each time we sampled anywhere from 90 to 100% of the stars on either side of 1 NGC 3201, NGC 6101 and NGC 7078 – see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 9 the limit to also observe the effect of randomly removing individual stars from each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 Chromosome Maps In addition to the 𝐶UBI colour distribution classification, for the HST photometry it is also possible to separate the populations us- ing chromosome maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Introduced by Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017), a chro- mosome map is a colour-colour plot that has been normalised in a way which allows efficient separation of sub-populations of dif- ferent abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' It uses the RGB width in a 𝐹275𝑊 − 𝐹814𝑊 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐹814𝑊 CMD, along with the RGB width of the pseudo-colour combination 𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐹814𝑊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Following the method in Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017), we defined a dividing line between populations in the Δ𝐹275𝑊 ,𝐹814𝑊 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Δ𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found that clusters such as NGC 2808 contained several distinct populations which can be split using a chromosome map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In these instances, the multiple populations tend to be easier to distinguish using a chromosome map, as they can become some- what blended together when using the Δ𝐶UBI distribution alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Therefore, by creating chromosome maps and then using the GMM method in two dimensions, as shown in Figure 12, we were able to directly compare the populations separated using a Δ𝐶UBI plot, against the populations separated by a chromosome map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The aim was to implement the same membership probability defined in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 of 𝑝 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 to cut out the ambiguous stars, shown in grey in Figure 12, before checking how the remaining stars were assigned to populations according to the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The HST photometry includes the UV filter F275W which has no ground-based equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We therefore relied on the 𝐶UBI distribution of the HST and ground-based photometry for a consis- tent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The HST F275W photometry was only used to confirm whether the𝐶UBI classification was consistent with the chromosome map method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To do this, the RGB stars of the 𝐶UBI distribution were separated into multiple populations with both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In Figure 13 we show the chromosome map of NGC 5024, where we colour code the stars classified as P1 and P2 with the Δ𝐶UBI distribution method in orange and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This figure shows that for the majority of the stars, the classification of different populations using Δ𝐶UBI was consistent with the classification based on the chromosome map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In all clusters, there was a small percentage of stars where the P1/P2 classification obtained using the chromosome map and Δ𝐶UBI disagree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We see that there are Δ𝐶UBI P1 stars in Figure 13 (blue) that inhabit the region in which the bulk of the P2 stars (orange) are located, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found the average fraction of stars that were classified differently by each method was ∼ 10% for the 28 GCs in our final sample, with a minimum of 4% and a maximum of 20% after the probability cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Clusters with high contamination percentages had heavily blended populations in the chromosome map, meaning the distribution of stars followed a more continuous distribution as opposed to distinct clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This caused difficulties in accurately determining the classification of popula- tions in one or both separation methods and therefore these clusters were excluded from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To further check the consistency of the population classification, we used overlapping stars that were covered by both (ground based and HST) photometric catalogues and had been independently classified into the different sub- populations using each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found consistent classifications of populations for stars common to both data sets, as demon- strated with large bold blue (P1) and orange (P2) points in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 F275W, F814W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 CF275W, F336W, F438W P1 P2 Probability cut 0 50 0 50 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Chromosome map using the HST photometry for NGC 5024, with Gaussian Mixture Models (GMMs) applied in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The lower left plot shows the chromosome map with populations P1 (blue) and P2 (orange) as defined by the two Gaussians in the top and right panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In grey are stars which lie in-between the two populations, with membership probabilites 𝑝 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 for either population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 F275W, F814W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 CF275W, F336W, F438W P2 P1 Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Chromosome map using the HST photometry for NGC 5024, as shown in Figure 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' however, we now use the Δ𝐶UBI separation to assign the stars into P1 (blue) and P2 (orange) populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Stars with membership probabilities 𝑝 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 are also removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' There is still a very good separation, as also shown in Figure 12, so we can see ‘contaminant’ stars by eye as blue points located in the orange clump and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bold circles indicate stars that overlap in both the HST and ground-based photometry, colour-coded to show the agreement between their independent classifications in each photometric catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' After ensuring consistent results between the different clas- sification methods/catalogues, we combined the HST and ground- based photometry by removing stars in the ground-based data which overlap with the HST field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' By doing this, we ensure the ground- based data begins at the same radius where the HST data ends, ensuring there are no gaps between the fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We then use this combined data set to study the behaviour of MPs across the full extent of these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 Radial Distributions of Different Populations A useful tool in understanding the behaviour of MPs as a function of radius is calculating the cumulative radial distribution of the stars in each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' If one population is more centrally concentrated within the cluster, we see a comparatively steeper slope in its cu- mulative radial distribution than we do for the other population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, if the populations are homogeneously mixed throughout the cluster, we see similar slopes for both distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The 𝐴+ parameter introduced by Alessandrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2016) is a way to quan- tify differing radial profiles, as it is an integration of the ‘area’ between the two distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The cumulative radial distributions in this work provide a spatially complete view of each cluster by combining the innermost region using HST photometry with the outer region using ground-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To calculate the cu- mulative radial distributions we used the method introduced and detailed by Alessandrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2016) and Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The 𝐴+ parameter considers the area between the cumulative ra- dial distributions of two populations, so for clusters exhibiting three distinct stellar populations such as NGC 1851, NGC 2808, NGC 6101 and NGC 7078, we combined the P2 and P3 stars into a single ‘enriched’ population, referred to as P2 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This clas- sification follows the logic of Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017), in which the primordial stars (P1) are identified as the group of stars aligning with Δ𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 = Δ𝐹275𝑊, 𝐹814𝑊 = 0 in a chro- mosome map, while P2 stars are any stellar populations located above the primordial stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We calculated a modified version of the 𝐴+ parameter using Equation 5 in order to characterize the weighted cumulative radial distributions of stars in each population using the total completeness fractions 𝑓𝑇 calculated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐴+(𝑅) = ∫ 𝑅 𝑅𝑚𝑖𝑛 �𝜙𝑃1(𝑅′) − 𝜙𝑃2(𝑅′)� 𝑑𝑅′ (5) Here, 𝜙 is the normalised, cumulative sum of the weights, 𝑤 = 1/ 𝑓𝑇 , of the stars in either the P1 or P2 population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our 𝐴+ parameter indi- cates whether a cluster has a P1 concentration in the center (𝐴+ > 0), a P2 concentration in the center (𝐴+ < 0), or a homogeneous mix of populations (𝐴+ ∼ 0) throughout the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The uncertainty in 𝐴+ was determined via bootstrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Briefly, the P1 and P2 stars of each cluster were sampled randomly for a total of 500 iterations using a sample size of 1000, with an 𝐴+ value calculated each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The final uncertainty for each cluster was calculated from the standard deviation of the 500 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Figure 14 shows the weighted and normalised cumulative ra- dial distributions of the two stellar populations found in NGC 5024 along the top panels, with the bottom panels showing the corre- sponding number ratio of enriched to total stars (P2/Ptotal) as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 5024 is an example of why the full ex- tent of the cluster should be analysed when considering the radial distributions of populations within a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The left panels show the behaviour of the cluster for only the HST field (1293 stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We already see by eye that both cumulative profiles are almost identical, which is also supported numerically by the parameter 𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The cumulative radial distribution of the HST photometry alone would suggest that the populations of this cluster are fully mixed and spatially indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, the middle panels show the result of this same analysis on the ground-based photometry (438 stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Here P2 is more centrally concentrat‹ed (𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='26), with the outer regions dominated by P1 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Finally, in the right panel, the full extent of the cluster is analysed by combining both the HST and ground-based stars, producing a value of 𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='11 and supporting the result that P2 is cen- trally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This information is lost when only observing the cluster center and using the resulting 𝐴+ parameter to describe the behaviour of the cluster as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' It is especially important to consider the outer regions of clusters, since dynamical mixing of the populations will affect the center of the cluster within shorter timescales than it does for the outer stars (Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To show the consistency of behaviour between the two photometric data sets, we plot the enriched star fraction P2/Ptotal as a function of radius in the lower panels of Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Here, the inner region also shows a mostly constant P2 concentration and the outer region shows a strong decline in P2 stars, supporting the result of the cu- mulative radial distributions while also showing agreement in the transition region between data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4 RESULTS For the 28 Galactic GCs in our sample we now investigate the trends associated with the 𝐴+ parameter and the enriched star fraction P2/Ptotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 we explore the global trends using the cumulative radial distributions, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 we explore the global trends using the enriched star fractions P2/Ptotal, and finally in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 we discuss individual notable clusters that have low dynamical ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Throughout this section we use cluster parameters provided by the Galactic Globular Cluster Database by Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), updated to the Gaia DR3 data as described by Vasiliev & Baumgardt (2021) and Baumgardt & Vasiliev (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We take the initial cluster mass and current cluster mass values, the former being calculated from the current cluster masses and cluster orbits using Equation 3 from Baumgardt & Makino (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The relaxation time (𝑇𝑅𝐻 ) of each cluster was also used, giving the time scale in which each cluster will become dynamically mixed, which was derived by Baumgardt & Hilker (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We define the dynamical age as the ratio of the age of a star cluster to its relaxation time and estimate the mass loss ratio (Mc/Mi) as the ratio of the current (Mc) and initial (Mi) mass of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also take the projected half-light radius (𝑅hlp), half-mass radius and orbital parameter values for each cluster from this database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The cluster ages are taken from the work of Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), while metallicity values are taken from Harris (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Previous work has found a clear correlation of the width of the RGB in clusters with MPs as a function of cluster metallic- ity [Fe/H], absolute visual magnitude 𝑀𝑉 and initial mass of the cluster (Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Since we have com- bined two independent photometric catalogues to get an extended spatial view, it was important that we replicated the well-established trends observed by others who used the same catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In par- ticular, we followed the method set out by Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) for the ground-based catalogue and determined the RGB widths (WRGB) in the same manner for the 28 Galactic GCs in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 2 4 6 8 10 Projected Radius [HLR] 200 400 600 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='26 0 2 4 6 8 10 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='11 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='60 200 400 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='35 0 200 400 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Cumulative radial distributions of different populations and enriched star fractions in NGC 5024 for the full radial range of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The scale shown for reference in the top panels is the distance from the centre of the cluster in units of projected half light radius [HLR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Upper left: The weighted, normalised, cumulative radial distribution of P1 (blue) and P2 (red) stars in the HST photometry within 𝑟 < 100′′ of the cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Upper middle: The ground-based photometry from 100′′ < 𝑟 < 740′′, analysed in the same way as the HST data in the upper left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Upper right: The 𝐴+ parameter for the combined data set, covering 0 < 𝑟 < 740′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The radius at which the HST photometry meets the ground-based photometry is shown by a black, dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We quote both 𝐴+ 4 for the calculated 𝐴+ value at a radial limit of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp and 𝐴+ total for the full radial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lower left: The fraction of P2 stars as a function of radius for the HST photometry (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Each bin has an equal number of stars, with the radial range of the bins illustrated at the bottom of the plot (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lower middle: The ground-based photometry analysed in the same way as the HST data in the lower left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lower right: The P2 fraction as a function of radius for the combined data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The total P2/Ptotal fractions are indicated in each panel for each corresponding radius range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found a strong correlation between WRGB and [Fe/H], with a Spearman correlation coefficient 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='693 and associated p- value = 4 × 10−5, as well as an anti-correlation between WRGB and 𝑀𝑉 , with 𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='331 and a p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For the HST data, we followed the method of Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017) and reproduced the correlation between WF275W,F814W and [Fe/H] for clusters with 𝑀𝑉 > −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3, providing a Spearman correlation coefficient of 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='704 and a p-value = 4 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also reproduced the trend between WF275W,F814W and 𝑀𝑉 , with 𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='104 and p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We conclude that our data exhibits the same well-established trends as previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 Global Trends using Cumulative Radial Distributions (𝐴+) We analysed large regions of the targets in our sample of 28 Galac- tic GCs and calculated the cumulative radial distribution parameters 𝐴+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We then identified clusters in which the 𝐴+ values indicated a high central concentration of either primordial or enriched stars at a significance larger than 3𝜎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These significantly segregated clus- ters will be discussed in detail in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The maximum radii for the outermost stars in the ground-based fields differed greatly for each cluster, so in order to make the results in different clusters comparable to each other, we analysed the spatial distribution of stars only out to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp in all clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We chose this limit as it was the minimum radius for our final sample of stars in NGC 6101, with most clusters extending beyond this radial limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The only clusters that did not reach this limit were NGC 3201, NGC 5053, NGC 6121 and NGC 6838 where the maximum radii for the ground-based photometry were in the range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5𝑅ℎ𝑙𝑝 (NGC 6838) < 𝑟𝑚𝑎𝑥 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0𝑅ℎ𝑙𝑝 (NGC 3201).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Limiting all clusters to this lower range would remove important information on the cluster proper- ties in the outermost regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Therefore, for these four clusters we assumed that the relative fraction of primordial and enriched stars is constant from the outermost radius covered by our photometry to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Since we extrapolate out to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp by sampling real stars in the outer radial bins, we do not expect that this will add significant uncertainty to the 𝐴+ parameters as we also propagate the uncertainties of these sampled stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Figure 15 shows the resulting 𝐴+ 4 parameters (calculated at a maximum radius of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp) as a function of dynamical age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found that dynamically old clusters all have 𝐴+ ∼ 0, in agreement with the idea that due to relaxation, populations become mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This also agrees with the findings of Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In dynamically young clusters, we found a larger range of 𝐴+ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Surprisingly, we not only found centrally concentrated MNRAS 000, 1–20 (2023) 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2 4 6 8 10 12 14 16 Age / Relaxation Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 A + 4 6101 5024 5272 3201 6809 5053 4590 288 5904 7089 2808 6205 7078 6752 6341 1261 7099 6254 5286 4833 6934 5986 1851 6981 6366 6218 6838 6121 Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The total cumulative radial distributions in terms of the 𝐴+ 4 parameter for the 28 Galactic GCs as a function of their dynamical age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' All clusters are limited to a radius equivalent to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp for direct comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Using this radius limit, clusters with an 𝐴+ value greater than 3-𝜎 significance from zero are displayed as labelled black points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' An 𝐴+ value close to zero indicates the MPs are spatially mixed throughout the analysed spatial extent of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Significantly positive 𝐴+ values indicate that the primordial (P1) population is more centrally concentrated, while negative values indicate the enriched (P2) population is more centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 for a special discussion on NGC 7078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' P2 populations (𝐴+ < 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 2808, NGC 5024, NGC 5272 and NGC 6809) consistent with the findings of Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), but also clusters with centrally concentrated P1 populations (𝐴+ > 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 3201 and NGC 6101), and clusters with full spatially mixed populations (𝐴+ ∼ 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 288, NGC 4590, NGC 5053, NGC 5904, NGC 70782 and NGC 7089) in the same small dynamical age range (age/relaxation time < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The central concentration of a primordial population seems to be in tension with the prediction of globular cluster formation models where P2 stars are preferentially concentrated towards the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also investigated the relationship between 𝐴+ 4 and the mass loss fraction (Mc/Mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Clusters that have lost >70% of their ini- tial masses due to dynamical evolution should be entirely mixed according to Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, given that their sim- ulations do not include the effects of stellar evolution, our present day masses cannot be directly compared with Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' To do this we need to take into account that star clusters lose ∼ 50% of their mass during a Hubble time due to stellar evolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' high mass stars dying first), so the Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) clusters that have lost >70% of their initial mass correspond to the clus- ters with Mc/Mi ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Therefore, clusters with Mc/Mi ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 are giving us a peek into the diversity of configu- rations the P1 and P2 populations of stars in globular clusters can display at the time of birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As expected, in Figure 16 we found that the clusters with significant central concentrations in either P1 or P2 have undergone the least amount of mass loss, with the exception of 2 See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 for a detailed discussion on NGC 7078 NGC 6809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Generally, as more mass is lost by a cluster, the initial concentrations of the multiple populations are also lost, as the stars become spatially mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We therefore concentrate our analysis on the clusters that should have retained the largest amount of their initial conditions in terms of dynamical age and mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' While 20 Galactic GCs were investigated by Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), our study overlaps with only 8 of these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We tested for consistency with their results by matching the constraints of their analysis and found all 8 overlapping clusters produce the same cumulative radial distributions as Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These constraints included limiting the HST field to 2 𝑅hlp in order to match the radial range covered by their analysis and only including the ground-based data for the analysis of NGC 288 within this same radial range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For our independent analysis, we included the ground-based photometry without restricting the radial range to 2 𝑅hlp and still found agreement with Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) for 7 out of the 8 overlapping clusters, since both the HST and ground- based photometry show 𝐴+ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The one cluster that did not agree with their results is NGC 6101, where we found P1 to be centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' When we considered only the HST photometry for NGC 6101, we found 𝐴+ ∼ 0 in agreement with Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), but with the inclusion of the ground-based photometry and therefore the outer region of the cluster, we found a P1 central concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This suggests that conclusions arrived at by studying only the inner regions of a cluster may be misleading, especially in dynamically young clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' A more extensive coverage of such clusters is required to obtain a full picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our results for the dynamically young clusters suggest that MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Mcurrent/Minitial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 A + 4 288 2808 3201 4590 5024 5272 5904 6101 6205 6809 7078 7089 5053 2 4 6 8 10 12 14 Age / Relaxation Time Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The total cumulative radial distributions in terms of the 𝐴+ 4 parameter (calculated at a maximum radius of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp) for the 28 Galactic GCs as a function of their mass loss ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Each cluster is also colour-coded by its dynamical age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Clusters categorised as ‘dynamically young’ (age/relaxation time < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5) are displayed as labelled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 for a special discussion on NGC 7078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' clusters are able to form with either enriched stars in the center, primordial stars in the center, or enriched and primordial stars dis- tributed in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This is an intriguing result, considering that the majority of globular cluster formation models will naturally produce clusters in which the P2 stars are centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our results therefore argue for the need of additional theories that can explain how clusters form with mixed stellar populations or centrally concentrated primordial stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 Global Trends using Enriched Star Fractions (P2 / Ptotal) For each of the 28 Galactic GCs in our sample we calculated the en- riched star fraction P2/Ptotal with associated standard errors, where enriched stars included both the P2 and P3 stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Un- like the cumulative radial distribution analysis, we did not imple- ment a radial limit of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp for each cluster, but instead calculated the P2/Ptotal fraction for the full possible extent of each cluster, tak- ing into account the total completeness fraction (see Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The top panel of Figure 17 shows the P2/Ptotal fraction as a function of the initial cluster mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We obtain a strong correlation between these two parameters with 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 and p-value = 1 × 10−9, similar to the correlation found by Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017) and Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2020) using the P1/Ptotal fraction against log(𝑀[𝑀⊙]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found no significant correlations for the global fraction P2/Ptotal as a func- tion of either metallicity or age (see Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' After removing the mass trend from our data, we similarly found that the residuals are uncorrelated with age or metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We neither found significant correlations with orbital parameters such as peri- and apogalactic distances and eccentricity, nor with the slope of the mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In order to test how young and low mass clusters fit into the global trends, we included an additional 7 Local Group clusters: NGC 121, NGC 336, NGC 416, NGC 1783, NGC 1978, Lindsay 1 and Fornax 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The Local Group clusters were separated into mul- tiple stellar populations using only HST photometry, but with the same method as outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' There is no need to combine HST and ground-based photometry for these clusters due to the fact that the half-light radius for each cluster is well within a single HST field, meaning the majority - if not all - stars are covered ny a single field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We only calculated the enriched star fractions P2/Ptotal for these additional clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In order to separate the populations, we used the narrow-band filter 𝐹343𝑁, which contains the NH ab- sorption line and can be used in the colour combination 𝐶UBUn = (𝑈 − 𝐵) − (𝐵 − 𝑈𝑛) = (𝐹336𝑊 − 𝐹438𝑊) − (𝐹438𝑊 − 𝐹343𝑁) as introduced by Niederhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In the same way we confirmed consistency between the Δ𝐶UBI distribution and chro- mosome maps, we also produced consistent results between the 𝐶UBUn and Δ𝐶UBI distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also included an additional 4 low-mass Milky Way clusters to our sample (Ruprecht 106, Palomar 12, Terzan 7 and E3), using previous work which performed spectroscopic analysis of stars and found no evidence of multiple populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' E3 and Ruprecht 106 do not contain enriched populations according to the analysis of MNRAS 000, 1–20 (2023) 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 log(Initial Mass) [M ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7 P2 / Ptotal rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 p-value: 1e-09 SMC LMC Fornax MW 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 [Fe/H] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7 P2 / Ptotal rs = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 p-value: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 Age [Gyr] 1e10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7 P2 / Ptotal rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='32 p-value: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The enriched stellar population fraction as a function of global parameters for Galactic GCs (black circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Added are SMC GCs (green triangles), LMC GCs (blue squares) and Fornax GCs (orange crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The large error bars plotted in grey for clusters with P2/Ptotal = 0 are due to the low number of stars with spectroscopic abundance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Top panel: The fraction of P2 stars as a function of the initial mass of each cluster shows a clear correlation between the two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Middle panel: There is no significant relationship between the enriched star fraction and metallicity of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bottom panel: There is no significant relationship between the enriched star fraction and the age of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Monaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Salinas & Strader (2015) and Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Frelijj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2021), respectively, and we therefore set them to P2/Ptotal = 0 with standard errors of 1/ √ 𝑁, where 𝑁 is the number of stars analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Similarly, the current consensus is that Terzan 7 and Palomar 12 do not contain multiple populations, based on the spectroscopic analysis of ≤ 5 RGB stars (Sbordone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Cohen 2004), and we therefore set P2/Ptotal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The standard errors for the enriched star fraction associated with Terzan 7 and Palomar 12 were comparatively much larger than for other clusters, in order to reflect the uncertainty of declaring a non-detection of MPs with a sample of only 5 RGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The age and metallicity of Lindsay 1 were taken from Glatt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2009), while those of E3 were taken from Forbes & Bridges (2010), and of Ruprecht 106 from Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), who averaged the values determined by Forbes & Bridges (2010) and Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2010, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' All other additional cluster ages and metallicities were taken from Usher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The addition of these 11 young and low-mass GCs to the sam- ple did not significantly influence the trends found for the P2/Ptotal fractions against global parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The initial mass correlation in Figure 17 is supported by the addition of these clusters, which continue the trend into the lower initial mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The relation- ship between P2/Ptotal and metallicity [Fe/H] previously showed a Spearman rank order coefficient of 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='11 and p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='58 for the original 28 Galactic GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' After the addition of the 11 young and low-mass GCs, this coefficient changed to 𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 with a p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01, showing a slight but ultimately inconclusive anti- correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For P2/Ptotal against age, the Spearman correlation only changes from 𝑟𝑠 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='21 with a p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='29 for the original 28 Galactic GCs, to 𝑟𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='32 with a p-value = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 for the full sample, again showing an inconclusive (weak) correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' There appears to be no significant trend between enriched star fractions and metallic- ity or age, but the addition of a larger sample of young and low-mass clusters may alter this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 Dynamically Young Clusters By ‘dynamically young’ we refer to the clusters in our sample with dynamical ages < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) found that dynamical age is a good indicator for the degree of dynamical mixing, with small dynamical ages corresponding to clusters which have retained the initial conditions of their formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Following this criterion, the clusters described in detail throughout this section are assumed to have preserved their initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We have divided this section into three parts, focusing on dynamically young clusters with: enriched (P2) populations concentrated in the center in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1, the primordial (P1) population in the center in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 and spatially mixed populations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The cumulative radial distribution plots for the covered extent of all dynamically young clusters can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 Clusters with centrally concentrated P2 stars In this section we discuss the individual results of the clus- ters NGC 2808, NGC 5024, NGC 5272 and NGC 6809, which contain a significant central concentration of the enriched (P2) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 2808 was separated into multiple stellar populations by Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2015) using a chromosome map with HST photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We find that the inner region covered by the HST field indicated that primordial and enriched stars are spatially mixed with 𝐴+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02, whereas Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) found 𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='029 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='001, in agreement with our results over the same approximate spatial range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2 𝑅hlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, the inclusion of stars in the ground-based photometry shows a significant P2 central concentration for the full range of the cluster, with 𝐴+ total = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Limiting the spatial range to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp resulted in a value of 𝐴+ 4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03, further strengthening the idea that omitting the outer stars from radially dependent analyses can hide the true properties of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 2808 contains the largest sample of stars from all 28 analysed MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 15 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Parameters for the 28 Galactic GCs studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The individual columns give the final number of stars in our sample after the analysis of Sections 2 and 3, split between the HST (NHST) and the ground-based (NGB) catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The total cumulative radial distribution parameter 𝐴+ 4 was calculated for all clusters at a maximum radius of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp in units of projected half-light radius, except for clusters specified in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also include the 𝐴+ total values calculated for the largest extent of each cluster covered by our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The enriched star fractions P2/Ptot,4 were also calculated for a radius of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp, and the full range (P2/Ptot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The next column gives the maximum analysed radius in each cluster in units of projected half-light radius (𝑟max [HLR]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The final columns give the dynamical ages (Age/Trh), mass loss fractions (Mc/Mi) and projected half-light radii (𝑅hlp) (see Section 4 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Cluster NHST NGB 𝐴+ 4 𝐴+ total P2/Ptot,4 P2/Ptot 𝑟max [HLR] Age / Trh Mc/Mi 𝑅hlp [pc] Fe/H NGC 288 190 356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='244 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='007 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='32 NGC 1261 903 153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='316 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='005 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27 NGC 1851 1241 256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='283 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='74 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='18 NGC 2808 3356 1401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='003 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='14 NGC 3201 187 363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='462 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='009 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='59 NGC 4590 217 144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='448 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='036 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='44 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='23 NGC 4833 535 336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='199 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='010 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='31 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='505 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='036 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 NGC 5053 0 181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='410 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='089 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27 NGC 5272 1259 619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='476 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='020 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 NGC 5286 1990 242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='56 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='283 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='010 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='69 NGC 5904 970 657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='59 ± 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='47 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='38 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='144 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='007 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='77 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='382 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='005 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='011 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='94 NGC 6838 135 213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='346 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='006 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='65 NGC 7099 295 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='63 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='231 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='010 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='54 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27 clusters, with 4757 stars in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' It presents a good opportunity for obtaining substantial amounts of individual spectra for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The final sample of the cluster contained 1323 P1 stars and 3433 P2 stars, which exacerbates the mass budget problem, especially considering that NGC 2808 has a young dy- namical age and should still retain Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='41 of its initial mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' A spectroscopic analysis of NGC 5024 was performed by Boberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2016) for 53 RGB stars within 500 arcseconds of the cluster center, discovering a centrally concentrated enriched population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This agrees with our cumulative radial distribution of 𝐴+ 4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05, which includes stars from the cluster center to 739 arcseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, for the two different methods used by Boberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2016), they find P2/Ptotal ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3, while our results for the total enriched fraction shows P2/Ptotal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Since Boberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2016) only used RGB stars with magnitudes 𝑉 < 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5, while our analysis includes the full RGB of stars with magnitudes 𝑉 < 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3, we argue that our enriched star fraction includes a larger and more complete sample and is therefore more indicative of the enriched star fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found NGC 5024 has the highest amount of remaining initial mass with Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='51, along with one of the lowest dynamical ages, meaning its initial conditions should not have changed significantly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' From Figure 14 we see that the photometry from the inner region alone provides a different picture than the combination of HST and ground-based photometry, supporting the idea that dynamical mixing of the populations affects the center of the cluster before the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Although the HST region contained 1293 stars and the ground-based photometry contained 438 stars, these outermost stars prove to be crucial in arriving at the full picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 5272 was previously analysed by Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), who used a combination of HST photometry and Ström- gren photometry from Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Additionally, Lardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2011) used SDSS photometry for RGB stars beyond 100 arcseconds from the cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Both discovered a centrally con- centrated enriched population, consistent with our cumulative radial distribution of 𝐴+ 4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 for stars within 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27 𝑅hlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' How- ever, we found that extending to the full possible extent of the cluster returned a value of 𝐴+ total = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10, showing a less signif- icant P2 central concentration overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found NGC 5272 has retained a high fraction of its initial mass, estimated to be close to Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='48, so we consider NGC 5272 to also largely preserve its initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our cumulative radial distribution for the HST photometry alone shows no dynamical mixing between the popula- tions with 𝐴+ ∼ 0, but the ground-based photometry indicates the outer regions are not yet mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Rain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) identified two populations in NGC 6809 based on 11 RGB stars using high resolution FLAMES/UVES spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Their spectroscopic identification of two populations is consis- tent with our photometric identification of two populations in both photometric data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found a centrally concentrated enriched population in both the HST and ground-based photometry, which indicates a lack of dynamical mixing within the center of the clus- ter when compared with NGC 5024 and NGC 5272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Interestingly, NGC 6809 is dynamically young but has lost a significant amount of its initial mass, with Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 6809 has the smallest galactocentric distance in our sample, with 𝑅𝐺𝐶 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 kpc MNRAS 000, 1–20 (2023) 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' and an escape velocity of 𝑣𝑒𝑠𝑐 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Tidal disruption affects clusters with smaller galactocentric distances more strongly (Baum- gardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2019) and the size of an accreted cluster in particular will respond to the tidal field of the MW upon accretion (Miholics et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As NGC 6809 is both suggested to be an accreted cluster (Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2019) and has a small galactocentric distance and relatively low escape velocity, we expect that although the cluster is dynamically young, tidal disruption after its accretion has affected its initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' It therefore becomes somewhat difficult to con- fidently conclude whether our discovery of a centrally concentrated P2 population is representative of its initial spatial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 Clusters with centrally concentrated P1 stars One of the most interesting results of this work is the centrally concentrated primordial populations found in NGC 3201 and NGC 6101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In order to test the validity of these findings, we present a more thorough analysis of the two clusters in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 3201 is considered dynamically young, but previous stud- ies of the cluster have produced complicated results that cause un- certainty around whether we can assume it maintains its initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 3201 is proposed to be an accreted cluster pre- viously belonging to Sequoia/Gaia-Enceladus (Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lucatello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2015) found that the P1 population in NGC 3201 hosts a higher fraction of binary stars than the P2 population, which they suggested to be due to the dense conditions of the central region that enhance the destruction and ejection of binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This result as- sumes that only P2 stars can be centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Kamann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2020) used HST photometry and MUSE spectroscopy and also found that NGC 3201 contains a higher binary fraction in the P1 population than it does for P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' They compare this result to sim- ulations suggesting P1 binaries are only overabundant outside the half-light radius (Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2015, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These simulations also assume a P2 central concentration, as they use this configuration for the initial conditions of their simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our discovery of a P1 concentration (𝐴+ 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12) therefore does not support the previous hypothesis proposed to describe the relative binary frac- tions between different sub-populations, but our result is not unique in that Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2022) also discovered a P1 central con- centration by combining HST photometry with photometry from the S-PLUS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bianchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) and Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2021) investigated the peculiar kinematics in the outskirts of NGC 3201, which contains tidal tails and exhibits flattened velocity dispersions in the outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' When analysing NGC 3201, we found that it suffered from sig- nificant differential reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, after correcting for its ef- fect (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2), the final spatial distribution of the populations showed no indication of problems due to differential reddening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In NGC 3201, we found that P1 stars had the highest concentration at intermediate radii around 150”, with P2 stars being dominant in the outer parts and also towards the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' A KS test showed that the central concentration of P2 was significant at a ∼ 2𝜎 level and significant at the 8𝜎 level towards the outer parts, leading to a U-shaped distribution in the relative fraction of P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In order to properly test the validity of the primordial central concentration discovery in the 𝐴+ parameter, we performed the probability cut and population limit tests outlined at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' By testing the effect of different limits in Δ𝐶UBI to separate the populations, we found 𝐴+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Similarly, by testing different probability thresholds for the membership of stars belonging to P1 and P2, we found 𝐴+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These tests confirm that the presence of a centrally concentrated primordial population is a consistent/robust result regardless of the method chosen to classify P1/P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our discovery of a centrally concentrated primordial population could indicate that the peculiar kinematics found by Bianchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) and Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2021) is driven by the enriched population of stars in the outskirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 3201 has intriguing characteristics and our discovery of a P1 central concentration further adds to these previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, it is difficult to describe the complexity of NGC 3201 using only the 𝐴+ parameter and future work would benefit from a parameter which incorporates both the radial spatial distributions between populations and the enriched star fraction for such clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) analysed NGC 6101 and found 𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='003 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='001, indicating the populations are homoge- neously mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In our analysis we found 𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 for 252 stars in the HST photometry, whereas 𝐴+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='19 was found using 229 stars in the ground-based photometry alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our combined cumulative radial distributions indicate a centrally concentrated primordial population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 6101 is the only case in our sample for which the HST and ground-based separations using the Δ𝐶UBI distributions returned a different number of populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The chromosome map returned two populations, as did the Δ𝐶UBI distribution for the HST photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, in the ground-based Δ𝐶UBI distribution, three populations were returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The blending of the populations was also somewhat present in the chromosome map, but two populations are nonetheless distinct enough for separation, as is also shown in Figure 7 of Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017) where the primordial population contains more stars than the enriched population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found NGC 6101 has retained almost half of its initial mass (Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='48) and has gone through the least amount of dynamical mixing of all 28 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' With a low metallicity of [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='98 dex (Harris 2010), the populations in a Δ𝐶UBI distribution are closer together than in more metal rich targets, since 𝐶UBI is most sensitive to molecular bands, which are weaker at low metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This leads to difficulties in separating the populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Due to that, we thoroughly tested how the separation of populations affected the final cumulative radial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The result of trying different probability thresholds for the memberships of stars belonging to P1 and P2 returned a value of 𝐴+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06, while the test of sampling arbitrary limits in the Δ𝐶UBI colour distributions returned 𝐴+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='19, showing a robust signal that P1 is concentrated in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Some simulations have studied the concept of an initially cen- trally concentrated population evolving over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For example, the simulations of Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) show that for a dynamically young cluster with an initial P2 central concentration, the P2 frac- tion as a function of radius will decrease significantly in the outer regions of the cluster, due to the slowing of two-body relaxation at larger distances from the cluster centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, we note that the same could be concluded if P1 were to have been formed more centrally concentrated, as there is no physical distinction between stars labeled P1 or P2 in these simulations, other than their ini- tial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Therefore, the behaviour we observe from the dynamically young clusters in our sample is indicative of the ini- tial conditions, where the P1 population was born initially more centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' When viewing only the inner HST region (𝑟 < 1𝑅hlp) of NGC 3201 (Figure A3), we found that the P2/Ptotal fraction decreases with increasing radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We performed a K-S test on the P1 and P2 distributions within this range to quantify this, based on the standard MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 17 two-sample test described in Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 of Monahan (2001), but modified to also include the weights (𝑤 = 1/ 𝑓𝑇 ) of each star, fol- lowing the method described in Equations 3-5 of Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The weighted K-S test showed the P1 and P2 distributions have a 2% probability of following the same distribution, meaning there is likely a P2 central concentration for the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' How- ever, if we consider stars beyond 1𝑅hlp the enriched star fraction increases for the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Figure 7 of Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) shows a simulated scenario in which the enriched star fraction as a function of radius could demonstrate similar U-shaped behaviour, however, it is not immediately clear that this represents the same phenomenon observed in NGC 3201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For example, the radius at which Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) expects this increase (𝑟 > 5𝑅hlp) is much larger than the radius at which we observe the increase (𝑟 ∼ 1𝑅hlp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Moreover, the dynamical ages (Age/Trh) at which the U-shaped behaviour occurs in the simulations is expected to be Age/Trh ≥ 5, whereas NGC 3201 has a dynamical age of Age/Trh = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Finally, Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2013) describes this increase as a "weak final rise" on the order of ∼ 10%, whereas in NGC 3201 we observe an ∼ 300% increase at an ∼ 8𝜎 significance between the minimum at ∼ 1𝑅hlp and the maximum at ∼ 4𝑅hlp of the enriched star fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Detailed simulations will be necessary to test how the initial conditions of NGC 3201 looked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 Spatially mixed populations We focus in this section on the dynamically young clusters that have retained most of their initial conditions but are nevertheless spatially mixed and do not contain one centrally concentrated population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These clusters include NGC 288, NGC 4590, NGC 5053, NGC 5904, NGC 6205, NGC 7078 and NGC 7089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 288 was analysed by Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) using HST photometry, in which they found that it contains spatially mixed populations with 𝐴+ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='045 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Similarly, we found two spatially mixed populations, with 𝐴+ 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 (P1 centrally concentrated only at < 3𝜎 level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Additionally, Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2022) used both, HST photometry and photometry from the S-PLUS survey, calculating cumulative radial distributions that show mixed populations in the central HST regions, but with a P2 central concentration in the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The discrepancies between our results in the outer regions - aside from the use of different photometric bands - appears to be due to differences in our analysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' More specifically, our sample of stars are corrected for photometric incompleteness, we exclude stars from our analysis in which the P1/P2 classifications are ambiguous (𝑝 > 80%), our limiting radius is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27 𝑅ℎ𝑙𝑝 compared to their 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 𝑅ℎ𝑙𝑝 and our sample includes an extra 116 stars in comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found that NGC 288 has retained only a fraction Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='24 of its initial mass, with an enriched fraction of P2/Ptotal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' At a glance, it seems plausible that mass loss is responsible for ejecting either primordial or enriched stars from the outer regions, resulting in spatially mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, it is also possible that NGC 288 formed with spatially mixed populations, as the initial configuration is difficult to determine due to the significant amount of mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found that NGC 4590 contains spatially mixed pop- ulations for both the HST and ground-based photometry, but based on a comparatively small sample size of 361 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Baum- gardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) found that NGC 4590 has large perigalactic (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 kpc) and apogalactic (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='42 kpc) distances, and Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) suggests one of the Helmi streams is the progenitor of this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found NGC 4590 retains approximately Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='45 of its initial mass and is one of the dynamically youngest clusters in our sample, but nonetheless contains fully spatially mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The large peri- and apogalactic distances suggest tidal stripping is unlikely to have removed a significant fraction of stars, but the accretion of NGC 4590 to the MW may have led to a stronger than predicted mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Previous work has found NGC 5053 to be dynamically complicated: it contains significant tidal tails (Jordi & Grebel 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lauchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2006) and a possible tidal bridge to NGC 5024 (Chun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Although NGC 5053 has one of the lowest dynamical ages and is predicted to retain a significant fraction of its initial mass with Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='41, we found its stellar populations are spatially mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 5053 was the only cluster for which we relied solely on the ground-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Due to the insufficient number of RGB stars in the HST photometry, the full extent of the ground-based photometry - including the cluster center - was used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The core of NGC 5053 has the lowest density of any cluster in our sample, and it has a large half-light radius, greatly reducing the blending effect in the cluster center that usually plagues ground-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As it is possible that NGC 5053 and NGC 5024 were accreted together within the same dwarf galaxy, we note that this event may have affected the mass loss of both clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The work of Lee (2019) using Strömgren photometry and the CUBI index found two populations in NGC 5904 with spatially mixed populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In a follow-up paper, Lee (2021) stated that this previously determined bimodal distribution could actually be separated further into three populations using Strömgren and Ca-CN-CH-NH photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' With this difference in classifications, their cumulative radial distributions changed from showing spatially mixed populations throughout the extent of the cluster - consistent with our results - to instead showing the most carbon-poor and nitrogen-rich population as centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Lardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2011) also separated NGC 5904 into two populations using SDSS photometry, which they refer to as UV-blue and UV-red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The resulting cumulative radial distributions from Lardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2011) show the UV-red stars are more centrally concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our final sample of NGC 5904 contains a large sample size of 1627 RGB stars and was consistent between the HST and ground-based photometry in identifying two stellar populations, exhibiting complete spatial mixing between populations and a consistent enriched fraction of P2/Ptotal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 throughout the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found that our results are consistent with only the initial findings of Lee (2019), as we did not find three populations within NGC 5904 using the combined HST and ground-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The introduction of spectroscopy to classify the populations based on chemical abundances such as carbon and nitrogen may help to check the validity of our photometrically separated populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 6205 was found to have a mass loss ratio close to Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 and is spatially mixed to its outermost regions at 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='19𝑅ℎ𝑙𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Similarly, we found NGC 7089 has a mass loss ratio of Mc/Mi ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='35 with spatially mixed populations extending out to 24𝑅ℎ𝑙𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Both clusters have large masses and are close to the upper limit of our definition of ‘dynamically young’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 6205 has previously been analysed by Savino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2018) using both HST and Strömgren photometry, in which they estimate MNRAS 000, 1–20 (2023) 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' an enriched fraction of approximately 80%, compared to our fraction of P2/Ptotal = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In terms of cumulative radial distributions, they also found no evidence for a centrally concentrated population in both the inner and outer regions of NGC 6205 (extending to approximately 700 arcseconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' NGC 7089 was analysed by Hartmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2022) using HST and S-PLUS survey photometry, discovering a P2 central concentration in both the HST field and outer region, which is at odds with our results of spatially mixed populations throughout the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We also find our results at odds with Lardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2011), who identified a centrally concentrated population using cumulative radial distributions from SDSS photometry for both NGC 6205 and NGC 7089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' If the dynamical age of NGC 6205 is long enough for dynamical mixing to occur throughout the entire cluster, we would expect this to occur for NGC 7078 and NGC 6809 as well, as per Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, we found clusters with similar dynamical ages have strongly varying spatial concentrations instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Previous photometric analysis of NGC 7078 has found contradictory results: Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2015) combined HST and SDSS photometry of RGB stars and discovered three stellar populations, which yielded a centrally concentrated primordial population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' however, Lardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2011) found only two popula- tions using SDSS photometry and consequentially discovered a centrally concentrated enriched population instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The 𝑚𝐹336𝑊 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐶𝐹275𝑊 ,𝐹336𝑊 ,𝐹438𝑊 plot of Piotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2015b) (Figure 22) shows at least two populations within NGC 7078 using HST photometry, while Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2017) distinctly separated the HST photometry into three populations using a chromosome map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found NGC 7078 contained one of the largest discrepancies for the P1 and P2 populations between the chromosome map and the Δ𝐶UBI distribution, with a contamination of approximately 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The low metallicity of NGC 7078 makes it difficult to separate the populations in the Δ𝐶UBI distribution, as the molecular bands responsible for the colour variations in Δ𝐶UBI become weaker, translating into smaller colour differences (see discussion in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Balbinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2022, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Additionally, this cluster suffers severely from differential reddening, which adds noise to the signal of the multiple populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Taking these caveats into account, we advise the reader to take the following results for NGC 7078 with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We checked other low metallicity clusters in our sample ([Fe/H] < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8) and found they did not suffer from this same confusion, and NGC 7078 is the only cluster in our sample affected by this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Because of the significant overlap between the Gaussians fitted by GMM to separate the populations in the Δ𝐶UBI distribution for NGC 7078, we took a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In the HST region, we rely on the chromosome map classification of populations, finding a resulting 𝐴+ value close to zero, which indicates the centre of the cluster is spatially mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Guided by the HST data, we establish colour cuts for the ground-based data which gave us relatively pure P1 and P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' More specifically, we selected only the extremes of the P1 (Δ𝐶UBI < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='7) and P2 (Δ𝐶UBI > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3) populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For reference we cross-matched our RGB stars with APOGEE DR17 (Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2022), where the stars in our final sample correspond to [Al/Fe] abundances of [Al/Fe]< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 for P1 stars and [Al/Fe]> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 for P2 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The final 𝐴+ 4 and 𝐴+ 𝑡𝑜𝑡𝑎𝑙 values quoted are therefore the combination of chromosome map classifications for the HST stars and our sample of extreme P1 and P2 stars selected as described above for the ground-based stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We find 𝐴+ 4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06, indicative of spatially mixed populations out to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27𝑅hlp, but with an overall signal indicating a P1 central concentration for the full extent (out to 15𝑅hlp) of the cluster (𝐴+ total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 Constraints on the loss of P1 stars Previous mass-loss scenarios involving internal enrichment aim to solve the mass budget problem by suggesting P1 stars are primarily located in the outskirts of GCs during formation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Krause et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' With P2 stars concentrated in the centre, mass-loss in the outskirts would then be responsible for the removal of P1 stars from the clusters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' D’Ercole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Bastian & Lardo (2015) explored this concept by analysing the correlations between enriched star fractions and cluster properties such as mass, metallicity and Galactocentric distance using literature data from 33 GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' For scenarios in which self-enrichment is responsible for the MP phenomenon, the enriched star fraction is expected to vary from the initial birth of the cluster to the present day, but was instead found to be constant throughout time, within errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' They concluded that the mass budget problem cannot be solved by assuming mass- loss in the outskirts of clusters, claiming that alternative theories are needed instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019) suggested a combination of polluting and diluting scenarios may explain the resulting chemical abundance spreads observed in GCs, with an emphasis that the inter- acting binaries theory (Vanbeveren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 2012) may be responsible for the ejection of stars in clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' According to their relative spa- tial distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 𝐴+), we have found varying behaviours for the initial spatial configurations of MPs in our sample of Galactic GCs, where dynamically young clusters in our sample show P1 centrally concentrated stars, as well as a homogeneous mix of populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, this by itself does not necessarily translate to the exacer- bation of the mass budget problem as one also needs to account for the relative number of P1 stars in the outskirts of the clusters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' where stars more likely to escape from the cluster reside).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our analysis has revealed that in the outer regions P1 stars do not constitute the majority of the stars, with the exception of NGC 5024 and NGC 6809 (see bottom right panels of Figure 14 and Figures in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This suggests that contrary to what is required by different models, during the dynamical evolution of these clusters P2 stars would be lost to the field population at a similar or higher rate than P1 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This would have important implications on the interpretations of the number of P2 stars found in the field, and their use to anchor the contribution of dissolved GCs to their host galaxy mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 5 CONCLUSIONS We have performed a spatially complete analysis of a large and diverse sample of 28 Galactic GCs, showing that GCs which still maintain their initial conditions can contain a central concentration of enriched or primordial stars, as well as a homogeneous mix of both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found centrally concentrated enriched populations in NGC 2808, NGC 5024, NGC 5272 and NGC 6809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' They can be explained with existing formation theories that involve internal polluters, such as SMS, FRMS or AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We can also rely on the notion of dynamical mixing to explain why GCs with large dynamical ages tend to have spatially homogeneous stellar populations over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' However, dynamically young GCs with a centrally concentrated pri- mordial population (NGC 3201 and NGC 6101) cannot be explained with current formation theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' These models cannot account ei- ther for dynamically young GCs that already contain fully spatially MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 19 mixed stellar populations such as NGC 288, NGC 4590, NGC 5053, NGC 5904, NGC 6205, NGC 7078 and NGC 7089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Furthermore, the existence of dynamically young clusters with fully mixed pop- ulations or a centrally concentrated P1, pose more challenges if P1 stars are required to be preferentially lost during the long term dynamical evolution of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Interpolations or simulations based off an incomplete view of clusters have previously been used to constrain the possible fractions of primordial or enriched stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' In our analysis, we used a spatially complete view of each cluster to calculate the enriched star fractions (P2 / Ptotal), which showed a clear correlation with the initial mass, but no clear correlations against other global parameters such as age and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Our sample of 28 Galactic GCs, 4 low-mass Galactic GCs and 7 Local Group GCs provided a range of 0 ≤ P2 / Ptotal < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 for the total enriched star fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We found that in some clusters, the enriched star fraction as a function of radius was constant across the extent of the cluster, while others exhibited either increasing or decreasing enriched star fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Current theories of GC formation and theoretical simulations have assumed the possibility of only a P2 central concentration, due in part to an analysis which limits itself to only the central regions of clusters and assumes conclusions on the properties of the full clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We argue the need for future theories and simulations to also consider alternative configurations of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The next stage of this research will explore the spectroscopic data available for our sample of 28 GCs in the same manner: combining data for the inner and outer regions of each cluster for a spatially complete view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We aim to check the validity of our photometric separations by spectroscopically separating the stellar populations based on chemical abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' We will also use our current classifications of populations to explore the kinematic differences, along with dif- ferences in chemical abundances and binary fractions in order to provide further observational information relating to the possible initial conditions and the final, dynamically mixed conditions of the clusters in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the referee for their insightful feedback on the manuscript, which improved the quality of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' The authors are very grateful to Florian Niederhofer for providing us an independent, HST based reddening map of NGC 7078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' This study was supported by the Klaus Tschira Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' DATA AVAILABILITY The Hubble Space Telescope UV Globular Cluster Survey (“HUGS”) photometric catalogue: https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' edu/prepds/hugs/ The wide-field, ground-based Johnson-Cousins UBVRI photomet- ric catalogue, courtesy of Stetson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' (2019), is available through the Canadian Astronomy Data center: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='cadc-ccda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' hia-iha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='nrc-cnrc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='ca/en/community/STETSON/ The Gaia Early Data Release 3 (EDR3) archive: https://gea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='int/archive/ The Galactic Globular Cluster Database Version 2: https: //people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='smp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} 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+page_content=', Langer N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=', Izzard R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=', 2009, A&A, 507, L1 APPENDIX A: CUMULATIVE RADIAL DISTRIBUTIONS OF DYNAMICALLY YOUNG CLUSTERS This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 5 10 15 20 Projected Radius [HLR] 200 400 600 800 1000 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='24 0 5 10 15 20 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='76 200 400 600 800 1000 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='66 0 200 400 600 800 1000 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='74 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 2808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 1 2 3 4 Projected Radius [HLR] 100 200 300 400 500 600 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0 1 2 3 4 Projected Radius [HLR] A + total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='09 A + 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 20 40 60 80 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='33 100 200 300 400 500 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='38 0 100 200 300 400 500 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 1 2 3 4 Projected Radius [HLR] 100 200 300 400 500 600 700 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='13 0 1 2 3 4 Projected Radius [HLR] A + total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 A + 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='12 0 20 40 60 80 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='53 100 200 300 400 500 600 700 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 0 200 400 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='51 Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 3201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 1 2 3 4 5 6 Projected Radius [HLR] 100 200 300 400 500 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='26 0 1 2 3 4 5 6 Projected Radius [HLR] A + total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 0 20 40 60 80 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='56 100 200 300 400 500 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='48 0 100 200 300 400 500 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 4590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 23 0 1 2 3 4 Projected Radius [HLR] 0 100 200 300 400 500 600 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 A + 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 0 1 2 3 4 Projected Radius [HLR] 0 100 200 300 400 500 600 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='45 Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 5053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' No HST photometry was used, so only the ground-based photometry was included in the final sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='50 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 2 4 6 8 10 12 14 Projected Radius [HLR] 200 400 600 800 1000 A + = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='23 0 2 4 6 8 10 12 14 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='70 200 400 600 800 1000 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='52 0 200 400 600 800 1000 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 Figure A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 5272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Projected Radius [HLR] 250 500 750 1000 1250 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='15 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='71 250 500 750 1000 1250 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='65 0 250 500 750 1000 1250 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='68 Figure A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 5904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 1 2 3 4 Projected Radius [HLR] 100 200 300 400 500 600 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='19 0 1 2 3 4 Projected Radius [HLR] A + total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='13 A + 4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='13 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='33 100 200 300 400 500 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='65 0 100 200 300 400 500 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='53 Figure A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 6101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) A Wide-Field View on Multiple Populations 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 2 4 6 8 10 Projected Radius [HLR] 200 400 600 800 1000 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='19 0 2 4 6 8 10 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='68 200 400 600 800 1000 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='66 0 200 400 600 800 1000 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='68 Figure A9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 6205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 1 2 3 4 Projected Radius [HLR] 200 400 600 800 A + = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='09 0 1 2 3 4 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='70 200 400 600 800 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='49 0 200 400 600 800 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='56 Figure A10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 6809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023) 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Leitinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 4 6 8 10 12 14 Projected Radius [HLR] 100 200 300 400 500 600 A + = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Projected Radius [HLR] A + total = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='10 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='63 100 200 300 400 500 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='67 0 100 200 300 400 500 600 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 Figure A11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 7078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' Refer to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='3 for a special discussion on NGC 7078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='5 Projected Radius [HLR] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 Cumulative Radial Distribution P1 P2 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='02 5 10 15 20 25 Projected Radius [HLR] 200 400 600 800 1000 1200 A + = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='44 0 5 10 15 20 25 Projected Radius [HLR] A + total = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='16 A + 4 = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='05 0 20 40 60 80 100 Projected Radius [arcsec] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='0 P2 / Ptotal P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='65 200 400 600 800 1000 1200 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='61 0 250 500 750 1000 1250 Projected Radius [arcsec] P2 / Ptotal: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content='64 Figure A12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' As in Figure 14, but for NGC 7089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} +page_content=' MNRAS 000, 1–20 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE2T4oBgHgl3EQf3AhK/content/2301.04166v1.pdf'} diff --git a/xNAyT4oBgHgl3EQfavfQ/vector_store/index.pkl b/xNAyT4oBgHgl3EQfavfQ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..26b15e5de68d1ae1413171a70df381eed2422217 --- /dev/null +++ b/xNAyT4oBgHgl3EQfavfQ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:993094723d176a4f659a3b460312cd971675031645afda8dabf47ca27c23bcac +size 121532 diff --git a/ydAzT4oBgHgl3EQfd_zp/content/tmp_files/2301.01431v1.pdf.txt b/ydAzT4oBgHgl3EQfd_zp/content/tmp_files/2301.01431v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..196b8ad7789e9aa7092acd1b0773c35c7915e5fc --- /dev/null +++ b/ydAzT4oBgHgl3EQfd_zp/content/tmp_files/2301.01431v1.pdf.txt @@ -0,0 +1,319 @@ +SEMI-MAE: MASKED AUTOENCODERS FOR SEMI-SUPERVISED +VISION TRANSFORMERS +Haojie Yu, Kang Zhao, Xiaoming Xu +Meituan Inc. +{yuhaojie02, zhaokang, xuxiaoming04}@meituan.com +ABSTRACT +Vision Transformer (ViT) suffers from data scarcity in semi-supervised learning (SSL). To alleviate +this issue, inspired by masked autoencoder (MAE), which is a data-efficient self-supervised learner, +we propose Semi-MAE, a pure ViT-based SSL framework consisting of a parallel MAE branch to +assist the visual representation learning and make the pseudo labels more accurate. The MAE branch +is designed as an asymmetric architecture consisting of a lightweight decoder and a shared-weights +encoder. We feed the weakly-augmented unlabeled data with a high masking ratio to the MAE branch +and reconstruct the missing pixels. Semi-MAE achieves 75.9% top-1 accuracy on ImageNet with +10% labels, surpassing prior state-of-the-art in semi-supervised image classification. In addition, +extensive experiments demonstrate that Semi-MAE can be readily used for other ViT models and +masked image modeling methods. +1 +Introduction +To date, Vision Transformers (ViT)[1] have achieved significant progress in supervised learning[1, 2, 3], self-supervised +learning[4, 5, 6, 7, 8], and various other computer vision tasks[9, 10, 11, 12]. ViTs have a weaker inductive bias than +CNNs[1], therefore a large amount of training data is often required to make ViTs generalize well. As a consequence, +the performance of ViTs is unsatisfactory in semi-supervised learning (SSL), where only a small number of labeled +data is provided and the rest are unlabeled. As discussed in [13], using FixMatch[14], one of the most popular SSL +approaches, to train a ViT presents an inferior performance than CNN architectures. To tackle this problem, [13] +proposes a joint semi-supervised training of CNN and ViT that outperforms CNN counterparts. Further, we continue to +explore pure ViTs in SSL, intending to obtain more accurate pseudo labels by improving the visual representation of +ViT itself. +In this work, we propose a simple yet efficient SSL paradigm of pure ViTs that surpasses previous CNN-based methods. +We consider the success of Transformers[15] in self-supervised learning[1], which is a promising solution to the data +scarcity issue by leveraging a large amount of unlabeled data. Specifically, we build on FixMatch[14] framework +except replace all CNNs with ViTs. Then for enhancing the learning of visual representations, we introduce a masked +autoencoder (MAE)[8] branch, which is parallel to the SSL framework and they share the same encoder. We mask +random patches from unlabeled data and design a lightweight decoder to reconstruct the input. Accordingly, the mean +squared error (MSE) between the reconstructed and original images contributes to the final loss. We call our method +Semi-MAE. Note that Semi-MAE is scalable to any other transformer-based model. +We perform extensive experiments to evaluate Semi-MAE. Notably, Semi-MAE with ViT-Small reaches 75.9% top-1 +accuracy on ImageNet with 10% labeled images. We demonstrate that pure ViTs can outperform CNN-based[14, +16, 17] and joint[13] SSL frameworks. Additionally, our MAE branch is a plug-and-play module that can improve +Semiformer[13] by 0.9%. We also show that other masked image modeling methods can further bring gains for +Semi-MAE, e.g., 76.0% top-1 accuracy with LoMaR[18]. +arXiv:2301.01431v1 [cs.CV] 4 Jan 2023 + +2 +Related Work +Vision Transformers +Transformers[15] have made substantial achievements in natural language processing +(NLP)[19], but in computer vision, convolutional neural networks (CNN) have dominated the past decade due to +their image-specific inductive bias. The appearance of [1], Vision Transformers (ViT) have finally addressed the archi- +tectural gap and have achieved success in image recognition[1, 20, 3], object detection[9, 10], segmentation[21, 22], +etc. However, ViTs have encountered obstacles when applied to semi-supervised learning (SSL), where the amount of +labeled data is insufficient for its training. In this work, we provide Semi-MAE to solve the aforementioned challenge. +Masked image modeling +Breakthroughs in masked language modeling (MLM)[19] in NLP have generated great +interest in the computer vision community, leading to the birth of masked image modeling (MIM) methods. [1] studied +the masked patch prediction objective and surprisingly found that self-training worked quite well on few-shot metrics. +iGPT[4] trained a sequence transformer to auto-regressively predict pixels on low-resolution ImageNet. BEiT[7] first +tokenized image patches into visual tokens via discrete VAE and then predicted randomly masked visual tokens by +the corrupted original image patches. Recently, masked autoencoder (MAE)[8] proposed an autoencoding approach, +whose objective was simply to reconstruct missing original patches in the pixel space given a partial observation. The +asymmetric design and high masking ratio yield a nontrivial task and help to learn well-generalized models while +leading to a significant reduction in computation. +Semi-supervised learning +Semi-supervised learning (SSL) has been shown to be a promising solution to exploit +unlabeled data. There are two classic strategies for SSL. One is pseudo labeling[23, 24] where model predictions are +converted to hard labels, the other is consistency regularization[25, 26] where models are trained to output consistent +results for different views of the input. FixMatch[14] integrated these two strategies: on unlabeled data, hard pseudo +labels are generated with weak augmentation as the target, and the model is fed a strongly-augmented version of the +same image. Motivated by ViTs’ success, [13] proposed a joint semi-supervised training of CNN and ViT. For the first +time, the application of ViTs in SSL achieves comparable performance against the CNN counterparts. On top of this, +we continue to explore the pure ViTs in SSL. +3 +Semi-MAE +Semi-MAE is a semi-supervised learning (SSL) framework of pure Vision Transformers[1] (ViTs) as illustrated in +Figure 1. Building on FixMatch[14], Semi-MAE introduces a masked autoencoder (MAE) branch to assist the encoder’s +visual representation learning. This branch parallels the original SSL framework and shares encoder weights. In this +section, we will first review FixMatch[14] algorithm and then elaborate proposed MAE branch. +Base architecture +FixMatch[14] is one of the most popular SSL frameworks in recent times. Its main contribution +lies in the combination of consistency regularization and pseudo-labeling. Specifically, the overall loss consists of two +cross-entropy losses: a supervised loss Ls and an unsupervised loss Lu. For a labeled sample {(xl +i, yl +i)}Nl +i=1, a weak +augmentation α(·) is applied to compute the supervised loss +Ls = 1 +Nl +Nl +� +i=1 +H(yi, f(α(xl +i))), +(1) +where H(·, ·) and f(·) are respectively the cross entropy loss function and the model forward function. As for an +unlabeled sample {xu +i }Nu +i=1, we first compute the output probabilities of its weakly-augmented version pi = f(α(xu +i )). +The pseudo label is produced by ˆyi = argmax(pi) with its confidence max(pi). The unsupervised loss is calculated +on the strong-augmented sample +Lu = +1 +Nu +Nu +� +i=1 +H(ˆyi, f(A(xu +i )))δ(max(pi) > τ), +(2) +where A(·) and δ(·) are respectively the strong augmentation and the indicator function. τ is the confidence threshold. +The overall loss is just L = Ls + λLu where λ denotes the relative weight of the unsupervised loss. In Semi-MAE, we +simply replace CNN models with ViTs. +MAE branch +As figured out in [13], the performance of ViT in SSL building on FixMatch[14] is inferior to CNN +counterparts. ViT generates inaccurate pseudo labels due to limited labeled data. Therefore we introduce a masked +autoencoder (MAE)[8] branch to enforce the visual representation learning and help ViT generate more accurate pseudo +2 + +Transformer +Unlabeled image +Share weights +Transformer +Transformer +Prediction +Prediction +Pseudo label +H(y, q) +Decoder +Weakly-augmented +Strongly-augmented +Masked input +Share weights +Reconstructed target +MAE branch +Figure 1: Overview of Semi-MAE. A weakly-augmented unlabeled sample is fed into the ViT, whose predictions with a +high confidence score can be converted to a one-hot pseudo label. The pseudo label is used for supervising the model’s +predictions of the strongly-augmented version of the same unlabeled sample. At the same time, the weakly-augmented +sample is divided into image patches. A random subset of patches is masked out and the rest are taken as the input of +the MAE branch. This branch shares the same ViT as the SSL framework. The latent representation and mask tokens +are processed by a small decoder to reconstruct the original image. +labels. Masked autoencoder learns visual representations from images with a high masking ratio and has demonstrated +its efficiency and effectiveness even only on ImageNet[27]. In particular, weakly-augmented unlabeled data also serve +as the original input of the MAE branch. The original input is first divided into patches, then we mask a high proportion +of patches and feed the rest to our encoder. This encoder is a ViT that shares the same weights as the encoder in the +SSL framework. Following the asymmetric design in [8], a small and independent decoder is used to reconstruct the +corrupted image from the latent representation and mask tokens. The reconstructed target is the pixel value for each +masked patch. The loss function is the mean square error (MSE) between the reconstructed and original images in the +pixel space. Eventually, our total loss function is +L = Ls + λLu + µLMAE +(3) +where µ is the trade-off for self-supervised loss weight. +4 +Experiments +4.1 +Experiment Settings +Datasets and evaluation metric +We conduct experiments on ImageNet[27], which contains ∼1.28M training and +50K validation images. Following [14], we sample 10% labeled images from the ImageNet training set and leave the +rest as unlabeled data. We select top-1 accuracy on the validation set as the evaluation metric. In addition, for a fair +comparison, we apply the same data augmentation as [13]. +Implementation details +We train the model from scratch. In detail, we first warm up the model for 100 epochs and +then train the model for 600 epochs with semi-supervision. We apply AdamW[28] as the optimizer with an initial +learning rate 10−3, which decays towards 10−5 using the cosine decay scheduler. The trade-offs λ and µ are respectively +3 + +10.0 and 5.0. In each batch, the ratio between labeled and unlabeled images is 1:7. We mainly use ViT-Small[1] as our +backbone. As for the MAE branch, we follow the default settings of [8]. +4.2 +Main Results +We compare Semi-MAE with the state-of-the-art semi-supervised methods. Results are presented in Table 1. Using +only ViT-Small, which has a smaller number of parameters than ResNet-50(22M v.s. 26M), Semi-MAE achieves 75.9% +top-1 accuracy that outperforms the prior state-of-the-art CNN-based methods. Semiformer[13] first introduces ViT to +SSL and achieves 75.5% top-1 accuracy with a joint framework. However, Semi-MAE with ViT-S alone can further +improve the performance over 0.4% than Semiformer[13]. These comparisons demonstrate that Semi-MAE achieves +state-of-the-art performance without additional data and more architectural improvement. +Table 1: The comparisons with state-of-the-art models. +Method +Architecture +Top-1 Acc(%) +UDA[29] +ResNet-50 +68.8% +FixMatch[14] +ResNet-50 +71.5% +S4L[16] +ResNet-50(4x) +73.2% +MPL[17] +ResNet-50 +73.9% +CowMix[30] +ResNet-50 +73.9% +Semiformer[13] +ViT-S+ResNet50 +75.5% +Semi-MAE (ours) +ViT-S +75.9% +4.3 +Ablation Studies +MAE branch +To prove the effectiveness and efficiency of our MAE branch, we further implement it into +Semiformer[13]. Results in Table 2 present that the MAE branch can bring marginal gains of 0.9% over the baseline. +Table 2: Results of Semiformer[13] with MAE branch. +Method +Architecture +MAE branch +Top-1 Acc(%) +Semiformer +ViT-S+ResNet50 +75.5% +✓ +76.4% +Other masked image modeling methods +Several masked image modeling methods[18, 7, 8] have demonstrated +their effectiveness to learn visual representations from images. Therefore, we investigate other MIM methods besides +MAE[8] and observe that LoMaR[18] can further boost the model performance by 0.1%, as shown in Table 3. +Table 3: Results with other masked image modeling methods. +MIM Method +Architecture +Top-1 Acc(%) +MAE +ViT-S +75.9% +LoMaR +ViT-S +76.0% +5 +Conclusion +We propose Semi-MAE, a pure Vision Transformer-based semi-supervised learning framework. By introducing a +masked autoencoder branch, Semi-MAE achieves substantial performance without extra training data. On ImageNet +with 10% labels, Semi-MAE can reach 75.9% top-1 accuracy, which surpasses the state-of-the-art CNN-based and joint +semi-supervised methods. This has proven that pure Vision Transformer is a promising solution for semi-supervised +learning. +4 + +References +[1] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, +Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: +Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020. +[2] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou. +Training data-efficient image transformers & distillation through attention. In International Conference on +Machine Learning, pages 10347–10357. PMLR, 2021. +[3] Ze Liu, Yutong Lin, Yue Cao, Han Hu, Yixuan Wei, Zheng Zhang, Stephen Lin, and Baining Guo. 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Milking cowmask for semi-supervised image classification. arXiv +preprint arXiv:2003.12022, 2020. +6 + diff --git a/ydAzT4oBgHgl3EQfd_zp/content/tmp_files/load_file.txt b/ydAzT4oBgHgl3EQfd_zp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ba420619b5bad0265f82e1b19bbebeba27e1992f --- /dev/null +++ b/ydAzT4oBgHgl3EQfd_zp/content/tmp_files/load_file.txt @@ -0,0 +1,243 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf,len=242 +page_content='SEMI-MAE: MASKED AUTOENCODERS FOR SEMI-SUPERVISED VISION TRANSFORMERS Haojie Yu, Kang Zhao, Xiaoming Xu Meituan Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' {yuhaojie02, zhaokang, xuxiaoming04}@meituan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='com ABSTRACT Vision Transformer (ViT) suffers from data scarcity in semi-supervised learning (SSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' To alleviate this issue, inspired by masked autoencoder (MAE), which is a data-efficient self-supervised learner, we propose Semi-MAE, a pure ViT-based SSL framework consisting of a parallel MAE branch to assist the visual representation learning and make the pseudo labels more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The MAE branch is designed as an asymmetric architecture consisting of a lightweight decoder and a shared-weights encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We feed the weakly-augmented unlabeled data with a high masking ratio to the MAE branch and reconstruct the missing pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Semi-MAE achieves 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% top-1 accuracy on ImageNet with 10% labels, surpassing prior state-of-the-art in semi-supervised image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In addition, extensive experiments demonstrate that Semi-MAE can be readily used for other ViT models and masked image modeling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' 1 Introduction To date, Vision Transformers (ViT)[1] have achieved significant progress in supervised learning[1, 2, 3], self-supervised learning[4, 5, 6, 7, 8], and various other computer vision tasks[9, 10, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' ViTs have a weaker inductive bias than CNNs[1], therefore a large amount of training data is often required to make ViTs generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' As a consequence, the performance of ViTs is unsatisfactory in semi-supervised learning (SSL), where only a small number of labeled data is provided and the rest are unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' As discussed in [13], using FixMatch[14], one of the most popular SSL approaches, to train a ViT presents an inferior performance than CNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' To tackle this problem, [13] proposes a joint semi-supervised training of CNN and ViT that outperforms CNN counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Further, we continue to explore pure ViTs in SSL, intending to obtain more accurate pseudo labels by improving the visual representation of ViT itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In this work, we propose a simple yet efficient SSL paradigm of pure ViTs that surpasses previous CNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We consider the success of Transformers[15] in self-supervised learning[1], which is a promising solution to the data scarcity issue by leveraging a large amount of unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Specifically, we build on FixMatch[14] framework except replace all CNNs with ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Then for enhancing the learning of visual representations, we introduce a masked autoencoder (MAE)[8] branch, which is parallel to the SSL framework and they share the same encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We mask random patches from unlabeled data and design a lightweight decoder to reconstruct the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Accordingly, the mean squared error (MSE) between the reconstructed and original images contributes to the final loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We call our method Semi-MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Note that Semi-MAE is scalable to any other transformer-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We perform extensive experiments to evaluate Semi-MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Notably, Semi-MAE with ViT-Small reaches 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% top-1 accuracy on ImageNet with 10% labeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We demonstrate that pure ViTs can outperform CNN-based[14, 16, 17] and joint[13] SSL frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Additionally, our MAE branch is a plug-and-play module that can improve Semiformer[13] by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We also show that other masked image modeling methods can further bring gains for Semi-MAE, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=', 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='0% top-1 accuracy with LoMaR[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='01431v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='CV] 4 Jan 2023 2 Related Work Vision Transformers Transformers[15] have made substantial achievements in natural language processing (NLP)[19], but in computer vision, convolutional neural networks (CNN) have dominated the past decade due to their image-specific inductive bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The appearance of [1], Vision Transformers (ViT) have finally addressed the archi- tectural gap and have achieved success in image recognition[1, 20, 3], object detection[9, 10], segmentation[21, 22], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' However, ViTs have encountered obstacles when applied to semi-supervised learning (SSL), where the amount of labeled data is insufficient for its training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In this work, we provide Semi-MAE to solve the aforementioned challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Masked image modeling Breakthroughs in masked language modeling (MLM)[19] in NLP have generated great interest in the computer vision community, leading to the birth of masked image modeling (MIM) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' [1] studied the masked patch prediction objective and surprisingly found that self-training worked quite well on few-shot metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' iGPT[4] trained a sequence transformer to auto-regressively predict pixels on low-resolution ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' BEiT[7] first tokenized image patches into visual tokens via discrete VAE and then predicted randomly masked visual tokens by the corrupted original image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Recently, masked autoencoder (MAE)[8] proposed an autoencoding approach, whose objective was simply to reconstruct missing original patches in the pixel space given a partial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The asymmetric design and high masking ratio yield a nontrivial task and help to learn well-generalized models while leading to a significant reduction in computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Semi-supervised learning Semi-supervised learning (SSL) has been shown to be a promising solution to exploit unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' There are two classic strategies for SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' One is pseudo labeling[23, 24] where model predictions are converted to hard labels, the other is consistency regularization[25, 26] where models are trained to output consistent results for different views of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' FixMatch[14] integrated these two strategies: on unlabeled data, hard pseudo labels are generated with weak augmentation as the target, and the model is fed a strongly-augmented version of the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Motivated by ViTs’ success, [13] proposed a joint semi-supervised training of CNN and ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' For the first time, the application of ViTs in SSL achieves comparable performance against the CNN counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' On top of this, we continue to explore the pure ViTs in SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' 3 Semi-MAE Semi-MAE is a semi-supervised learning (SSL) framework of pure Vision Transformers[1] (ViTs) as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Building on FixMatch[14], Semi-MAE introduces a masked autoencoder (MAE) branch to assist the encoder’s visual representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' This branch parallels the original SSL framework and shares encoder weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In this section, we will first review FixMatch[14] algorithm and then elaborate proposed MAE branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Base architecture FixMatch[14] is one of the most popular SSL frameworks in recent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Its main contribution lies in the combination of consistency regularization and pseudo-labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Specifically, the overall loss consists of two cross-entropy losses: a supervised loss Ls and an unsupervised loss Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' For a labeled sample {(xl i, yl i)}Nl i=1, a weak augmentation α(·) is applied to compute the supervised loss Ls = 1 Nl Nl � i=1 H(yi, f(α(xl i))), (1) where H(·, ·) and f(·) are respectively the cross entropy loss function and the model forward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' As for an unlabeled sample {xu i }Nu i=1, we first compute the output probabilities of its weakly-augmented version pi = f(α(xu i )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The pseudo label is produced by ˆyi = argmax(pi) with its confidence max(pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The unsupervised loss is calculated on the strong-augmented sample Lu = 1 Nu Nu � i=1 H(ˆyi, f(A(xu i )))δ(max(pi) > τ), (2) where A(·) and δ(·) are respectively the strong augmentation and the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' τ is the confidence threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The overall loss is just L = Ls + λLu where λ denotes the relative weight of the unsupervised loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In Semi-MAE, we simply replace CNN models with ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' MAE branch As figured out in [13], the performance of ViT in SSL building on FixMatch[14] is inferior to CNN counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' ViT generates inaccurate pseudo labels due to limited labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Therefore we introduce a masked autoencoder (MAE)[8] branch to enforce the visual representation learning and help ViT generate more accurate pseudo 2 Transformer Unlabeled image Share weights Transformer Transformer Prediction Prediction Pseudo label H(y, q) Decoder Weakly-augmented Strongly-augmented Masked input Share weights Reconstructed target MAE branch Figure 1: Overview of Semi-MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' A weakly-augmented unlabeled sample is fed into the ViT, whose predictions with a high confidence score can be converted to a one-hot pseudo label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The pseudo label is used for supervising the model’s predictions of the strongly-augmented version of the same unlabeled sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' At the same time, the weakly-augmented sample is divided into image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' A random subset of patches is masked out and the rest are taken as the input of the MAE branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' This branch shares the same ViT as the SSL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The latent representation and mask tokens are processed by a small decoder to reconstruct the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Masked autoencoder learns visual representations from images with a high masking ratio and has demonstrated its efficiency and effectiveness even only on ImageNet[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In particular, weakly-augmented unlabeled data also serve as the original input of the MAE branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The original input is first divided into patches, then we mask a high proportion of patches and feed the rest to our encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' This encoder is a ViT that shares the same weights as the encoder in the SSL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Following the asymmetric design in [8], a small and independent decoder is used to reconstruct the corrupted image from the latent representation and mask tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The reconstructed target is the pixel value for each masked patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The loss function is the mean square error (MSE) between the reconstructed and original images in the pixel space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Eventually, our total loss function is L = Ls + λLu + µLMAE (3) where µ is the trade-off for self-supervised loss weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='1 Experiment Settings Datasets and evaluation metric We conduct experiments on ImageNet[27], which contains ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='28M training and 50K validation images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Following [14], we sample 10% labeled images from the ImageNet training set and leave the rest as unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We select top-1 accuracy on the validation set as the evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In addition, for a fair comparison, we apply the same data augmentation as [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Implementation details We train the model from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In detail, we first warm up the model for 100 epochs and then train the model for 600 epochs with semi-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We apply AdamW[28] as the optimizer with an initial learning rate 10−3, which decays towards 10−5 using the cosine decay scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' The trade-offs λ and µ are respectively 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='0 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In each batch, the ratio between labeled and unlabeled images is 1:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' We mainly use ViT-Small[1] as our backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' As for the MAE branch, we follow the default settings of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='2 Main Results We compare Semi-MAE with the state-of-the-art semi-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Results are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Using only ViT-Small, which has a smaller number of parameters than ResNet-50(22M v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' 26M), Semi-MAE achieves 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% top-1 accuracy that outperforms the prior state-of-the-art CNN-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Semiformer[13] first introduces ViT to SSL and achieves 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='5% top-1 accuracy with a joint framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' However, Semi-MAE with ViT-S alone can further improve the performance over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='4% than Semiformer[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' These comparisons demonstrate that Semi-MAE achieves state-of-the-art performance without additional data and more architectural improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Table 1: The comparisons with state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Method Architecture Top-1 Acc(%) UDA[29] ResNet-50 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='8% FixMatch[14] ResNet-50 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='5% S4L[16] ResNet-50(4x) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='2% MPL[17] ResNet-50 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% CowMix[30] ResNet-50 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% Semiformer[13] ViT-S+ResNet50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='5% Semi-MAE (ours) ViT-S 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='3 Ablation Studies MAE branch To prove the effectiveness and efficiency of our MAE branch, we further implement it into Semiformer[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Results in Table 2 present that the MAE branch can bring marginal gains of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Table 2: Results of Semiformer[13] with MAE branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Method Architecture MAE branch Top-1 Acc(%) Semiformer ViT-S+ResNet50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='5% ✓ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='4% Other masked image modeling methods Several masked image modeling methods[18, 7, 8] have demonstrated their effectiveness to learn visual representations from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Therefore, we investigate other MIM methods besides MAE[8] and observe that LoMaR[18] can further boost the model performance by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='1%, as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Table 3: Results with other masked image modeling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' MIM Method Architecture Top-1 Acc(%) MAE ViT-S 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% LoMaR ViT-S 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='0% 5 Conclusion We propose Semi-MAE, a pure Vision Transformer-based semi-supervised learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' By introducing a masked autoencoder branch, Semi-MAE achieves substantial performance without extra training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' On ImageNet with 10% labels, Semi-MAE can reach 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='9% top-1 accuracy, which surpasses the state-of-the-art CNN-based and joint semi-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' This has proven that pure Vision Transformer is a promising solution for semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' 4 References [1] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' An image is worth 16x16 words: Transformers for image recognition at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content='11929, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' [2] Hugo Touvron, Matthieu Cord, Matthijs Douze, Francisco Massa, Alexandre Sablayrolles, and Hervé Jégou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' Training data-efficient image transformers & distillation through attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 10347–10357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydAzT4oBgHgl3EQfd_zp/content/2301.01431v1.pdf'} +page_content=' PMLR, 2021.' metadata={'source': 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161799 diff --git a/ztE1T4oBgHgl3EQfRQMb/content/tmp_files/2301.03049v1.pdf.txt b/ztE1T4oBgHgl3EQfRQMb/content/tmp_files/2301.03049v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..28671b9ad9ba9b7e95ea4ddc7e51963dbb8dd542 --- /dev/null +++ b/ztE1T4oBgHgl3EQfRQMb/content/tmp_files/2301.03049v1.pdf.txt @@ -0,0 +1,2524 @@ +AutoAC: Towards Automated Attribute Completion +for Heterogeneous Graph Neural Network +Guanghui Zhu, Zhennan Zhu, Wenjie Wang, Zhuoer Xu, Chunfeng Yuan, Yihua Huang +State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China +Department of Computer Science and Technology, Nanjing University, Nanjing, China +{zhuzhennan, wenjie.wang, zhuoer.xu}@smail.nju.edu.cn, {zgh, cfyuan, yhuang}@nju.edu.cn +Abstract—Many real-world data can be modeled as het- +erogeneous graphs that contain multiple types of nodes and +edges. Meanwhile, due to excellent performance, heterogeneous +graph neural networks (GNNs) have received more and more +attention. However, the existing work mainly focuses on the +design of novel GNN models, while ignoring another important +issue that also has a large impact on the model performance, +namely the missing attributes of some node types. The hand- +crafted attribute completion requires huge expert experience +and domain knowledge. Also, considering the differences in +semantic characteristics between nodes, the attribute completion +should be fine-grained, i.e., the attribute completion operation +should be node-specific. Moreover, to improve the performance +of the downstream graph learning task, attribute completion +and the training of the heterogeneous GNN should be jointly +optimized rather than viewed as two separate processes. To +address the above challenges, we propose a differentiable at- +tribute completion framework called AutoAC for automated +completion operation search in heterogeneous GNNs. We first +propose an expressive completion operation search space, includ- +ing topology-dependent and topology-independent completion +operations. Then, we propose a continuous relaxation schema +and further propose a differentiable completion algorithm where +the completion operation search is formulated as a bi-level +joint optimization problem. To improve the search efficiency, +we leverage two optimization techniques: discrete constraints +and auxiliary unsupervised graph node clustering. Extensive +experimental results on real-world datasets reveal that AutoAC +outperforms the SOTA handcrafted heterogeneous GNNs and the +existing attribute completion method. +Index Terms—heterogeneous graph, graph neural network, +attribute completion, differentiable search +I. INTRODUCTION +Graph-structured data are ubiquitous, such as social net- +works [1], scholar networks [2], biochemical networks [3], +and knowledge graphs [4]. Meanwhile, many real-world graph +data are heterogeneous [5]. Unlike the homogeneous graph +with only one node type and one edge type, the hetero- +geneous graph [6] consists of multiple types of nodes and +edges associated with attributes in different feature spaces. +For example, the IMDB dataset is a typical heterogeneous +graph, which contains three node types (movie, actor, director) +and two edge types (movie-actor, movie-director), as shown in +Figure 1(a). Due to containing rich information and semantics, +heterogeneous graphs have drawn more and more attention. +Recently, graph neural networks (GNNs) [7], [8] have +demonstrated powerful representation learning ability on +graph-structured data [9]. Meanwhile, many heterogeneous +GNNs (HGNNs) have been proposed for heterogeneous +graphs [10] [11] [12] [13] [14] [15] [16] [17]. However, the +existing work on heterogeneous graphs mainly focuses on the +construction of novel GNN models, while ignoring another +important issue that also has a large impact on the model +performance, namely the attributes of some types of nodes are +missing [18]. Missing node attributes is a common problem +because collecting the attributes of all nodes is prohibitively +expensive or even impossible due to privacy concerns. Since +the attributes of all nodes are required in the GNN-based +heterogeneous models, some handcrafted ways are employed +to deal with the problem of missing attributes. For example, +the missing attribute vector can be the sum or the mean of +directly connected nodes’ attribute vectors. Besides, the one- +hot representations of a certain node type can also be used +to replace the missing attributes. However, the handcrafted +ways require huge expert experience and domain knowledge. +Also, the topological relationships in the graph are not taken +into account. Recently, an attention-based method [18] was +proposed to complete each no-attribute node by weighted +aggregation of the attributes from the directly neighboring +attributed nodes. Such an attribute completion method only +considers the attributes of 1-hop neighbors without exploiting +the attributes of higher-order neighbors. +Moreover, existing attribute completion methods are all +coarse-grained. That is, for a specific node type without +attributes, they adopt the same attribute completion operation +for all nodes without considering the differences in semantic +characteristics between nodes. In practice, fine-grained at- +tribute completion is more reasonable. The attribute comple- +tion operations for the nodes with different semantics should +be different. Take the IMDB dataset as an example. The target +type of nodes (i.e., movie nodes) has attributes, and the other +types of nodes (i.e., actor nodes and director nodes) have no +attributes. As shown in Figure 1(b), there exist three attribute +completion operations, including 1) For actors (e.g. Jackie +Chan) who are involved in movies that mostly belong to the +same genre (Kung Fu movies), average attribute aggregation +of local (i.e., 1-hop) neighboring nodes should be used. 2) +For actors who have strong collaborative relationships with +other actors and directors, the message-passing based multi- +hop attribute aggregation is more suitable. 3) For guest actors +without representative movies, we can directly use the simple +one-hot encoding to complete attributes. +arXiv:2301.03049v1 [cs.LG] 8 Jan 2023 + +Movie +(with attribute) +director +(attribute missing) +actor +(attribute missing) +Romance? +Thriller? +Action? +…? +classification +node +(a) +Jackie Chan +Ruch Hour +Police Story +CZ12 +The Foreigner +Project A +actor +Ng Man-Tat +actor +Stephen +Chow +actor +A Chinese Odyssey +Li Gong +actor +Red Sorghum +director +Yi-Mou Zhang +2-hop +2-hop +Faye Wong +actor +Chungking Express +Alan Tam +actor +We Are Family +singer (guest actor) +(b) +Fig. 1. (a) Example of heterogeneous graphs with incomplete attributes, i.e., the IMDB dataset. (b) Different attribute completion operations for the actor +node, i.e., local attribute aggregation, message-passing based multi-hop attribute aggregation, and one-hot representation. +For the IMDB dataset, the number of actor nodes that have +no attributes is 6124. Manually differentiating the semantic +characteristics of all no-attribute nodes and then selecting +the most suitable completion operations according to seman- +tic characteristics is infeasible. Thus, an automated attribute +completion method that can search the optimal completion +operations efficiently is required. Moreover, to improve the +performance of the downstream graph learning task, the au- +tomated attribute completion and the training of the heteroge- +neous GNN should be jointly optimized rather than viewed as +two separate processes. +To address the above challenges, we propose a differentiable +attribute completion framework called AutoAC1 for automated +completion operation search in heterogeneous GNNs. AutoAC +is a generic framework since it can integrate different hetero- +geneous GNNs flexibly. By revisiting the existing attribute +completion methods, we first propose an expressive com- +pletion operation search space, including topology-dependent +and topology-independent completion operations. Instead of +searching over the discrete space (i.e., candidate completion +operations for each no-attribute node), we propose a con- +tinuous relaxation scheme by placing a weighted mixture of +candidate completion choices, which turns the search task into +an optimization problem regarding the weights of choices (i.e., +completion parameters). Thus, due to the continuous search +space, the search process becomes differentiable and we can +perform completion operation searching via gradient descent. +To further improve the search efficiency, we formulate the +search of attribute completion operations and the training of +GNN as a constrained bi-level joint optimization problem. +Specifically, we keep the search space continuous in the +optimization process of completion parameters (i.e., upper- +level optimization) but enforce attribute completion choices +being discrete in the optimization process of weights in the +heterogeneous GNN (i.e., lower-level optimization). In this +way, there is only one activated completion operation for each +no-attribute node during the training of GNN, removing the +need to perform all candidate completion operations. Inspired +by NASP [19], we employ proximal iteration to solve the +constrained optimization problem efficiently. +Finally, to reduce the dimension of the attribute completion +parameters, we further leverage an auxiliary unsupervised +1AutoAC is available at https://github.com/PasaLab/AutoAC +graph node clustering task with the spectral modularity func- +tion during the process of GNN training. +To summarize, the main contributions of this paper can be +highlighted as follows: +• We are the first, to the best of our knowledge, to model +the attribute completion problem as an automated search +problem for the optimal completion operation of each no- +attribute node. +• We propose an expressive completion operation search +space and further propose a differentiable attribute com- +pletion framework where the completion operation search +is formulated as a bi-level joint optimization problem. +• To improve search efficiency, we enforce discrete con- +straints on completion parameters in the training of +heterogeneous GNN. Moreover, we leverage an auxiliary +unsupervised graph node clustering task to reduce the +dimension of the attribute completion parameters. +• Extensive experimental results on real-world datasets +reveal that AutoAC is effective to boost the performance +of heterogeneous GNNs and outperforms the SOTA at- +tribute completion method in terms of performance and +efficiency. +II. RELATED WORK +A. Heterogeneous Graph Neural Network +Graph neural network [8] [20] [1] [21] [22] [9] aims +to extend neural networks to graphs. Since heterogeneous +graphs are more common in the real world [5], heterogeneous +GNNs have been proposed recently. Part of the work is based +on meta-paths. HAN [10] leverages the semantics of meta- +paths and uses hierarchical attention to aggregate neighbors. +MAGNN [14] utilizes RotatE [23] to encode intermediate +nodes along each meta-path and mix multiple meta-paths using +hierarchical attention. Another part of the work chooses to +extract rich semantic information in heterogeneous graphs. +GTN [11] learns a soft selection of edge types and com- +posite relations for generating useful multi-hop connections. +HetGNN [13] uses Bi-LSTM to aggregate node features for +each type and among types. As the state-of-the-art model, +SimpleHGN [17] revisits existing methods and proposes a +simple framework using learnable edge-type embedding and +residual connections for both nodes and edges. Recently, + +AS-GCN [24] employs the heterogeneous GNN to mine the +semantics for text-rich networks. +Different from the above methods, HGNN-AC [18] notices +that most of the nodes in the real heterogeneous graph have +missing attributes, which could cause great harm to the per- +formance of heterogeneous models, and proposes an attention- +based attribute completion method. However, HGNN-AC +needs to get node embeddings based on network topology +using metapath2vec [25], which is a time-consuming process. +Moreover, the attribute completion in HGNN-AC is coarse- +grained and supports only one completion operation for all +no-attribute nodes. HGCA [26] unifies attribute completion +and representation learning in an unsupervised heterogeneous +network. MRAP [27] performs node attribute competition in +knowledge graphs with multi-relational propagation. +B. Neural Architecture Search (NAS) +NAS [28] that designs effective neural architectures auto- +matically has received more attention. The core components +of NAS contain search space, search algorithm, and perfor- +mance estimation strategy. Recently, many works use NAS +to design GNN models due to the complexity of GNN [29]. +PolicyGNN [30] uses reinforcement learning to train meta- +strategies and then adaptively determines the choice of ag- +gregation layers for each node. SANE [31] and SNAG [31] +search for aggregation functions using microscope-based +and reinforcement learning-based strategies, respectively. The +architecture-level approaches such as GraphNAS [32], Au- +toGNN [33], and PSP [34] aim to search for architectural +representations of each layer, including sampling functions, at- +tention computation functions, aggregation functions, and acti- +vation functions. The above works are based on homogeneous +graphs. Due to the rich semantic and structural information in +heterogeneous graphs, applying NAS to heterogeneous graphs +is more challenging. Recently, there exist some excellent +attempts. GEMS [35] uses the evolutionary algorithm to search +for meta-graphs between source and target nodes. DiffMG [36] +uses differentiable methods to find the best meta-structures in +heterogeneous graphs. However, the above works only focus +on the GNN model and ignore the heterogeneous graph data +itself, which is even more important in practice. +C. Proximal Iteration +Proximal iteration [37] is used to handle the optimization +problem with a constraint C, i.e., minx f(x), s.t. x ∈ C, where +f is a differentiable objective function. The proximal step is: +x(k+1) = proxC +� +x(k) − ϵ∇f +� +x(k)�� +proxC(x) = arg min +z +1 +2(∥z − x∥)2, s.t. z ∈ C +(1) +where ϵ is the learning rate. Due to the excellent theoretical +guarantee and good empirical performance, proximal iteration +has been applied to many deep learning problems (e.g., archi- +tecture search [19]). +III. PRELIMINARIES +Heterogeneous Graph. Given a graph G = ⟨V, E⟩ where V +and E denote the node set and the edge set respectively, G +is heterogeneous when the number of node and edge types +exceeds 2. Each node v ∈ V and each edge e ∈ E are +associated with a node type and an edge type respectively. +Attribute Missing in Heterogeneous Graph. Let xv ∈ Rd +denote the original d-dimensional attribute vector in the node +v. In practice, the attributes of some types of nodes are not +available. Thus, the node set V in G can be divided into two +subsets, i.e., V + and V −, which denote the attributed node-set +and no-attribute node-set. +Attribute Completion. Let X = {xv | v ∈ V +} denote the +input attribute set. Attribute completion aims to complete the +attribute for each no-attribute node v ∈ V − by leveraging the +available attribute information X and the topological structure +of G. Let xC +v denote the completed attribute. Thus, after com- +pletion, the node attributes for the training of heterogeneous +GNN is Xnew = X ∪XC = {xv | v ∈ V +}∪{xc +v | v ∈ V −}. +In this paper, we aim to search for the optimal completion +operation for each no-attribute node to improve the prediction +performance of GNN models. +IV. THE PROPOSED METHODOLOGY +In this section, We first present the proposed completion +operation search space and then introduce the differentiable +search strategy. Moreover, we introduce the optimization +techniques including discrete constraints and the auxiliary +unsupervised graph node clustering task for further improving +the search efficiency. +A. Search Space of Attribute Completion Operation +Due to the semantic differences between nodes, using a +single attribute completion operation for all no-attribute nodes +belonging to the same node type is not reasonable. The avail- +able completion operations should be diverse and we can select +the most suitable completion operation for each node with +missing attributes. Thus, to capture both the node semantics +and the topological structure information during the attribute +completion process, we first propose an expressive completion +operation search space, which consists of topology-dependent +and topology-independent operations. +Specifically, the topology-dependent operations employ the +topology information of the graph to guide the attribute +completion. Inspired by the node aggregation operations in +typical GNNs (e.g., GraphSage [1], GCN [8], APPNP [38]), +we design three topology-dependent attribute completion op- +erations, i.e., mean, GCN-based, PPNP-based operations. In +contrast, the topology-independent operation directly uses one- +hot encoding to replace the missing attribute. AutoAC aims to +search the optimal operation for each no-attribute node from +the general and scalable search space where we can draw +on more node aggregation operations in GNNs as attribute +completion operations. + +1) Topology-Dependent Completion Operation: Such type +of completion operations can be further divided into two +categories: local attribute aggregation and global (i.e., multi- +hop) attribute aggregation. +Local Attribute Aggregation. Similar to the node aggregation +in GraphSage [1], we first propose mean attribute aggregation. +Mean Attribute Aggregation. For the node v ∈ V −, we +calculate the mean of neighbors’ attributes to complete the +missing attribute. The completed attribute xC +v is as follows: +xC +v = W · mean +� +xu, ∀u ∈ N + +v +� +(2) +where N + +v denotes the local (i.e, 1-hop) neighbors of node v +in set V +. W is the trainable transformation matrix. +GCN-based Attribute Aggregation. Similar to spectral graph +convolutions in GCN [8], we complete the missing attribute +with the following renormalized graph convolution form. +xC +v = +� +u∈N + +v +(deg(v) · deg(u))−1/2 · xu · W +(3) +Global Attribute Aggregation. Motivated by the node aggre- +gation in APPNP [38], we propose PPNP-based completion +operation for global attribute aggregation. +PPNP-based Attribute Aggregation. Besides the GCN-based +attribute completion, we use another popular node aggregation +method PPNP (i.e., Personalized PageRank [38]) for attribute +completion. Specifically, let A ∈ Rn×n denote the adjacency +matrix of the graph G. ˜A = A+In denotes the adjacency ma- +trix with added self-loops. The form of PPNP-based attribute +completion is: +Xppnp = α +� +In − (1 − α ˆ˜A) +�−1 +· X′, X′ = X · W +XC = {Xppnp +i +| ∀i ∈ V −} +(4) +where ˆ˜A = ˜D−1/2 ˜A ˜D−1/2 is the symmetrically normalized +adjacency matrix with self-loops, with the diagonal degree +matrix ˜D. α ∈ (0, 1] is the restart probability. Note that the +missing attributes are filled with zeros in X. After PPNP-based +attribute aggregation, we complete the attributes of the nodes +in V − with Xppnp. +2) Topology-Independent Completion Operation: For the +no-attribute nodes that have few neighbors or are less af- +fected by the neighbor information, we can directly use one- +hot encoding to replace the missing attributes. The one-hot +representation of a specific node type is also a commonly +used handcrafted attribute completion method [17]. For ex- +ample, there are K distinct actors in IMDB. The one-hot +representation for the actor node is a K-dimensional vector. +For a specific actor, the element in the corresponding index +is 1 and the others are 0. Then, the one-hot representation is +transformed linearly for dimension alignment. +3) Search Space Size Analysis: In summary, the proposed +search space O contains a diverse set of attribute completion +operations. Let N − denote the total number of nodes with +missing attributes. Thus, the space size can be calculated by +|O|N − +, which is exponential to N −. In practice, the attribute +missing of some node types is a common problem, leading +to huge search space. Thus, the block-box optimization-based +search method (e.g., evolutionary algorithm) over a discrete +search space is infeasible. To address this issue, we propose +a differentiable search strategy to find the optimal completion +operations efficiently. +B. Differentiable Search Strategy +In this section, we first introduce a continuous relaxation +scheme for the completion operation search space to make the +search process to be differentiable. Then, we introduce the dif- +ferentiable search algorithm and two optimization techniques +to improve the search efficiency. +1) Continuous Relaxation and Optimization: Inspired by +the success of the differentiable NAS, we first design a contin- +uous search space and then perform differentiable completion +operation search via gradient descent. +As shown in Equation 5, instead of searching over the +discrete space, we view the completion operation as a weighted +mixture of candidate choices. +xC +v = +� +o∈O +exp +� +α(v) +o +� +� +o′∈O exp +� +α(v) +o′ +�o (v) +(5) +where v denotes the node with the missing attribute, o +denotes the candidate operation in the search space O, o (v) +denotes the completed attribute of node v with o. α(v) indicates +the mixing weight vector of dimension |O| for node v. +Furthermore, we refer to α = {α(v) | v ∈ V −} ∈ RN −×|O| +as the completion parameters. +After continuous relaxation, the search objective becomes +the learning of the completion parameters α. To this end, we +formulate the search problem as an optimization problem that +can jointly learn the completion parameters α and the weights +w in the heterogeneous GNN by gradient descent. Let Ltrain +and Lval denote the training loss and validation loss respec- +tively. Since both losses are determined by the completion +parameters α and the weights w, the search objective is a bi- +level optimization problem. +min +α Lval (ω∗, α) +s.t. ω∗ = argminw Ltrain(ω, α) +(6) +where the upper-level optimization is for the optimal comple- +tion parameters α and the lower-level optimization is for the +optimal weights w in the GNN model. +2) Overview: Figure 2 shows the overall framework of au- +tomated attribute completion for heterogeneous graphs. First, +we perform a continuous relaxation of the search space by +placing a mixture of candidate completion operations. Then, +the completion parameters α are optimized. After determin- +ing the attribute completion operations for each no-attribute +node, we view the completed attributes together with the raw +attributes as the initial embedding for the training of the graph +neural network. + +one-hot +mean +GCN +PPNP +Discrete search space for +automated attribute completion +Node with missing attributes +? +? +? +? +one-hot +mean +GCN +PPNP +𝜶𝟐 +𝜶𝟏 +𝜶𝟑 +𝜶𝟒 +one-hot +mean +GCN +PPNP +× +× +× +Continuous relaxation +by placing a mixture of +candidate choices +Adding discrete constraints +when training GNN +Graph +Neural +Network +Softmax +Layer +Softmax +Layer +Supervised +Classification +Loss +𝑪𝟏 +𝑪𝟐 +… +𝑪𝑴 +Auxiliary +Unsupervised +Clustering +Loss +Joint +Loss +GNN Training +𝑪𝟏 +𝑪𝟐 +… +𝑪𝑵 +𝑪𝟏 +𝑪𝟐 +… +𝑪𝑴 +𝑪𝟑 +𝑪𝑴−𝟏 +Graph Clustering +𝑪𝟏 +𝑪𝟐 +𝑪𝟑 +𝑪𝑴 +𝑪𝑴−𝟏 +node +attributes +attributes +after completion +Attribute Completion +attributes +missing +Fig. 2. +The overall workflow of automated attribute completion for the heterogeneous graph neural network. +Why not use the weighted mixture. Although the continuous +relaxation allows the search of completion operations to be dif- +ferentiable, there still exist following limitations when directly +using the weighted mixture of all completion operations: +1) High computational overhead: After continuous relax- +ation, we need to perform all candidate completion +operations for each no-attribute node when training het- +erogeneous GNNs, leading to huge computational over- +head. Also, solving the bi-level optimization problem in +Equation 6 incurs significant computational overhead. +2) Performance gap: At the end of the search, con- +tinuous parameters α needs to be discretized, i.e., +argmaxo∈O α(v) +o , resulting in inconsistent performance +between searched and final completion operations. +3) Large dimension of α: The dimension of completion +parameters α is N − × |O|, which is proportional to the +total number of nodes with missing attributes. The large +dimension of α leads to a slow convergence rate and +low search efficiency. +To address the first two issues (i.e., reducing computational +overhead and avoiding performance gap), we first propose an +efficient search algorithm with discrete constraints. Specifi- +cally, for each no-attribute node v, the completion parameters +satisfy the following constraints: α(v) ∈ C = C1 ∩ C2, where +C1 = {α(v) | ∥α(v)∥0 = 1}, C2 = {α(v) | 0 ≤ α(v) +i +≤ 1}. +The constraint C2 allows α to be optimized continuously, and +C1 keeps the choices of completion operation to be discrete +when training GNN. As shown in Figure 2, there is only one +activated edge for each choice when training GNN, removing +the need to perform all candidate completion operations. The +final completion operation is derived from the learned com- +pletion parameter α. For node v, the edge with the maximum +completion parameter will be kept. We leverage proximal +iteration [37] to solve the constrained optimization problem. +Moreover, proximal iteration can improve the computational +efficiency of optimizing α without second-order derivative. +Moreover, to address the third issue (i.e., reducing the +dimension of α), we propose an auxiliary unsupervised clus- +tering task. In practice, the no-attribute nodes with similar +semantic characteristics may have the same completion opera- +tion. Take the actor nodes in the IMDB dataset as an example. +For the actors with a large number of representative movies, +the average attribute aggregation operation is more suitable. +Thus, we can cluster all no-attribute nodes into M clusters, +where the nodes in each cluster have the same completion +operation. The optimization goal becomes to search for the +optimal attribute completion operation for each cluster. In this +way, the size of the completion parameters α is reduced from +N − × |O| to M × |O|, M ≪ N −. As shown in Figure 2, the +auxiliary unsupervised clustering loss can be jointly optimized +with the node classification loss (i.e., cross-entropy). +The proposed framework AutoAC is composed of multiple +iterations. In each iteration, the completion parameters α and +the weights in the GNN are optimized alternatively. Next, we +introduce the search algorithm with discrete constraints and +the auxiliary unsupervised clustering task in detail. +C. Search Algorithm with Discrete Constraints +Equation 6 implies a bi-level optimization problem with α +as the upper-level variable and w as the lower-level variable. +Following the commonly used methods in meta learning [39] +and NAS [40], we use a one-step gradient approximation to +the optimal internal weight parameters ω∗ to improve the + +Algorithm 1 Search Algorithm in AutoAC +1: Initialize completion parameters α according to defined +search space O; +2: while not converge do +3: +Get discrete choices of attribute completion operations: +¯α(k) = proxc1(α(k)) +4: +Update +α +for +continuous +variables: +α(k+1) += +proxc2(α(k) − ϵ∇¯α(k)Lval(ω(k), ¯α(k))) +5: +Refine discrete choices after updating: +¯α(k+1) += +proxc1(α(k+1)) +6: +Update ω(k) by ∇ω(k)Ltrain(ω(k), ¯α(k+1)) +7: end while +efficiency. Thus, the gradient of the completion parameters +α is as follows (we omit the step index k for brevity): +∇αLval (ω∗, α) +≈∇αLval (ω − ξ∇ωLtrain(ω, α), α) +=∇αLval (ω′, α) − ξ∇2 +α,ωLtrain(ω, α)∇ω′Lval (ω′, α) +(7) +where ω is the weights of the GNN, ξ is the learning rate +of internal optimization, and ω′ = ω − ξ∇ωLtrain(ω, α) +indicates the weights for a one-step forward model. we update +the completion parameters α to minimize the validation loss. +In Equation 7, there exists a second-order derivative, which +is expensive to compute due to a large number of param- +eters. Also, the continuous relaxation trick further leads to +huge computational overhead since all candidate completion +operations need to be performed when training the GNN. +Moreover, the overall search process is divided into two stages: +search and evaluation. In the evaluation stage, the continuous +completion parameters α need to be discretized for replacing +every mixed choice as the most likely operation by taking the +argmax, leading to performance gap between the search and +evaluation stage. +To optimize α efficiently and avoid the performance gap, +we propose a search algorithm with discrete constraints when +optimizing completion parameters α. For the no-attribute node +v, let the feasible space of α(v) be C = {α(v) | ∥α(v)∥0 = 1 ∧ +0 ≤ α(v) +i +≤ 1}. We denote it as the intersection of two feasible +spaces (i.e., C = C1 ∩ C2), where C1 = {α(v) | ∥α(v)∥0 = 1}, +C2 = {α(v) | 0 ≤ α(v) +i +≤ 1}. The optimization problem under +constraints can be solved by the proximal iterative algorithm. +Proposition 1: proxC(z) = proxC2(proxC1(z)) +Inspired by Proposition 1 +[19], [37], in the k-th proximal +iteration, we first get discrete variables constrained by C1, +i.e., ¯α(k) = proxC1(α(k)) (the node notation v is omitted for +brevity). Then, we derive gradients w.r.t ¯α(k) and keep α to +be optimized as continuous variables but constrained by C2. +α(k+1) = proxc2(α(k) − ϵ∇¯α(k)Lval(¯α(k))) +(8) +The detailed search algorithm is described in Algorithm 1. +First, we get a discrete representation of α by proximal step +(Line 3). Then, we view ω(k) as constants and optimize α(k+1) +for continuous variables (Line 4). Since there is no need to +compute the second-order derivative, the efficiency of updating +α can be improved significantly. After updating α, we further +refine discrete choices and get ¯α(k+1) for updating ω(k) on the +training dataset, which contributes to reducing the performance +gap caused by discretizing completion parameters α from con- +tinuous variables. Moreover, since only one candidate choice +is activated for each no-attribute node, the computational +overhead can also be reduced. The computational efficiency +of updating α can be significantly improved. +D. Auxiliary Unsupervised Clustering Task +As mentioned before, the dimension of the completion +parameters α is N − × |O| (|O| ≪ N −, |O| = 4). Take the +DBLP dataset as an example, the number of nodes with +missing attributes is about 1.2 × 104, leading to a large +dimension of completion parameters α. As a result, optimizing +α with a limited size of validation dataset is very difficult. +Inspired by the observation that the no-attribute nodes +with similar explicit topological structure or implicit semantic +characteristics, we further propose an auxiliary unsupervised +clustering task to divide all no-attribute nodes into M clusters. +In each cluster, all nodes share the same completion operation. +In this way, the dimension of the completion parameters α can +be reduced to M ×|O|, M ≪ N −, and optimizing α becomes +feasible and efficient. +It is well known that the EM algorithm [41] is a commonly +used method (e.g., K-Means [42]) to solve the problem of un- +supervised clustering. In the scenario of graph node clustering, +let hv denote the hidden node representation learned by the +heterogeneous GNN. The E-step is responsible for assigning +the optimal cluster for each node v by calculating the distances +between hv and all cluster centers. The M-step is used to +update the centers of all clusters. The E-step and M-step are +performed alternately until convergence. +Although the EM algorithm has a convergence guarantee, +it is sensitive to the initial values, making it difficult to apply +to the proposed automated completion framework. The main +reason is that the bi-level optimization problem defined in +Equation 6 is iterative. In the early optimization process, +the weights of the GNN have not yet converged and the +node representations learned in the GNN are less informative. +Such low-quality representations lead to inaccurate clustering, +which has a negative impact on the subsequent clustering +quality and further leads to a deviation from the overall +optimization direction. +To address this issue, we first formulate the problem of +unsupervised node clustering as a form of soft classification, +and use the assignment matrix C to record the probability of +each node belonging to each cluster. Moreover, as shown in +Figure 2, we embed the clustering process into the bi-level +iterative optimization process. +Motivated by graph pooling and graph module partitioning, +we introduce the Spectral Modularity Function Q [43] [44]. +From a statistical perspective, this function can reflect the clus- + +tering quality of graph node modules through the assignment +matrix C [45]: +Q = +1 +2 |E| +� +ij +� +Aij − didj +2 |E| +� +δ (ci, cj) +(9) +where |E| is the number of edges in the graph, δ(ci, cj) = 1 +only if nodes i and j are in the same cluster, otherwise 0. di +and dj represent the degrees of node i and node j respectively. +It can be known that in a random graph, the probability that +node i and node j are connected is didj +2|E| [45]. +Then, the optimization goal is converted into maximizing +the spectral modularity function Q, but it is an NP-hard +problem. Fortunately, this function can be represented by an +approximate spectral domain relaxation form: +Q = +1 +2 |E| Tr +� +C⊤BC +� +(10) +where Cij ∈ [0, 1] denotes the cluster probability. B is the +modular matrix B = A − dd⊤ +2|E|. Finding the optimal solution +of the assignment matrix C is to maximize Q. To prevent +falling into local optimum (i.e., all nodes tend to be in the +same cluster), we further add the collapse regularization term. +The assignment matrix C should be amortized as adaptively +as possible, so as to skip the local optimum. +Let LGmoC denote the unsupervised clustering loss, which +can be expressed as: +LGmoC = − +1 +2 |E| Tr +� +C⊤BC +� +� +�� +� +modularity loss ++ +√ +M +|V | +����� +� +i +C⊤ +i +����� +F +� +�� +� +collapse regularization +(11) +where |V | is the number of nodes, M is the number of +clusters, ∥ · ∥F represents the Frobenius norm of the matrix. +Note that LGmoC can be jointly optimized with the supervised +classification loss. Specifically, LGmoC can be used as an aux- +iliary task for the bi-level optimization problem in Equation 6. +The unsupervised clustering loss is added to Ltrain for joint +optimization. Let λ denote the loss-weighted coefficient. The +optimization objective is updated as: +min +α Lval (w∗, α) +s.t. w∗ = arg min +w (Ltrain(w, α) + λLGmoC) +(12) +E. Complexity Analysis +In the heterogeneous graph G = ⟨V, E⟩, the total number of +nodes is N, the total number of nodes with missing attributes is +N −, and the embedding dimension is k. In each iteration of 12, +we can divide the search process of AutoAC into three phases, +i.e., attribute completion phase, upper-level optimization for +completion parameters α, and lower-level optimization for +weights ω. We first analyze the computational complexity. +Since discrete constraints are performed, only one candidate +completion operation is activated for each no-attribute node. +The computational complexity of each completion operation +TABLE I +STATISTICS OF THE DATASETS +Datasets #Nodes #Node +Types +#Nodes under +Each Type +#Edges Target Node/Edge +Type +Attribute +DBLP +26128 +4 +author(A):4057 +239566 +author +A:Missing +paper(P):14328 +paper-author +P:Raw +term(T):7723 +T:Missing +venue(V):20 +V:Missing +ACM +10942 +4 +paper(P):3025 +547872 +paper +P:Raw +author(A):5959 +A:Missing +subject(S):56 +S:Missing +term(T):1902 +T:Missing +IMDB +21420 +4 +movie(M):4932 +86642 +movie +M:Raw +director(D):2393 +movie-keyword +D:Missing +actor(A):6124 +A:Missing +keyword(K):7971 +K:Missing +LastFM +20612 +3 +user(U):1892 +141521 +user-artist +U:Missing +artist(A):17632 +A:Raw +tag(T):2980 +T:Missing +is as follows: Mean attribute aggregation: O(N − × k2), +GCN-based attribute aggregation: O(N − × k2), PPNP-based +attribute aggregation: O(N×k2), one-hot attribute completion: +O(1). Thus, the computational complexity of the attribute +completion phase is O(N×k2). In the upper-level optimization +phase, the complexity is O(CH + |O| × M × bα), where +CH denotes the forward computation overhead of the het- +erogeneous GNN, bα the gradient computation overhead for +each completion parameter. For brevity, we omit the difference +between the validation and training datasets. The lower-level +optimization phase contains the optimization of weights and +unsupervised clustering. The complexity of optimizing ω is +O(CH +|ω|×bω), where bω is the gradient computation over- +head for each weight parameter. The complexity of calculating +the clustering loss LGmoC is O(d2 × N + |E|) [45], where d +is the average degree. Overall, the computational complexity +of each iteration is, O(N × k2) + O(CH + |O| × M × bα) + +|ω| × bω) + O(d2 × N + |E|). +Next, we analyze the space complexity of AutoAC. For the +attribute completion phase, the space complexity is O(k2). For +the optimization phase, the space complexity is O(N × k + +|O| × M + |ω| + N × M), where O(N × M) is the space +complexity in the unsupervised clustering. +V. EXPERIMENTS +A. Experimental Setup +1) Experimental Setting: We use the recently proposed +Heterogeneous Graph Benchmark (HGB) [17] to conduct all +experiments, which offers a fair way to compare heterogeneous +GNN models. HGB gives a set of standard benchmark datasets +and unified strategies for feature preprocessing and data split. +In the node classification task, all edges are available during +training, and node labels are split according to 24% for +training, 6% for validation, and 70% for test in each dataset. +In the link prediction task, we mask 10% edges of the target +link type and the negative edges are randomly sampled The +statistics of the four datasets are summarized in Table I. More +details of datasets can be seen in Appendix A. + +Moreover, the handcrafted attribute completion methods for +existing heterogeneous GNNs are provided by HGB. Micro-F1 +and Macro-F1 are provided to evaluate the node classification +performance, while the MRR and ROC-AUC metrics are used +for link prediction. The evaluation metrics are obtained by +submitting predictions to the HGB website2. +B. Implementation Details +All experiments are performed in the transductive setting. +We employ the Adam optimizer [46] to optimize both ω and +α. For optimizing ω, the learning rate and the weight decay +are 5e-4 and 1e-4 respectively. For optimizing α, the learning +rate and the weight decay are 5e-3 and 1e-5 respectively. +We implement AutoAC based on the widely-used hetero- +geneous GNNs, i.e., MAGNN [14] and SimpleHGN [17]. +The loss weighted coefficient λ and the number of clusters +M are two hyperparameters of AutoAC. For MAGNN, we +empirically set λ to 0.5 for all datasets, M to 4 for the DBLP +and ACM datasets, 16 for the IMDB dataset. For SimpleHGN, +λ is 0.4 for all datasets, and M is 8 for the DBLP dataset, +12 for the ACM and IMDB datasets. Moreover, all the GNN +models are implemented with PyTorch. All experiments are +run on a single GPU (NVIDIA Tesla V100) five times and the +average performance and standard deviation are reported. +C. Effectiveness of AutoAC +1) Performance comparison with humancrafted heteroge- +neous GNNs: Depending on whether or not the meta-path is +used, we divide the humancrafted heterogeneous GNNs into +two categories: +• GNNs with meta-path: HAN [13], GTN [11], Het- +SANN [16], MAGNN [14], HGCA [26]. +• GNNs without meta-path: HGT [15], GATNE [47], Het- +GNN [13], GCN [8] and GAT [20] (two commonly used +general-purpose GNNs), as well as the current SOTA +GNN model SimpleHGN [17]. +The configurations of baselines can be seen in Appendix B. +Since AutoAC is general and thus can be integrated into +different GNNs. We select two representative GNN models +from the two categories (i.e., MAGNN and SimpleHGN) from +the perspective of performance and computational efficiency. +Then, we combine AutoAC with the two models, denoted by +MAGNN-AutoAC and SimpleHGN-AutoAC respectively. +Table II shows the performance comparison between Au- +toAC and existing heterogeneous GNNs on node classifica- +tion. AutoAC can improve the performance of MAGNN and +SimpleHGN stably on all datasets. The performance gain +obtained by AutoAC over MAGNN is around 0.7%-3% and +the error rate is reduced by 2.87%-11.69%. Also, SimpleHGN- +AutoAC outperforms SimpleHGN by 1%-3% and reduces the +error rate by 1.59%-22.09%. By combining with the SOTA +model SimpleHGN, SimpleHGN-AutoAC can achieve the best +performance in all models. +2https://www.biendata.xyz/competition/hgb-1/ +Moreover, Table II shows that AutoAC can bring signif- +icant performance improvement on the datasets where the +classification target nodes have no raw attributes (e.g., DBLP). +Besides, for the datasets where the target nodes already have +raw attributes (e.g., ACM and IMDB), completing other non- +target nodes using AutoAC can still promote the classification +accuracy of target nodes. Especially, for the IMDB dataset, +since there are too many non-target nodes with missing at- +tributes (i.e., 77% of all nodes), the performance improvement +with AutoAC is more significant. +Note that the performance of MAGNN without attribute +completion is not as good as other models, such as GTN +and GAT. However, MAGNN-AutoAC performs better than +GTN on DBLP and ACM, and outperforms GAT on DBLP +and IMDB, which indicates that effective attribute completion +for heterogeneous graphs can compensate for the performance +gap introduced by the GNN model. By unifying attribute +completion and representation learning in an unsupervised +heterogeneous network, the recently proposed HGCA can also +achieve competitive performance on DBLP and ACM. Such +experimental results further verify the necessity of AutoAC. +2) Performance comparison with the existing attribute com- +pletion method HGNN-AC: As the current SOTA attribute +completion method, HGNN-AC [18] uses the attention mech- +anism to aggregate the attributes of the direct neighbors for +the nodes with missing attributes. The attention information +is calculated by the pre-learning of topological embedding. +To be fair, both AutoAC and HGNN-AC are evaluated under +the unified HGB benchmark. And, we also combine HGNN- +AC with MAGNN and SimpleHGN, denoted by MAGNN- +HGNNAC and SimpleHGN-HGNNAC respectively. +Table III shows that AutoAC outperforms HGNN-AC on +all datasets. Specifically, MAGNN-AutoAC achieves 1%- +4% performance improvement over MAGNN-HGNNAC. For +the SimpleHGN model, SimpleHGN-AutoAC outperforms +SimpleHGN-HGNNAC by 0.4%-2%. Moreover, the perfor- +mance improvement of HGNN-AC for attribute completion +is not stable. As shown in Table III, after attribute completion +with HGNN-AC, MAGNN-HGNNAC is instead inferior to +MAGNN on the three datasets, while MAGNN-AutoAC can +achieve significant performance improvement with attribute +completion. Similarly, there is a degradation in performance +on the DBLP dataset compared to SimpleHGN. +3) Performance comparison on link prediction: To verify +the effectiveness of AutoAC on different downstream tasks, we +further conduct link prediction in Table V. AutoAC can greatly +improve the performance of heterogeneous GNNs, especially +on IMDB. With AutoAC, MRR and ROC-AUC of SimpleHGN +are increased by 9.7% and 28%, respectively. +In summary, AutoAC achieves better performance and more +stable performance improvement, indicating the effectiveness +of searching for the most suitable attribute completion opera- +tions for no-attribute nodes from a diverse search space. + +TABLE II +PERFORMANCE AND RUNTIME (CLOCK TIME IN SECONDS) COMPARISON BETWEEN AUTOAC AND SOTA HUMANCRAFTED HETEROGENEOUS GNNS ON +NODE CLASSIFICATION. THE BOLD AND THE UNDERLINE INDICATE THE BEST AND THE SECOND BEST IN EACH CATEGORY (I.E., USING AND NOT USING +META-PATH). * INDICATES THE GLOBAL BEST IN ALL MODELS. p-VALUE INDICATES THE STATISTICALLY SIGNIFICANT IMPROVEMENT (I.E., T-TEST WITH +p < 0.05) OVER THE BEST BASELINE. +Dataset +DBLP +ACM +IMDB +Macro-F1 +Micro-F1 +Runtime +(Total) +Runtime +(Per epoch) +Macro-F1 +Micro-F1 +Runtime +(Total) +Runtime +(Per epoch) +Macro-F1 +Micro-F1 +Runtime +(Total) +Runtime +(Per epoch) +HAN +93.17±0.19 +93.64±0.17 +44 +0.23 +87.68±1.94 +87.73±1.81 +31 +0.25 +59.70±0.90 +65.61±0.54 +13 +0.08 +GTN +93.52±0.55 +93.97±0.54 +13600 +340 +91.63±1.27 +91.53±1.30 +3234 +77 +59.26±0.84 +64.07±0.65 +9960 +249 +HetSANN +84.08±1.01 +84.96±0.88 +201 +0.93 +90.09±1.06 +90.00±1.02 +470 +1.50 +49.25±0.57 +57.47±1.12 +520 +1.13 +HGCA +93.05±0.46 +93.62±0.41 +495 +55 +91.75±0.54 +91.67±0.56 +30 +1.5 +43.54±1.17 +53.44±1.00 +56 +2.8 +MAGNN +93.16±0.38 +93.65±0.34 +401 +19 +91.06±1.44 +90.95±1.43 +230 +23 +56.92±1.76 +65.11±0.59 +108 +9.8 +MAGNN-AutoAC +93.95±0.30 +94.39±0.25 +432 +21 +91.84±0.45 +91.77±0.45 +684 +25 +58.96±1.31 +66.11±0.53 +576 +11 +HGT +92.77±0.35 +93.44±0.31 +131 +1.87 +90.27±0.55 +90.14±0.51 +545 +7.07 +63.02±0.80 +67.01±0.36 +257 +3.38 +HetGNN +92.77±0.24 +93.23±0.23 +20580 +98 +84.93±0.78 +84.83±0.76 +25410 +121 +47.87±0.33 +50.83±0.26 +18270 +87 +GCN +90.54±0.27 +91.18±0.25 +29 +0.09 +92.63±0.23 +92.60±0.22 +26 +0.08 +59.95±0.72 +65.35±0.35 +10 +0.11 +GAT +92.96±0.35 +93.46±0.35 +14 +0.14 +92.41±0.84 +92.39±0.84 +29 +0.14 +56.95±1.55 +64.24±0.55 +10 +0.21 +SimpleHGN +93.83±0.18 +94.25±0.19 +43 +0.39 +92.92±0.67 +92.85±0.68 +42 +0.47 +62.98±1.66 +67.42±0.42 +25 +0.36 +SimpleHGN-AutoAC +95.15±0.29* +95.52±0.26* +72 +0.58 +93.86±0.18* +93.80±0.18* +108 +0.62 +64.92±0.58* +67.94±0.41* +72 +0.55 +p-value +2.9 × e − 8 +3.3 × e − 9 +- +- +1.6 × e − 6 +2.9 × e − 6 +- +- +1.4 × e − 6 +9.8 × e − 6 +TABLE III +PERFORMANCE COMPARISON BETWEEN AUTOAC AND HGNNAC.THE BOLD AND THE UNDERLINED INDICATE THE BEST AND THE SECOND BEST IN +EACH CATEGORY. p-VALUE INDICATES THE STATISTICALLY SIGNIFICANT IMPROVEMENT (I.E., T-TEST WITH p < 0.05) OVER THE BEST BASELINE. +Dataset +DBLP +ACM +IMDB +Model \ Metrics +Macro-F1 +Micro-F1 +Macro-F1 +Micro-F1 +Macro-F1 +Micro-F1 +MAGNN +93.16±0.38 +93.65±0.34 +91.06±1.44 +90.95±1.43 +56.92±1.76 +65.11±0.59 +MAGNN-HGNNAC +92.97±0.72 +93.43±0.69 +90.89±0.87 +90.83±0.87 +56.63±0.81 +63.85±0.85 +MAGNN-AutoAC +93.95±0.30 +94.39±0.25 +91.84±0.45 +91.77±0.45 +58.96±1.31 +66.11±0.53 +SimpleHGN +93.83±0.18 +94.25±0.19 +92.92±0.67 +92.85±0.68 +62.98±1.66 +67.42±0.42 +SimpleHGN-HGNNAC +93.24±0.49 +93.73±0.45 +93.16±0.24 +93.09±0.23 +64.44±1.13 +67.67±0.39 +SimpleHGN-AutoAC +95.15±0.29 +95.52±0.26 +93.86±0.18 +93.80±0.18 +64.92±0.58 +67.94±0.41 +p-value +2.9 × e−8 +3.3 × e−9 +7.3 × e−7 +1.1 × e−6 +4 × e−3 +1 × e−3 +TABLE IV +THE OVERALL RUNTIME OVERHEAD (CLOCK TIME IN SECONDS) OF AUTOAC AND HGNN-AC. / INDICATES THAT THE STAGE IS NOT INVOLVED. +Datasets +Models +End-to-End Runtime Overhead (Seconds) +Speedup +Pre-learn +Search +Train/Retrain +Total +DBLP +SimpleHGN-HGNNAC +33048 +/ +432 +33480 +465× +SimpleHGN-AutoAC +/ +36 +36 +72 +MAGNN-HGNNAC +33048 +/ +900 +33948 +78× +MAGNN-AutoAC +/ +72 +360 +432 +ACM +SimpleHGN-HGNNAC +3888 +/ +432 +4320 +40× +SimpleHGN-AutoAC +/ +72 +36 +108 +MAGNN-HGNNAC +3888 +/ +1260 +5148 +7.5× +MAGNN-AutoAC +/ +432 +252 +684 +IMDB +SimpleHGN-HGNNAC +8568 +/ +324 +8892 +123× +SimpleHGN-AutoAC +/ +36 +36 +72 +MAGNN-HGNNAC +8568 +/ +180 +8748 +15× +MAGNN-AutoAC +/ +504 +72 +576 +D. Efficiency Study +Besides the effectiveness, we also evaluate the efficiency +of AutoAC in the terms of runtime overhead. Table II and V +show the runtime of AutoAC and other handcrafted HGNNs +on node classification and link prediction tasks. Although the +attribute completion and GNN training are jointly optimized +in AutoAC, the computational efficiency of AutoAC is still +competitive compared to other baselines. +Also, we compare AutoAC with the existing attribute com- +pletion method HGNN-AC. Table IV shows the efficiency +comparison between AutoAC and HGNN-AC. AutoAC con- +tains the search and retraining stages, and HGNN-AC contains +the pre-learning and training stages. We can see that AutoAC is +much more efficient than HGNN-AC. The end-to-end runtime +overhead of AutoAC can be reduced by 15× to 465×. The +main reason why HGNN-AC is inefficient is that the pre- +leaning stage that learns a topological embedding for each +node is very time-consuming. Especially for the DBLP dataset +with a large number of nodes, the pre-learning overhead +is up to 9 GPU hours. In contrast, there is no additional +pre-leaning stage in AutoAC. Moreover, by introducing the +discrete constraints and auxiliary unsupervised clustering task, +the search efficiency can be improved significantly. +In summary, AutoAC can not only achieve better perfor- +mance but also demonstrate higher computational efficiency. + +TABLE V +PERFORMANCE AND RUNTIME (CLOCK TIME IN SECONDS) COMPARISON ON LINK PREDICTION. THE BOLD AND THE UNDERLINE INDICATE THE BEST +AND THE SECOND BEST. p-VALUE INDICATES THE STATISTICALLY SIGNIFICANT IMPROVEMENT (I.E., T-TEST WITH p < 0.05) OVER THE BEST BASELINE. +Dataset +LastFM +DBLP +IMDB +Model \ Metrics +ROC-AUC +MRR +Runtime +(Total) +Runtime +(Per epoch) +ROC-AUC +MRR +Runtime +(Total) +Runtime +(Per epoch) +ROC-AUC +MRR +Runtime +(Total) +Runtime +(Per epoch) +GATNE +66.87±0.16 +85.93±0.63 +75960 +15435 +71.94±2.00 +87.23±0.76 +92160 +16278 +47.45±6.48 +74.58±3.34 +71280 +14269 +HetGNN +62.09±0.01 +85.56±0.14 +20580 +98 +88.89±0.40 +94.39±0.62 +22050 +105 +56.55±0.83 +78.10±0.56 +19950 +95 +GCN +59.17±0.31 +79.38±0.65 +13 +0.13 +80.48±0.81 +90.99±0.56 +31 +0.12 +51.90±1.10 +76.99±1.87 +28 +0.11 +GAT +58.56±0.66 +77.04±2.11 +10 +0.12 +72.89±3.09 +82.56±3.35 +32 +0.15 +48.30±1.35 +76.74±2.00 +12 +0.10 +SimpleHGN +67.16±0.37 +86.73±0.27 +46 +0.35 +94.61±0.11 +97.21 ±0.16 +58 +0.75 +57.92±2.32 +79.09 ±1.40 +28 +0.44 +SimpleHGN-AutoAC +67.72±0.17 +87.10±0.19 +42 +0.43 +95.87±0.66 +98.21±0.21 +61 +0.87 +74.14±0.73 +86.27±0.45 +32 +0.49 +p-value +9.3 × e−4 +9.5 × e−4 +- +- +4.2 × e−5 +8.2 × e−7 +- +- +6.7 × e−9 +7.2 × e−10 +- +- +E. Ablation Study +1) Study on the necessity of searching attribute completion +operations from a diverse search space: We compare AutoAC +with the following two methods: +• Single-operation attribute completion: We complete +all no-attribute nodes with the same single completion +operation (i.e., GCN AC, PPNP AC, MEAN AC, and +One-hot AC). +• Random attribute completion: For each no-attribute +node, we randomly select an attribute completion opera- +tion from the search space. +Table VI and Table VII show the completion operation +ablation study on SimpleHGN and MAGNN. Due to the +differences in the data characteristics, there is no single +completion operation that can perform well on all datasets. By +searching the optimal attribute completion operations AutoAC +can achieve the best performance on all datasets. +Take SimpleHGN shown in Table VI for example. GCN AC +is more effective on DBLP and IMDB, while PPNP AC +performs better on ACM. Moreover, for a specific attribute +completion operation, the performance is related to the dataset +and the chosen GNN model. We take DBLP as an example. +GCN AC performs better on SimpleHGN. However, when +the GNN model becomes MAGNN, GCN AC is not as good +as MEAN AC. Additionally, the performance of the random +attribute completion is not stable and can be even worse than +the baseline model. Choosing an inappropriate completion +operation can have a negative effect on the final performance. +2) Study on the search algorithm with discrete constraints: +When optimizing the attribute completion parameters α, we +enforce discrete constraints on α and solve the bi-level op- +timization problem with proximal iteration. To verify the +effectiveness of discrete constraints, we further run AutoAC +with and without discrete constraints in Table VIII. +The search algorithm with discrete constraints can achieve +better performance with less search time overhead on all +datasets. Additionally, proximal iteration allows removing +the need for second-order derivative in solving the bi-level +optimization problem. Thus, the memory overhead can also +be reduced significantly.As shown in Table VIII, the memory +overhead of MAGNN-AutoAC without discrete constraints is +huge and the out-of-memory error occurs on DBLP. +3) Study on the auxiliary unsupervised clustering: To re- +duce the dimension of the completion parameters α, we +(a) SimpleHGN +(b) MAGNN +Fig. 3. Performance comparison between different clustering methods. +(a) DBLP +(b) ACM +(c) IMDB +Fig. 4. Convergence of LGmoC on three datasets. +leverage an auxiliary unsupervised clustering task. Figure 3 +shows the performances of different clustering methods. +• w/o cluster: We directly search the attribute completion +operations for each no-attribute node without clustering. +• EM: After each iteration of the optimization process, we +adopt the EM algorithm for clustering according to node +representation learned by the GNN model. +• EM with warmup: a variant of the EM algorithm, which +adds a warm-up process at the beginning of the clustering. +In Figure 3, AutoAC can achieve the best performance on all +datasets. Searching completion operations without clustering +yields relatively poor performance. Reducing the dimension of +α with unsupervised clustering is very necessary. Moreover, + +Ittt Baseline +Performance Comparison +w/o Cluster +EM +X +EM with Warmup +AutoAC +DBLP-Macro-F1 ACM-Macro-F1 IMDB-Macro-F1 +DBLP-Micro-F1ACM-Micro-F1 +IMDB-Micro-F1 +Dataset-MetricIttt Baseline +Performance Comparison +/ / w/o Cluster +EM +X +EM with Warmup +AutoAC +IMDB-Micro-F1 +Dataset-MetricDBLPACMIMDBTABLE VI +COMPLETION OPERATION ABLATION STUDY ON SIMPLEHGN. BOLD INDICATES THE GLOBAL BEST. UNDERLINE INDICATES THE BEST AMONG ALL +SINGLE ATTRIBUTE COMPLETION OPERATIONS. +Dataset +DBLP +ACM +IMDB +Model \ Metrics +Macro-F1 +Micro-F1 +Macro-F1 +Micro-F1 +Macro-F1 +Micro-F1 +Baseline (SimpleHGN) +93.83±0.18 +94.25±0.19 +92.92±0.67 +92.85±0.68 +62.98±1.66 +67.42±0.42 +GCN AC +94.23±0.21 +94.88±0.23 +93.25±0.45 +93.18±0.47 +64.67±0.94 +67.96±0.53 +PPNP AC +85.76±2.24 +86.58±2.23 +93.42±0.46 +93.34±0.48 +53.36±19.31 +61.68±11.76 +MEAN AC +90.91±0.72 +91.53±0.67 +92.99±0.60 +92.90±0.62 +63.73±0.94 +67.61±0.30 +One-hot AC +93.80±0.13 +94.30±0.14 +93.38±0.16 +93.31±0.15 +64.17±0.83 +67.89±0.24 +Random AC +91.28±1.63 +91.77±1.55 +93.02±0.29 +92.95±0.31 +64.03±0.68 +67.43±0.33 +AutoAC +95.15±0.29 +95.52±0.26 +93.86±0.18 +93.80±0.18 +64.92±0.58 +67.94±0.41 +TABLE VII +COMPLETION OPERATION ABLATION STUDY ON MAGNN. BOLD INDICATES THE GLOBAL BEST. UNDERLINE INDICATES THE BEST AMONG ALL SINGLE +ATTRIBUTE COMPLETION OPERATIONS. +Dataset +DBLP +ACM +IMDB +Model \ Metrics +Macro-F1 +Micro-F1 +Macro-F1 +Micro-F1 +Macro-F1 +Micro-F1 +Baseline (MAGNN) +93.16±0.38 +93.65±0.34 +91.06±1.44 +90.95±1.43 +56.92±1.76 +65.11±0.59 +GCN AC +93.74±0.34 +94.16±0.34 +90.96±0.77 +90.87±0.76 +57.96±1.11 +65.71±0.50 +PPNP AC +93.46±0.32 +93.94±0.29 +90.38±0.67 +90.28±0.67 +58.46±1.17 +65.97±0.56 +MEAN AC +93.89±0.12 +94.33±0.13 +90.97±0.48 +90.86±0.49 +57.60±0.71 +65.42±0.38 +One-hot AC +93.73±0.32 +94.15±0.28 +91.04±0.69 +90.92±0.70 +58.12±1.71 +65.43±0.68 +Random AC +93.38±0.25 +93.87±0.19 +91.09±0.61 +90.98±0.63 +57.97±1.15 +65.57±0.77 +AutoAC +93.95±0.30 +94.39±0.25 +91.84±0.45 +91.77±0.45 +58.96±1.31 +66.11±0.53 +TABLE VIII +ABLATION STUDY ON DISCRETE CONSTRAINTS. / INDICATES MEMORY OVERFLOW. +Dataset +DBLP +ACM +IMDB +Model \ Metrics +Macro-F1 +Micro-F1 +Search Time +(Seconds) +Macro-F1 +Micro-F1 +Search Time +(Seconds) +Macro-F1 +Micro-F1 +Search Time +(Seconds) +SimpleHGN-AutoAC +95.15±0.29 +95.52±0.26 +32 +93.86±0.18 +93.80±0.18 +72 +64.92±0.58 +67.94±0.41 +36 +w/o Discrete constraints +95.12±0.27 +95.49±0.25 +216 +93.43±0.74 +93.34±0.76 +360 +64.74±0.68 +67.85±0.52 +180 +MAGNN-AutoAC +93.95±0.30 +94.39±0.25 +72 +91.84±0.45 +91.77±0.45 +432 +58.96±1.31 +66.11±0.53 +504 +w/o Discrete constraints +/ +/ +/ +91.24±0.67 +91.45±0.68 +1800 +58.44±1.12 +65.65±0.34 +1908 +Fig. 5. Distribution of searched attribute completion operations. +(a) Author +(b) Subject +(c) Term +Fig. 6. Detailed distribution of searched completion operations for each no- +attribute node type on the ACM dataset using SimpleHGN-AutoAC. +the proposed unsupervised clustering method outperforms +EM and its variant, indicating the effectiveness of the joint +(a) Actor +(b) Director +(c) Keyword +Fig. 7. Detailed distribution of searched completion operations for each no- +attribute node type on the IMDB dataset using SimpleHGN-AutoAC. +optimization of the unsupervised clustering loss and the clas- +sification loss. Figure 4 also shows the convergence of the +unsupervised clustering loss LGmoC, which exhibits a stable +decreasing trend during the optimization process. +F. Distribution of Searched Completion Operations +Figure 5 shows the proportion of attribute completion +operations searched by SimpleHGN-AutoAC and MAGNN- +AutoAC. For different models and datasets, the proportions +of searched completion operations are quite different. In +SimpleHGN-AutoAC, DBLP tends to select GCN AC, while +ACM prefers PPNP AC. For the same dataset, different +GNNs also result in different distributions. Take DBLP as an +example. MAGNN-AutoAC is more inclined to MEAN AC + +51.67% +14.57% +18.68% +15.09%12.50% +64.29% +14.29% +8.93%GCN_AC +PPNP_AC +MEAN AC +One-hot Ac +94.74% +2:63% +2.10%57.45% +8.83% +14.94% +18.78%56.87% +9.03% +12.70% +21.40%GCN AC +PPNP AC +MEAN AC +One-hot_AC +62.90% +7.13% +18.76% +11.22%GCN_AC +PPNP_AC +MEAN_AC +One-hot AC +80 +Percentage(%) +60 +40 +P +20 +0 +SimpleHGN_ACM +SimpleHGN_IMDB +MAGNN_DBLP +MAGNN_IMDB +SimpleHGN DBLP +MAGNN_ACM +Model Dataset(a) DBLP +(b) ACM +(c) IMDB +Fig. 8. Performance comparison under different M. +than GCN AC compared to SimpleHGN-AutoAC. The results +further indicate the necessity of searching for suitable attribute +completion operations under different datasets and GNNs. +Figure 6 and Figure 7 show the proportion of searched +completion operations for each no-attribute node type on ACM +and IMDB. For ACM, multiple different completion operations +are selected even for the same node type. Specifically, more +than half of the author and subject nodes choose PPNP AC, +while the proportions of other three operations are quite sim- +ilar. Most term nodes are assigned PPNP AC (i.e., 94.74%), +indicating that the term type is more likely to capture the +global information. The main reason is that the target node +type (i.e., paper) with raw attributes in ACM contains only the +paper title. The high-order PPNP AC operations are preferred. +In contrast, GCN AC accounts for the majority of completion +operations on IMDB. This is because that the target node +type (i.e., movie) has raw attributes and contains rich features, +such as length, country, language, likes of movies, and ratings. +Thus, the local completion operation GCN AC is appropriate. +Next, we analyze the completion operations of concrete +actor nodes. In IMDB, node No.10797 is the actor Leonardo +DiCaprio, who has starred in 22 movies, and the neighbor- +hood information is very rich. As a result, AutoAC chooses +GCN AC for him. In contrast, node No.10799 is the actor +Leonie Benesch, who has appeared in only one movie. Thus, +one-hot AC is automatically selected by AutoAC. +G. Hyperparameter Sensitivity +1) Effect of the number of clusters M: Figure 8 shows the +performance of AutoAC under different M. Both SimpleHGN- +AutoAC and MAGNN-AutoAC can achieve stable perfor- +mance, showing that AutoAC has sufficient robustness to M. +2) Effect of the loss weighted coefficient λ: We further +evaluate the weighted coefficient λ of the auxiliary unsuper- +vised clustering loss. The available values of λ are set to +[0.1, 0.2, 0.3, 0.4, 0.5]. Figure 9 shows the performances of +AutoAC under different λ. IMDB is very robust to λ, and the +performance change is very insignificant. For DBLP, λ = 0.4 +and λ = 0.5 are suitable for SimpleHGN and MAGNN, +respectively. For ACM, the choice of λ is slightly sensitive. +The effects of the learning rate and the weight decay can +be seen in Appendix C. +H. Impacts of Attribute Missing Rates and Masked Edge Rates +1) Study on the performance of the same dataset with +varying attribute missing rates in the node classification task: +Table IX shows the performance of SimpleHGN-AutoAC with +varying attribute missing rates. We change attribute miss- +ing rates by completing the missing attributes with one-hot +(a) DBLP +(b) ACM +(c) IMDB +Fig. 9. Performance comparison under different λ +TABLE IX +PERFORMANCE OF SIMPLEHGN-AUTOAC WITH VARYING ATTRIBUTE +MISSING RATES IN THE NODE CLASSIFICATION TASK. +Datasets +Attribute +Missing Rates +Node Types with +Missing attributes +Macro-F1 +Micro-F1 +DBLP +0% +/ +93.83±0.18 +94.25±0.19 +15% +author +94.35±0.17 +94.72±0.16 +30% +term, venue +95.09±0.13 +95.47±0.12 +45% +author, term, venue +95.15±0.29 +95.52±0.26 +ACM +0% +/ +92.92±0.67 +92.85±0.68 +17% +subject, term +93.10±0.27 +93.14±0.26 +54% +author, subject +93.55±0.20 +93.47±0.21 +69% +author, subject, term +93.86±0.18 +93.80±0.18 +IMDB +0% +/ +62.98±1.66 +67.42±0.42 +37% +keyword +63.65±0.57 +67.52±0.36 +67% +actor, keyword +64.59±0.53 +67.86±0.42 +76% +director, actor, keyword +64.92±0.58 +67.94±0.41 +TABLE X +PERFORMANCE OF SIMPLEHGN-AUTOAC WITH VARYING MASKED EDGE +RATES IN THE LINK PREDICTION TASK. +Datasets +Masked Edge Rates +Models +ROC-AUC +MRR +DBLP +5% +SimpleHGN +95.92±0.56 +97.16±0.44 +SimpleHGN-AutoAC +97.62±0.36 +99.02±0.24 +10% +SimpleHGN +94.61±0.11 +97.21±0.16 +SimpleHGN-AutoAC +95.87±0.66 +98.21±0.21 +20% +SimpleHGN +91.34±0.61 +95.65±0.41 +SimpleHGN-AutoAC +94.08±0.72 +97.61±0.33 +30% +SimpleHGN +88.76±0.66 +95.39±0.24 +SimpleHGN-AutoAC +91.11±0.67 +97.42±0.44 +IMDB +5% +SimpleHGN +64.89±0.58 +81.86±0.94 +SimpleHGN-AutoAC +86.57±1.36 +92.75±0.84 +10% +SimpleHGN +57.92±2.32 +79.09±1.40 +SimpleHGN-AutoAC +74.14±0.73 +86.27±0.45 +20% +SimpleHGN +58.21±0.39 +79.71±0.34 +SimpleHGN-AutoAC +73.75±0.82 +86.25±0.32 +30% +SimpleHGN +54.13±0.79 +77.57±0.67 +SimpleHGN-AutoAC +65.81±0.31 +83.23±0.21 +encoding, which is a commonly used handcrafted attribute +completion method. A missing rate of 0% means that all +missing attributes are completed manually. 45%, 69%, and +76% are inherent attribute missing rates of DBLP, ACM, and +IMDB, respectively, i.e., only one node type has raw attributes. +From Table IX, we can see that SimpleHGN-AutoAC performs +better with higher missing rates, indicating that AutoAC is +capable of searching for the suitable completion operation for +each no-attribute node and the searched completion operations +are superior to the handcrafted completion method. +2) Study on the performance of the same dataset with +varying masked edge rates in the link prediction task: Table X +shows the performance of SimpleHGN-AutoAC with varying +masked edge rates. The edges are masked randomly. We +can see that SimpleHGN-AutoAC achieves better performance +than SimpleHGN at different masked edge rates, especially on +the IMDB dataset. Moreover, the performance of both models +decreases as the masked edge rate increases. + +95.5 +95.0 +Peformance(%) +SimpeHGN-MacroE1 +94.5 +SimpleHGN-Micro_F1 +MAGNN-Macro F1 +MAGNN-Macro F1 +94.0 +93.5 +4 +8 +12 +16 +M93.5 +Peformance(%) +93.0 +SimpleHGN-Macro F1 +SimpleHGN-Micro_F1 +92.5 +MAGNN-Macro F1 +MAGNN-Macro F1 +92.0 +91.5 +4 +8 +12 +16 +M68 +66 +Peformance(%) +64 +SimpleHGN-MacroE1 +SimpleHGN-Micro_F1 +MAGNN-Macro F1 +62 +MAGNN-MacroF1 +60 +58 +4 +8 +12 +16 +MSimpleHGN-MacroF1 +SimpleHGN-Micro F1 +MAGNN-MacroF1 +MAGNN-MacroF1★SimpleHGN-Macro F1 +SimpleHGN-Micro_F1 +MAGNN-Macro_F1 +MAGNN-Macro F1SimpleHGN-Macro F1 +SimpleHGN-Micro_F1 +MAGNN-Macro_F1 +MAGNN-Macro_F1VI. CONCLUSION +In this paper, we proposed a differentiable attribute com- +pletion framework called AutoAC for automated completion +operation search in heterogeneous GNNs. 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LastFM is extracted from last.fm with +timestamps from January 2015 to June 2015. We use the subset +released by [48]. The target is to predict whether a user likes +a certain artist. The raw attribute of the artist node is the one- +hot encoding. For the DBLP dataset, the attributes of the target +nodes are missing. For the ACM and IMDB datasets, the target +nodes have raw attributes. +B. Implementations and Configurations of Baselines +We use the HGB benchmark to evaluate the performance +of all baselines. In HGB, implementations of baselines are +based on their official codes to avoid errors introduced by +re-implementation. Next, we present the configurations of +baselines in the node classification and link prediction tasks, +respectively. For brevity, we denote the dimension of node +embedding as d, the dimension of edge embedding as de, the +dimension of attention vector (if exists) as da, the number of +GNN layers as L, the number of attention heads as nh, the +negative slope of LeakyReLU as s. +1) Node Classification: The baselines in the node classifi- +cation task contain HAN, GTN, HetSANN, MAGNN, HGCA, +HGT, HetGNN, GCN, GAT, SimpleHGN, and HGNN-AC. +• HAN: We set d = 8, da = 128, nh = 8, and L = 2 for +all datasets. +• GTN: The adaptive learning rate is employed for all +datasets. We set d = 64 and the number of GTN channels +to 2. For DBLP and ACM, we set L = 2. For IMDB, we +set L = 3. +• HetSANN: For ACM, we set d = 64, L = 3, and nh = 8. +For IMDB, we set d = 32, L = 2, and nh = 4. For DBLP, +we set d = 64, L = 2, and nh = 4. +• MAGNN: For DBLP and ACM, we set the batch size to 8, +and the number of neighbor samples to 100. For IMDB, +we use the full batch training. +• HGCA: We set d = 64, the temperature parameter τ = +0.5, and the loss coefficient λ = 0.5. +• HGT: We use the layer normalization in each layer, and +set d = 64 and nh = 8 for all datasets. L is set to 2, 3, +5 for ACM, DBLP and IMDB,respectively. +• HetGNN: We set d = 128, and the batch size to 200 for +all datasets. For random walk, we set the walk length to +30 and the window size to 5. +3https://dblp.uni-trier.de/ +4http://dl.acm.org/ +5https://www.imdb.com +(a) DBLP +(b) ACM +(c) IMDB +Fig. 10. Performance comparison under different learning rates +(a) DBLP +(b) ACM +(c) IMDB +Fig. 11. Performance comparison under different weight decay values +• GCN: We set d = 64 for all datasets. We set L = 3 for +DBLP and ACM, and L = 4 for IMDB. +• GAT: We set d = 64 and nh = 8 for all datasets. For +DBLP and ACM, we set s = 0.05 and L = 3. For IMDB, +we set s = 0.1 and L = 5. +• SimpleHGN: We set d = de = 64, nh = 8, and the edge +residual β = 0.05 for all datasets. For DBLP and ACM, +we set L = 3 and s = 0.05. For IMDB, we set L = 6 +and s = 0.1. +• HGNN-AC: We set d = 64, nh = 8, the divided ratio +α of N + to 0.3, and the loss weighted coefficient λ to +0.5 for all datasets, which are consistent with the original +paper. +2) Link prediction: The baselines in the link prediction task +contain GATNE, HetGNN, GCN, GAT, and SimpleHGN. +• GATNE: We set d = 200, de = 10, and da = 20 for all +datasets. For the random walk, we set the walk length to +30 and the window size to 5. For neighbor sampling, we +set the number of negative samples for optimization to 5 +and the number of neighbor samples for aggregation to +10. +• HetGNN: We set d = 128, and the batch size to 200 for +all datasets. For random walk, we set the walk length to +30 and the window size to 5. +• GCN: We set d = 64 and L = 2 for all datasets. +• GAT: For LastFM, We set d = 64, nh = 4, L = 3, and +s = 0.1. For DBLP, we set d = 64, nh = 8, L = 3, and +s = 0.05. For IMDB, we set d = 64, nh = 4, L = 5, +and s = 0.1. +• SimpleHGN: We set d = 64, de = 32, nh = 2, the edge +residual β = 0, and s = 0.01 for all datasets. For DBLP, +we set L = 3. For LastFM, we set L = 4. For IMDB, +we set L = 6. +C. Effects of the learning rate and the weight decay +We further evaluate the effect of the learning rate and +weight decay when optimizing the completion parameters α. +The available learning rates are set to [3e-3, 4e-3, 5e-3, 6e- +3, 7e-3]. The available weight decay values are set to [5e- + +95.6 +95.4 +95.2 +Peformance(%) +95.0 +SimpleHGN-MacroE1 +94.8 +SimpleHGN-Micro F1 +MAGNN-Macro F1 +94.6 +MAGNN-Macro F1 +94.4 +94.2 +94.0 +0.003 +0.004 +0.005 +0.006 +0.007 +The learning rate of94.0 +93.5 +93.0 +Peformance(%) +SimpeHGN-MacroE1 +92.5 +SimpleHGN-Micro F1 +MAGNN-Macro F1 +MAGNN-Macro F1 +92.0 +91.5 +91.0 +0.003 +0.004 +0.005 +0.006 +0.007 +The learning rate of68 +66 +Peformance(%) +64 +SimpleHGN-Macro F1 +SimpleHGN-Micro_F1 +62 +MAGNN-Macro F1 +MAGNN-Macro F1 +60 +58 +0.003 +0.004 +0.005 +0.006 +0.007 +The learning rate of95.50 +95.25 +95.00 +Peformance(%) +SimpleHGN-Macro F1 +94.75 +SimpleHGN-Micro_F1 +MAGNN-Macro F1 +94.50 +MAGNN-Macro F1 +94.25 +94.00 +93.75 +0.5 +1.0 +2.0 +3.0 +4.0 +The weight decay of a +1e-594.0 +93.5 +Peformance(%) +93.0 +SimpleHGN-Macro F1 +SimpleHGN-Micro_F1 +92.5 +MAGNN-Macro_F1 +MAGNN-Macro F1 +92.0 +91.5 +0.5 +1.0 +2.0 +3.0 +4.0 +The weight decay of a +1e-568 +66 +Peformance(%) +64 +SimpleHGN-Macro F1 +SimpleHGN-Micro F1 +62 +MAGNN-Macro_F1 +MAGNN-Macro F1 +60 +58 +0.5 +1.0 +2.0 +3.0 +4.0 +The weight decay of a +1e-56,1e-5, 2e-5, 3e-5, 4e-3]. Figure 10 and Figure 11 show +the performances of AutoAC with different learning rates +and different weight decay, respectively. The green and blue +lines represent SimpleHGN-AutoAC and MAGNN-AutoAC, +respectively. From Figure 10 and Figure 11, we can see that +AutoAC is very robust to the learning rate and the weight +decay. + diff --git a/ztE1T4oBgHgl3EQfRQMb/content/tmp_files/load_file.txt b/ztE1T4oBgHgl3EQfRQMb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca4bd7fac3491ceb767983e270763782122b2e70 --- /dev/null +++ b/ztE1T4oBgHgl3EQfRQMb/content/tmp_files/load_file.txt @@ -0,0 +1,1948 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf,len=1947 +page_content='AutoAC: Towards Automated Attribute Completion for Heterogeneous Graph Neural Network Guanghui Zhu, Zhennan Zhu, Wenjie Wang, Zhuoer Xu, Chunfeng Yuan, Yihua Huang State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing, China Department of Computer Science and Technology, Nanjing University, Nanjing, China {zhuzhennan, wenjie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='wang, zhuoer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='xu}@smail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='cn, {zgh, cfyuan, yhuang}@nju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='cn Abstract—Many real-world data can be modeled as het- erogeneous graphs that contain multiple types of nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Meanwhile, due to excellent performance, heterogeneous graph neural networks (GNNs) have received more and more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, the existing work mainly focuses on the design of novel GNN models, while ignoring another important issue that also has a large impact on the model performance, namely the missing attributes of some node types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The hand- crafted attribute completion requires huge expert experience and domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Also, considering the differences in semantic characteristics between nodes, the attribute completion should be fine-grained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', the attribute completion operation should be node-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, to improve the performance of the downstream graph learning task, attribute completion and the training of the heterogeneous GNN should be jointly optimized rather than viewed as two separate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To address the above challenges, we propose a differentiable at- tribute completion framework called AutoAC for automated completion operation search in heterogeneous GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We first propose an expressive completion operation search space, includ- ing topology-dependent and topology-independent completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, we propose a continuous relaxation schema and further propose a differentiable completion algorithm where the completion operation search is formulated as a bi-level joint optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To improve the search efficiency, we leverage two optimization techniques: discrete constraints and auxiliary unsupervised graph node clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Extensive experimental results on real-world datasets reveal that AutoAC outperforms the SOTA handcrafted heterogeneous GNNs and the existing attribute completion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Index Terms—heterogeneous graph, graph neural network, attribute completion, differentiable search I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' INTRODUCTION Graph-structured data are ubiquitous, such as social net- works [1], scholar networks [2], biochemical networks [3], and knowledge graphs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Meanwhile, many real-world graph data are heterogeneous [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Unlike the homogeneous graph with only one node type and one edge type, the hetero- geneous graph [6] consists of multiple types of nodes and edges associated with attributes in different feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For example, the IMDB dataset is a typical heterogeneous graph, which contains three node types (movie, actor, director) and two edge types (movie-actor, movie-director), as shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Due to containing rich information and semantics, heterogeneous graphs have drawn more and more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Recently, graph neural networks (GNNs) [7], [8] have demonstrated powerful representation learning ability on graph-structured data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Meanwhile, many heterogeneous GNNs (HGNNs) have been proposed for heterogeneous graphs [10] [11] [12] [13] [14] [15] [16] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, the existing work on heterogeneous graphs mainly focuses on the construction of novel GNN models, while ignoring another important issue that also has a large impact on the model performance, namely the attributes of some types of nodes are missing [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Missing node attributes is a common problem because collecting the attributes of all nodes is prohibitively expensive or even impossible due to privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Since the attributes of all nodes are required in the GNN-based heterogeneous models, some handcrafted ways are employed to deal with the problem of missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For example, the missing attribute vector can be the sum or the mean of directly connected nodes’ attribute vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Besides, the one- hot representations of a certain node type can also be used to replace the missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, the handcrafted ways require huge expert experience and domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Also, the topological relationships in the graph are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Recently, an attention-based method [18] was proposed to complete each no-attribute node by weighted aggregation of the attributes from the directly neighboring attributed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Such an attribute completion method only considers the attributes of 1-hop neighbors without exploiting the attributes of higher-order neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, existing attribute completion methods are all coarse-grained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' That is, for a specific node type without attributes, they adopt the same attribute completion operation for all nodes without considering the differences in semantic characteristics between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In practice, fine-grained at- tribute completion is more reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The attribute comple- tion operations for the nodes with different semantics should be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Take the IMDB dataset as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The target type of nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', movie nodes) has attributes, and the other types of nodes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', actor nodes and director nodes) have no attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As shown in Figure 1(b), there exist three attribute completion operations, including 1) For actors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Jackie Chan) who are involved in movies that mostly belong to the same genre (Kung Fu movies), average attribute aggregation of local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', 1-hop) neighboring nodes should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) For actors who have strong collaborative relationships with other actors and directors, the message-passing based multi- hop attribute aggregation is more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3) For guest actors without representative movies, we can directly use the simple one-hot encoding to complete attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='03049v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='LG] 8 Jan 2023 Movie (with attribute) director (attribute missing) actor (attribute missing) Romance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thriller?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Action?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' …?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' classification node (a) Jackie Chan Ruch Hour Police Story CZ12 The Foreigner Project A actor Ng Man-Tat actor Stephen Chow actor A Chinese Odyssey Li Gong actor Red Sorghum director Yi-Mou Zhang 2-hop 2-hop Faye Wong actor Chungking Express Alan Tam actor We Are Family singer (guest actor) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' (a) Example of heterogeneous graphs with incomplete attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', the IMDB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' (b) Different attribute completion operations for the actor node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', local attribute aggregation, message-passing based multi-hop attribute aggregation, and one-hot representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the IMDB dataset, the number of actor nodes that have no attributes is 6124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Manually differentiating the semantic characteristics of all no-attribute nodes and then selecting the most suitable completion operations according to seman- tic characteristics is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, an automated attribute completion method that can search the optimal completion operations efficiently is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, to improve the performance of the downstream graph learning task, the au- tomated attribute completion and the training of the heteroge- neous GNN should be jointly optimized rather than viewed as two separate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To address the above challenges, we propose a differentiable attribute completion framework called AutoAC1 for automated completion operation search in heterogeneous GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' AutoAC is a generic framework since it can integrate different hetero- geneous GNNs flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' By revisiting the existing attribute completion methods, we first propose an expressive com- pletion operation search space, including topology-dependent and topology-independent completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Instead of searching over the discrete space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', candidate completion operations for each no-attribute node), we propose a con- tinuous relaxation scheme by placing a weighted mixture of candidate completion choices, which turns the search task into an optimization problem regarding the weights of choices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', completion parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, due to the continuous search space, the search process becomes differentiable and we can perform completion operation searching via gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To further improve the search efficiency, we formulate the search of attribute completion operations and the training of GNN as a constrained bi-level joint optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifically, we keep the search space continuous in the optimization process of completion parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', upper- level optimization) but enforce attribute completion choices being discrete in the optimization process of weights in the heterogeneous GNN (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', lower-level optimization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In this way, there is only one activated completion operation for each no-attribute node during the training of GNN, removing the need to perform all candidate completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Inspired by NASP [19], we employ proximal iteration to solve the constrained optimization problem efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Finally, to reduce the dimension of the attribute completion parameters, we further leverage an auxiliary unsupervised 1AutoAC is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='com/PasaLab/AutoAC graph node clustering task with the spectral modularity func- tion during the process of GNN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To summarize, the main contributions of this paper can be highlighted as follows: We are the first, to the best of our knowledge, to model the attribute completion problem as an automated search problem for the optimal completion operation of each no- attribute node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We propose an expressive completion operation search space and further propose a differentiable attribute com- pletion framework where the completion operation search is formulated as a bi-level joint optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To improve search efficiency, we enforce discrete con- straints on completion parameters in the training of heterogeneous GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, we leverage an auxiliary unsupervised graph node clustering task to reduce the dimension of the attribute completion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Extensive experimental results on real-world datasets reveal that AutoAC is effective to boost the performance of heterogeneous GNNs and outperforms the SOTA at- tribute completion method in terms of performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Heterogeneous Graph Neural Network Graph neural network [8] [20] [1] [21] [22] [9] aims to extend neural networks to graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Since heterogeneous graphs are more common in the real world [5], heterogeneous GNNs have been proposed recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Part of the work is based on meta-paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HAN [10] leverages the semantics of meta- paths and uses hierarchical attention to aggregate neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' MAGNN [14] utilizes RotatE [23] to encode intermediate nodes along each meta-path and mix multiple meta-paths using hierarchical attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Another part of the work chooses to extract rich semantic information in heterogeneous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GTN [11] learns a soft selection of edge types and com- posite relations for generating useful multi-hop connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HetGNN [13] uses Bi-LSTM to aggregate node features for each type and among types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As the state-of-the-art model, SimpleHGN [17] revisits existing methods and proposes a simple framework using learnable edge-type embedding and residual connections for both nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Recently, AS-GCN [24] employs the heterogeneous GNN to mine the semantics for text-rich networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Different from the above methods, HGNN-AC [18] notices that most of the nodes in the real heterogeneous graph have missing attributes, which could cause great harm to the per- formance of heterogeneous models, and proposes an attention- based attribute completion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, HGNN-AC needs to get node embeddings based on network topology using metapath2vec [25], which is a time-consuming process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, the attribute completion in HGNN-AC is coarse- grained and supports only one completion operation for all no-attribute nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HGCA [26] unifies attribute completion and representation learning in an unsupervised heterogeneous network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' MRAP [27] performs node attribute competition in knowledge graphs with multi-relational propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Neural Architecture Search (NAS) NAS [28] that designs effective neural architectures auto- matically has received more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The core components of NAS contain search space, search algorithm, and perfor- mance estimation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Recently, many works use NAS to design GNN models due to the complexity of GNN [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' PolicyGNN [30] uses reinforcement learning to train meta- strategies and then adaptively determines the choice of ag- gregation layers for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' SANE [31] and SNAG [31] search for aggregation functions using microscope-based and reinforcement learning-based strategies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The architecture-level approaches such as GraphNAS [32], Au- toGNN [33], and PSP [34] aim to search for architectural representations of each layer, including sampling functions, at- tention computation functions, aggregation functions, and acti- vation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The above works are based on homogeneous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Due to the rich semantic and structural information in heterogeneous graphs, applying NAS to heterogeneous graphs is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Recently, there exist some excellent attempts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GEMS [35] uses the evolutionary algorithm to search for meta-graphs between source and target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' DiffMG [36] uses differentiable methods to find the best meta-structures in heterogeneous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, the above works only focus on the GNN model and ignore the heterogeneous graph data itself, which is even more important in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Proximal Iteration Proximal iteration [37] is used to handle the optimization problem with a constraint C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', minx f(x), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' x ∈ C, where f is a differentiable objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The proximal step is: x(k+1) = proxC � x(k) − ϵ∇f � x(k)�� proxC(x) = arg min z 1 2(∥z − x∥)2, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' z ∈ C (1) where ϵ is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Due to the excellent theoretical guarantee and good empirical performance, proximal iteration has been applied to many deep learning problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', archi- tecture search [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' PRELIMINARIES Heterogeneous Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Given a graph G = ⟨V, E⟩ where V and E denote the node set and the edge set respectively, G is heterogeneous when the number of node and edge types exceeds 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Each node v ∈ V and each edge e ∈ E are associated with a node type and an edge type respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Attribute Missing in Heterogeneous Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let xv ∈ Rd denote the original d-dimensional attribute vector in the node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In practice, the attributes of some types of nodes are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the node set V in G can be divided into two subsets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', V + and V −, which denote the attributed node-set and no-attribute node-set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Attribute Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let X = {xv | v ∈ V +} denote the input attribute set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Attribute completion aims to complete the attribute for each no-attribute node v ∈ V − by leveraging the available attribute information X and the topological structure of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let xC v denote the completed attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, after com- pletion, the node attributes for the training of heterogeneous GNN is Xnew = X ∪XC = {xv | v ∈ V +}∪{xc v | v ∈ V −}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In this paper, we aim to search for the optimal completion operation for each no-attribute node to improve the prediction performance of GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' THE PROPOSED METHODOLOGY In this section, We first present the proposed completion operation search space and then introduce the differentiable search strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, we introduce the optimization techniques including discrete constraints and the auxiliary unsupervised graph node clustering task for further improving the search efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Search Space of Attribute Completion Operation Due to the semantic differences between nodes, using a single attribute completion operation for all no-attribute nodes belonging to the same node type is not reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The avail- able completion operations should be diverse and we can select the most suitable completion operation for each node with missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, to capture both the node semantics and the topological structure information during the attribute completion process, we first propose an expressive completion operation search space, which consists of topology-dependent and topology-independent operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifically, the topology-dependent operations employ the topology information of the graph to guide the attribute completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Inspired by the node aggregation operations in typical GNNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', GraphSage [1], GCN [8], APPNP [38]), we design three topology-dependent attribute completion op- erations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', mean, GCN-based, PPNP-based operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In contrast, the topology-independent operation directly uses one- hot encoding to replace the missing attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' AutoAC aims to search the optimal operation for each no-attribute node from the general and scalable search space where we can draw on more node aggregation operations in GNNs as attribute completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 1) Topology-Dependent Completion Operation: Such type of completion operations can be further divided into two categories: local attribute aggregation and global (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', multi- hop) attribute aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Local Attribute Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Similar to the node aggregation in GraphSage [1], we first propose mean attribute aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Mean Attribute Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the node v ∈ V −, we calculate the mean of neighbors’ attributes to complete the missing attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The completed attribute xC v is as follows: xC v = W · mean � xu, ∀u ∈ N + v � (2) where N + v denotes the local (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e, 1-hop) neighbors of node v in set V +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' W is the trainable transformation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GCN-based Attribute Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Similar to spectral graph convolutions in GCN [8], we complete the missing attribute with the following renormalized graph convolution form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' xC v = � u∈N + v (deg(v) · deg(u))−1/2 · xu · W (3) Global Attribute Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Motivated by the node aggre- gation in APPNP [38], we propose PPNP-based completion operation for global attribute aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' PPNP-based Attribute Aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Besides the GCN-based attribute completion, we use another popular node aggregation method PPNP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', Personalized PageRank [38]) for attribute completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifically, let A ∈ Rn×n denote the adjacency matrix of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ˜A = A+In denotes the adjacency ma- trix with added self-loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The form of PPNP-based attribute completion is: Xppnp = α � In − (1 − α ˆ˜A) �−1 X′, X′ = X · W XC = {Xppnp i | ∀i ∈ V −} (4) where ˆ˜A = ˜D−1/2 ˜A ˜D−1/2 is the symmetrically normalized adjacency matrix with self-loops, with the diagonal degree matrix ˜D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' α ∈ (0, 1] is the restart probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Note that the missing attributes are filled with zeros in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' After PPNP-based attribute aggregation, we complete the attributes of the nodes in V − with Xppnp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Topology-Independent Completion Operation: For the no-attribute nodes that have few neighbors or are less af- fected by the neighbor information, we can directly use one- hot encoding to replace the missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The one-hot representation of a specific node type is also a commonly used handcrafted attribute completion method [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For ex- ample, there are K distinct actors in IMDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The one-hot representation for the actor node is a K-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For a specific actor, the element in the corresponding index is 1 and the others are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, the one-hot representation is transformed linearly for dimension alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3) Search Space Size Analysis: In summary, the proposed search space O contains a diverse set of attribute completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let N − denote the total number of nodes with missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the space size can be calculated by |O|N − , which is exponential to N −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In practice, the attribute missing of some node types is a common problem, leading to huge search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the block-box optimization-based search method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', evolutionary algorithm) over a discrete search space is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To address this issue, we propose a differentiable search strategy to find the optimal completion operations efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Differentiable Search Strategy In this section, we first introduce a continuous relaxation scheme for the completion operation search space to make the search process to be differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, we introduce the dif- ferentiable search algorithm and two optimization techniques to improve the search efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 1) Continuous Relaxation and Optimization: Inspired by the success of the differentiable NAS, we first design a contin- uous search space and then perform differentiable completion operation search via gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As shown in Equation 5, instead of searching over the discrete space, we view the completion operation as a weighted mixture of candidate choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' xC v = � o∈O exp � α(v) o � � o′∈O exp � α(v) o′ �o (v) (5) where v denotes the node with the missing attribute, o denotes the candidate operation in the search space O, o (v) denotes the completed attribute of node v with o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' α(v) indicates the mixing weight vector of dimension |O| for node v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Furthermore, we refer to α = {α(v) | v ∈ V −} ∈ RN −×|O| as the completion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' After continuous relaxation, the search objective becomes the learning of the completion parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To this end, we formulate the search problem as an optimization problem that can jointly learn the completion parameters α and the weights w in the heterogeneous GNN by gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let Ltrain and Lval denote the training loss and validation loss respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Since both losses are determined by the completion parameters α and the weights w, the search objective is a bi- level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' min α Lval (ω∗, α) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ω∗ = argminw Ltrain(ω, α) (6) where the upper-level optimization is for the optimal comple- tion parameters α and the lower-level optimization is for the optimal weights w in the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Overview: Figure 2 shows the overall framework of au- tomated attribute completion for heterogeneous graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' First, we perform a continuous relaxation of the search space by placing a mixture of candidate completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, the completion parameters α are optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' After determin- ing the attribute completion operations for each no-attribute node, we view the completed attributes together with the raw attributes as the initial embedding for the training of the graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' one-hot mean GCN PPNP Discrete search space for automated attribute completion Node with missing attributes ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='one-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='PPNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝜶𝟐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝜶𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝜶𝟑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝜶𝟒 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='one-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='GCN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='PPNP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Continuous relaxation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='by placing a mixture of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='candidate choices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Adding discrete constraints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='when training GNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Neural ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Softmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Softmax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Supervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝑴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Auxiliary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Unsupervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Joint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='GNN Training ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝑵 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝑴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝑴−𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Graph Clustering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝟑 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝑴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='𝑪𝑴−𝟏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='attributes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='attributes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='after completion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Attribute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Completion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='attributes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='missing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The overall workflow of automated attribute completion for the heterogeneous graph neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Why not use the weighted mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Although the continuous relaxation allows the search of completion operations to be dif- ferentiable, there still exist following limitations when directly using the weighted mixture of all completion operations: 1) High computational overhead: After continuous relax- ation, we need to perform all candidate completion operations for each no-attribute node when training het- erogeneous GNNs, leading to huge computational over- head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Also, solving the bi-level optimization problem in Equation 6 incurs significant computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Performance gap: At the end of the search, con- tinuous parameters α needs to be discretized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', argmaxo∈O α(v) o , resulting in inconsistent performance between searched and final completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3) Large dimension of α: The dimension of completion parameters α is N − × |O|, which is proportional to the total number of nodes with missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The large dimension of α leads to a slow convergence rate and low search efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To address the first two issues (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', reducing computational overhead and avoiding performance gap), we first propose an efficient search algorithm with discrete constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifi- cally, for each no-attribute node v, the completion parameters satisfy the following constraints: α(v) ∈ C = C1 ∩ C2, where C1 = {α(v) | ∥α(v)∥0 = 1}, C2 = {α(v) | 0 ≤ α(v) i ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The constraint C2 allows α to be optimized continuously, and C1 keeps the choices of completion operation to be discrete when training GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As shown in Figure 2, there is only one activated edge for each choice when training GNN, removing the need to perform all candidate completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The final completion operation is derived from the learned com- pletion parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For node v, the edge with the maximum completion parameter will be kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We leverage proximal iteration [37] to solve the constrained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, proximal iteration can improve the computational efficiency of optimizing α without second-order derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, to address the third issue (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', reducing the dimension of α), we propose an auxiliary unsupervised clus- tering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In practice, the no-attribute nodes with similar semantic characteristics may have the same completion opera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Take the actor nodes in the IMDB dataset as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the actors with a large number of representative movies, the average attribute aggregation operation is more suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, we can cluster all no-attribute nodes into M clusters, where the nodes in each cluster have the same completion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The optimization goal becomes to search for the optimal attribute completion operation for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In this way, the size of the completion parameters α is reduced from N − × |O| to M × |O|, M ≪ N −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As shown in Figure 2, the auxiliary unsupervised clustering loss can be jointly optimized with the node classification loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', cross-entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The proposed framework AutoAC is composed of multiple iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In each iteration, the completion parameters α and the weights in the GNN are optimized alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Next, we introduce the search algorithm with discrete constraints and the auxiliary unsupervised clustering task in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Search Algorithm with Discrete Constraints Equation 6 implies a bi-level optimization problem with α as the upper-level variable and w as the lower-level variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Following the commonly used methods in meta learning [39] and NAS [40], we use a one-step gradient approximation to the optimal internal weight parameters ω∗ to improve the Algorithm 1 Search Algorithm in AutoAC 1: Initialize completion parameters α according to defined search space O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2: while not converge do 3: Get discrete choices of attribute completion operations: ¯α(k) = proxc1(α(k)) 4: Update α for continuous variables: α(k+1) = proxc2(α(k) − ϵ∇¯α(k)Lval(ω(k), ¯α(k))) 5: Refine discrete choices after updating: ¯α(k+1) = proxc1(α(k+1)) 6: Update ω(k) by ∇ω(k)Ltrain(ω(k), ¯α(k+1)) 7: end while efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the gradient of the completion parameters α is as follows (we omit the step index k for brevity): ∇αLval (ω∗, α) ≈∇αLval (ω − ξ∇ωLtrain(ω, α), α) =∇αLval (ω′, α) − ξ∇2 α,ωLtrain(ω, α)∇ω′Lval (ω′, α) (7) where ω is the weights of the GNN, ξ is the learning rate of internal optimization, and ω′ = ω − ξ∇ωLtrain(ω, α) indicates the weights for a one-step forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' we update the completion parameters α to minimize the validation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In Equation 7, there exists a second-order derivative, which is expensive to compute due to a large number of param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Also, the continuous relaxation trick further leads to huge computational overhead since all candidate completion operations need to be performed when training the GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, the overall search process is divided into two stages: search and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In the evaluation stage, the continuous completion parameters α need to be discretized for replacing every mixed choice as the most likely operation by taking the argmax, leading to performance gap between the search and evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To optimize α efficiently and avoid the performance gap, we propose a search algorithm with discrete constraints when optimizing completion parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the no-attribute node v, let the feasible space of α(v) be C = {α(v) | ∥α(v)∥0 = 1 ∧ 0 ≤ α(v) i ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We denote it as the intersection of two feasible spaces (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', C = C1 ∩ C2), where C1 = {α(v) | ∥α(v)∥0 = 1}, C2 = {α(v) | 0 ≤ α(v) i ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The optimization problem under constraints can be solved by the proximal iterative algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Proposition 1: proxC(z) = proxC2(proxC1(z)) Inspired by Proposition 1 [19], [37], in the k-th proximal iteration, we first get discrete variables constrained by C1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', ¯α(k) = proxC1(α(k)) (the node notation v is omitted for brevity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, we derive gradients w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='t ¯α(k) and keep α to be optimized as continuous variables but constrained by C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' α(k+1) = proxc2(α(k) − ϵ∇¯α(k)Lval(¯α(k))) (8) The detailed search algorithm is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' First, we get a discrete representation of α by proximal step (Line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, we view ω(k) as constants and optimize α(k+1) for continuous variables (Line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Since there is no need to compute the second-order derivative, the efficiency of updating α can be improved significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' After updating α, we further refine discrete choices and get ¯α(k+1) for updating ω(k) on the training dataset, which contributes to reducing the performance gap caused by discretizing completion parameters α from con- tinuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, since only one candidate choice is activated for each no-attribute node, the computational overhead can also be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The computational efficiency of updating α can be significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Auxiliary Unsupervised Clustering Task As mentioned before, the dimension of the completion parameters α is N − × |O| (|O| ≪ N −, |O| = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Take the DBLP dataset as an example, the number of nodes with missing attributes is about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2 × 104, leading to a large dimension of completion parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As a result, optimizing α with a limited size of validation dataset is very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Inspired by the observation that the no-attribute nodes with similar explicit topological structure or implicit semantic characteristics, we further propose an auxiliary unsupervised clustering task to divide all no-attribute nodes into M clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In each cluster, all nodes share the same completion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In this way, the dimension of the completion parameters α can be reduced to M ×|O|, M ≪ N −, and optimizing α becomes feasible and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' It is well known that the EM algorithm [41] is a commonly used method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', K-Means [42]) to solve the problem of un- supervised clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In the scenario of graph node clustering, let hv denote the hidden node representation learned by the heterogeneous GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The E-step is responsible for assigning the optimal cluster for each node v by calculating the distances between hv and all cluster centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The M-step is used to update the centers of all clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The E-step and M-step are performed alternately until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Although the EM algorithm has a convergence guarantee, it is sensitive to the initial values, making it difficult to apply to the proposed automated completion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The main reason is that the bi-level optimization problem defined in Equation 6 is iterative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In the early optimization process, the weights of the GNN have not yet converged and the node representations learned in the GNN are less informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Such low-quality representations lead to inaccurate clustering, which has a negative impact on the subsequent clustering quality and further leads to a deviation from the overall optimization direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To address this issue, we first formulate the problem of unsupervised node clustering as a form of soft classification, and use the assignment matrix C to record the probability of each node belonging to each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, as shown in Figure 2, we embed the clustering process into the bi-level iterative optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Motivated by graph pooling and graph module partitioning, we introduce the Spectral Modularity Function Q [43] [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' From a statistical perspective, this function can reflect the clus- tering quality of graph node modules through the assignment matrix C [45]: Q = 1 2 |E| � ij � Aij − didj 2 |E| � δ (ci, cj) (9) where |E| is the number of edges in the graph, δ(ci, cj) = 1 only if nodes i and j are in the same cluster, otherwise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' di and dj represent the degrees of node i and node j respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' It can be known that in a random graph, the probability that node i and node j are connected is didj 2|E| [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, the optimization goal is converted into maximizing the spectral modularity function Q, but it is an NP-hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Fortunately, this function can be represented by an approximate spectral domain relaxation form: Q = 1 2 |E| Tr � C⊤BC � (10) where Cij ∈ [0, 1] denotes the cluster probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' B is the modular matrix B = A − dd⊤ 2|E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Finding the optimal solution of the assignment matrix C is to maximize Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To prevent falling into local optimum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', all nodes tend to be in the same cluster), we further add the collapse regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The assignment matrix C should be amortized as adaptively as possible, so as to skip the local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let LGmoC denote the unsupervised clustering loss, which can be expressed as: LGmoC = − 1 2 |E| Tr � C⊤BC � � �� � modularity loss + √ M |V | ����� � i C⊤ i ����� F � �� � collapse regularization (11) where |V | is the number of nodes, M is the number of clusters, ∥ · ∥F represents the Frobenius norm of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Note that LGmoC can be jointly optimized with the supervised classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifically, LGmoC can be used as an aux- iliary task for the bi-level optimization problem in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The unsupervised clustering loss is added to Ltrain for joint optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Let λ denote the loss-weighted coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The optimization objective is updated as: min α Lval (w∗, α) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' w∗ = arg min w (Ltrain(w, α) + λLGmoC) (12) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Complexity Analysis In the heterogeneous graph G = ⟨V, E⟩, the total number of nodes is N, the total number of nodes with missing attributes is N −, and the embedding dimension is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In each iteration of 12, we can divide the search process of AutoAC into three phases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', attribute completion phase, upper-level optimization for completion parameters α, and lower-level optimization for weights ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We first analyze the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Since discrete constraints are performed, only one candidate completion operation is activated for each no-attribute node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='The computational complexity of each completion operation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='TABLE I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='STATISTICS OF THE DATASETS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Datasets #Nodes #Node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Types ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='#Nodes under ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Each Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='#Edges Target Node/Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='Attribute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='DBLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='26128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='author(A):4057 ' 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+page_content='T:Missing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='is as follows: Mean attribute aggregation: O(N − × k2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GCN-based attribute aggregation: O(N − × k2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' PPNP-based attribute aggregation: O(N×k2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' one-hot attribute completion: O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the computational complexity of the attribute completion phase is O(N×k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In the upper-level optimization phase, the complexity is O(CH + |O| × M × bα), where CH denotes the forward computation overhead of the het- erogeneous GNN, bα the gradient computation overhead for each completion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For brevity, we omit the difference between the validation and training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The lower-level optimization phase contains the optimization of weights and unsupervised clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The complexity of optimizing ω is O(CH +|ω|×bω), where bω is the gradient computation over- head for each weight parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The complexity of calculating the clustering loss LGmoC is O(d2 × N + |E|) [45], where d is the average degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Overall, the computational complexity of each iteration is, O(N × k2) + O(CH + |O| × M × bα) + |ω| × bω) + O(d2 × N + |E|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Next, we analyze the space complexity of AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the attribute completion phase, the space complexity is O(k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the optimization phase, the space complexity is O(N × k + |O| × M + |ω| + N × M), where O(N × M) is the space complexity in the unsupervised clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Experimental Setup 1) Experimental Setting: We use the recently proposed Heterogeneous Graph Benchmark (HGB) [17] to conduct all experiments, which offers a fair way to compare heterogeneous GNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HGB gives a set of standard benchmark datasets and unified strategies for feature preprocessing and data split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In the node classification task, all edges are available during training, and node labels are split according to 24% for training, 6% for validation, and 70% for test in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In the link prediction task, we mask 10% edges of the target link type and the negative edges are randomly sampled The statistics of the four datasets are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' More details of datasets can be seen in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, the handcrafted attribute completion methods for existing heterogeneous GNNs are provided by HGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Micro-F1 and Macro-F1 are provided to evaluate the node classification performance, while the MRR and ROC-AUC metrics are used for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The evaluation metrics are obtained by submitting predictions to the HGB website2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Implementation Details All experiments are performed in the transductive setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We employ the Adam optimizer [46] to optimize both ω and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For optimizing ω, the learning rate and the weight decay are 5e-4 and 1e-4 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For optimizing α, the learning rate and the weight decay are 5e-3 and 1e-5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We implement AutoAC based on the widely-used hetero- geneous GNNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', MAGNN [14] and SimpleHGN [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The loss weighted coefficient λ and the number of clusters M are two hyperparameters of AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For MAGNN, we empirically set λ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 for all datasets, M to 4 for the DBLP and ACM datasets, 16 for the IMDB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For SimpleHGN, λ is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4 for all datasets, and M is 8 for the DBLP dataset, 12 for the ACM and IMDB datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, all the GNN models are implemented with PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' All experiments are run on a single GPU (NVIDIA Tesla V100) five times and the average performance and standard deviation are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Effectiveness of AutoAC 1) Performance comparison with humancrafted heteroge- neous GNNs: Depending on whether or not the meta-path is used, we divide the humancrafted heterogeneous GNNs into two categories: GNNs with meta-path: HAN [13], GTN [11], Het- SANN [16], MAGNN [14], HGCA [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GNNs without meta-path: HGT [15], GATNE [47], Het- GNN [13], GCN [8] and GAT [20] (two commonly used general-purpose GNNs), as well as the current SOTA GNN model SimpleHGN [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The configurations of baselines can be seen in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Since AutoAC is general and thus can be integrated into different GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We select two representative GNN models from the two categories (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', MAGNN and SimpleHGN) from the perspective of performance and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Then, we combine AutoAC with the two models, denoted by MAGNN-AutoAC and SimpleHGN-AutoAC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Table II shows the performance comparison between Au- toAC and existing heterogeneous GNNs on node classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' AutoAC can improve the performance of MAGNN and SimpleHGN stably on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The performance gain obtained by AutoAC over MAGNN is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='7%-3% and the error rate is reduced by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='87%-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='69%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Also, SimpleHGN- AutoAC outperforms SimpleHGN by 1%-3% and reduces the error rate by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='59%-22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='09%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' By combining with the SOTA model SimpleHGN, SimpleHGN-AutoAC can achieve the best performance in all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='biendata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='xyz/competition/hgb-1/ Moreover, Table II shows that AutoAC can bring signif- icant performance improvement on the datasets where the classification target nodes have no raw attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', DBLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Besides, for the datasets where the target nodes already have raw attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', ACM and IMDB), completing other non- target nodes using AutoAC can still promote the classification accuracy of target nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Especially, for the IMDB dataset, since there are too many non-target nodes with missing at- tributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', 77% of all nodes), the performance improvement with AutoAC is more significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Note that the performance of MAGNN without attribute completion is not as good as other models, such as GTN and GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, MAGNN-AutoAC performs better than GTN on DBLP and ACM, and outperforms GAT on DBLP and IMDB, which indicates that effective attribute completion for heterogeneous graphs can compensate for the performance gap introduced by the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' By unifying attribute completion and representation learning in an unsupervised heterogeneous network, the recently proposed HGCA can also achieve competitive performance on DBLP and ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Such experimental results further verify the necessity of AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Performance comparison with the existing attribute com- pletion method HGNN-AC: As the current SOTA attribute completion method, HGNN-AC [18] uses the attention mech- anism to aggregate the attributes of the direct neighbors for the nodes with missing attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The attention information is calculated by the pre-learning of topological embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To be fair, both AutoAC and HGNN-AC are evaluated under the unified HGB benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' And, we also combine HGNN- AC with MAGNN and SimpleHGN, denoted by MAGNN- HGNNAC and SimpleHGN-HGNNAC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Table III shows that AutoAC outperforms HGNN-AC on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifically, MAGNN-AutoAC achieves 1%- 4% performance improvement over MAGNN-HGNNAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the SimpleHGN model, SimpleHGN-AutoAC outperforms SimpleHGN-HGNNAC by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4%-2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, the perfor- mance improvement of HGNN-AC for attribute completion is not stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As shown in Table III, after attribute completion with HGNN-AC, MAGNN-HGNNAC is instead inferior to MAGNN on the three datasets, while MAGNN-AutoAC can achieve significant performance improvement with attribute completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Similarly, there is a degradation in performance on the DBLP dataset compared to SimpleHGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3) Performance comparison on link prediction: To verify the effectiveness of AutoAC on different downstream tasks, we further conduct link prediction in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' AutoAC can greatly improve the performance of heterogeneous GNNs, especially on IMDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' With AutoAC, MRR and ROC-AUC of SimpleHGN are increased by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='7% and 28%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In summary, AutoAC achieves better performance and more stable performance improvement, indicating the effectiveness of searching for the most suitable attribute completion opera- tions for no-attribute nodes from a diverse search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' TABLE II PERFORMANCE AND RUNTIME (CLOCK TIME IN SECONDS) COMPARISON BETWEEN AUTOAC AND SOTA HUMANCRAFTED HETEROGENEOUS GNNS ON NODE CLASSIFICATION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' THE BOLD AND THE UNDERLINE INDICATE THE BEST AND THE SECOND BEST IN EACH CATEGORY (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', USING AND NOT USING META-PATH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' * INDICATES THE GLOBAL BEST IN ALL MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' p-VALUE INDICATES THE STATISTICALLY SIGNIFICANT IMPROVEMENT (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', T-TEST WITH p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='05) OVER THE BEST BASELINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Dataset DBLP ACM IMDB Macro-F1 Micro-F1 Runtime (Total) Runtime (Per epoch) Macro-F1 Micro-F1 Runtime (Total) Runtime (Per epoch) Macro-F1 Micro-F1 Runtime (Total) Runtime (Per epoch) HAN 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='19 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='64±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='17 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='23 87.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='8 × e − 6 TABLE III PERFORMANCE COMPARISON BETWEEN AUTOAC AND HGNNAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='THE BOLD AND THE UNDERLINED INDICATE THE BEST AND THE SECOND BEST IN EACH CATEGORY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' p-VALUE INDICATES THE STATISTICALLY SIGNIFICANT IMPROVEMENT (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', T-TEST WITH p < 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='58 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='41 p-value 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='9 × e−8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='3 × e−9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='3 × e−7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='1 × e−6 4 × e−3 1 × e−3 TABLE IV THE OVERALL RUNTIME OVERHEAD (CLOCK TIME IN SECONDS) OF AUTOAC AND HGNN-AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' / INDICATES THAT THE STAGE IS NOT INVOLVED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Datasets Models End-to-End Runtime Overhead (Seconds) Speedup Pre-learn Search Train/Retrain Total DBLP SimpleHGN-HGNNAC 33048 / 432 33480 465× SimpleHGN-AutoAC / 36 36 72 MAGNN-HGNNAC 33048 / 900 33948 78× MAGNN-AutoAC / 72 360 432 ACM SimpleHGN-HGNNAC 3888 / 432 4320 40× SimpleHGN-AutoAC / 72 36 108 MAGNN-HGNNAC 3888 / 1260 5148 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5× MAGNN-AutoAC / 432 252 684 IMDB SimpleHGN-HGNNAC 8568 / 324 8892 123× SimpleHGN-AutoAC / 36 36 72 MAGNN-HGNNAC 8568 / 180 8748 15× MAGNN-AutoAC / 504 72 576 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Efficiency Study Besides the effectiveness, we also evaluate the efficiency of AutoAC in the terms of runtime overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Table II and V show the runtime of AutoAC and other handcrafted HGNNs on node classification and link prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Although the attribute completion and GNN training are jointly optimized in AutoAC, the computational efficiency of AutoAC is still competitive compared to other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Also, we compare AutoAC with the existing attribute com- pletion method HGNN-AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Table IV shows the efficiency comparison between AutoAC and HGNN-AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' AutoAC con- tains the search and retraining stages, and HGNN-AC contains the pre-learning and training stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We can see that AutoAC is much more efficient than HGNN-AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The end-to-end runtime overhead of AutoAC can be reduced by 15× to 465×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The main reason why HGNN-AC is inefficient is that the pre- leaning stage that learns a topological embedding for each node is very time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Especially for the DBLP dataset with a large number of nodes, the pre-learning overhead is up to 9 GPU hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In contrast, there is no additional pre-leaning stage in AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, by introducing the discrete constraints and auxiliary unsupervised clustering task, the search efficiency can be improved significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In summary, AutoAC can not only achieve better perfor- mance but also demonstrate higher computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' TABLE V PERFORMANCE AND RUNTIME (CLOCK TIME IN SECONDS) COMPARISON ON LINK PREDICTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' THE BOLD AND THE UNDERLINE INDICATE THE BEST AND THE SECOND BEST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' p-VALUE INDICATES THE STATISTICALLY SIGNIFICANT IMPROVEMENT (I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', T-TEST WITH p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='05) OVER THE BEST BASELINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Dataset LastFM DBLP IMDB Model \\ Metrics ROC-AUC MRR Runtime (Total) Runtime (Per epoch) ROC-AUC MRR Runtime (Total) Runtime (Per epoch) ROC-AUC MRR Runtime (Total) Runtime (Per epoch) GATNE 66.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='30±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='35 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='74±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='00 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='10 SimpleHGN 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='37 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='27 46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='35 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='11 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='16 58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='75 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='32 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='09 ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='40 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='44 SimpleHGN-AutoAC 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='17 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='19 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='43 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='66 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21 61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='87 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='73 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='49 p-value 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='3 × e−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 × e−4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2 × e−5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2 × e−7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='7 × e−9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2 × e−10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Ablation Study 1) Study on the necessity of searching attribute completion operations from a diverse search space: We compare AutoAC with the following two methods: Single-operation attribute completion: We complete all no-attribute nodes with the same single completion operation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', GCN AC, PPNP AC, MEAN AC, and One-hot AC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Random attribute completion: For each no-attribute node, we randomly select an attribute completion opera- tion from the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Table VI and Table VII show the completion operation ablation study on SimpleHGN and MAGNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Due to the differences in the data characteristics, there is no single completion operation that can perform well on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' By searching the optimal attribute completion operations AutoAC can achieve the best performance on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Take SimpleHGN shown in Table VI for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GCN AC is more effective on DBLP and IMDB, while PPNP AC performs better on ACM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, for a specific attribute completion operation, the performance is related to the dataset and the chosen GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We take DBLP as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GCN AC performs better on SimpleHGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' However, when the GNN model becomes MAGNN, GCN AC is not as good as MEAN AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Additionally, the performance of the random attribute completion is not stable and can be even worse than the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Choosing an inappropriate completion operation can have a negative effect on the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Study on the search algorithm with discrete constraints: When optimizing the attribute completion parameters α, we enforce discrete constraints on α and solve the bi-level op- timization problem with proximal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To verify the effectiveness of discrete constraints, we further run AutoAC with and without discrete constraints in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The search algorithm with discrete constraints can achieve better performance with less search time overhead on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Additionally, proximal iteration allows removing the need for second-order derivative in solving the bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the memory overhead can also be reduced significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='As shown in Table VIII, the memory overhead of MAGNN-AutoAC without discrete constraints is huge and the out-of-memory error occurs on DBLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3) Study on the auxiliary unsupervised clustering: To re- duce the dimension of the completion parameters α, we (a) SimpleHGN (b) MAGNN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Performance comparison between different clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' (a) DBLP (b) ACM (c) IMDB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Convergence of LGmoC on three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' leverage an auxiliary unsupervised clustering task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Figure 3 shows the performances of different clustering methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' w/o cluster: We directly search the attribute completion operations for each no-attribute node without clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' EM: After each iteration of the optimization process, we adopt the EM algorithm for clustering according to node representation learned by the GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' EM with warmup: a variant of the EM algorithm, which adds a warm-up process at the beginning of the clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In Figure 3, AutoAC can achieve the best performance on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Searching completion operations without clustering yields relatively poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Reducing the dimension of α with unsupervised clustering is very necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, Ittt Baseline Performance Comparison w/o Cluster EM X EM with Warmup AutoAC DBLP-Macro-F1 ACM-Macro-F1 IMDB-Macro-F1 DBLP-Micro-F1ACM-Micro-F1 IMDB-Micro-F1 Dataset-MetricIttt Baseline Performance Comparison / / w/o Cluster EM X EM with Warmup AutoAC IMDB-Micro-F1 Dataset-MetricDBLPACMIMDBTABLE VI COMPLETION OPERATION ABLATION STUDY ON SIMPLEHGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' BOLD INDICATES THE GLOBAL BEST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' UNDERLINE INDICATES THE BEST AMONG ALL SINGLE ATTRIBUTE COMPLETION OPERATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Dataset DBLP ACM IMDB Model \\ Metrics Macro-F1 Micro-F1 Macro-F1 Micro-F1 Macro-F1 Micro-F1 Baseline (SimpleHGN) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='19 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='68 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='98±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='66 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42 GCN AC 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='88±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='23 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='47 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='94 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='53 PPNP AC 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='76±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='24 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='58±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='23 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='46 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='48 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='36±19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='31 61.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='61 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='98±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='63 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='97±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='15 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='77 AutoAC 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='30 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='96±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='31 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='53 TABLE VIII ABLATION STUDY ON DISCRETE CONSTRAINTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' / INDICATES MEMORY OVERFLOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Dataset DBLP ACM IMDB Model \\ Metrics Macro-F1 Micro-F1 Search Time (Seconds) Macro-F1 Micro-F1 Search Time (Seconds) Macro-F1 Micro-F1 Search Time (Seconds) SimpleHGN-AutoAC 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='29 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='26 32 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 72 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='58 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='41 36 w/o Discrete constraints 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='27 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25 216 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='43±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='74 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='76 360 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='68 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='52 180 MAGNN-AutoAC 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='30 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25 72 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 432 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='96±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='31 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='53 504 w/o Discrete constraints / / / 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='68 1800 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='44±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='12 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='34 1908 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Distribution of searched attribute completion operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' (a) Author (b) Subject (c) Term Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Detailed distribution of searched completion operations for each no- attribute node type on the ACM dataset using SimpleHGN-AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' the proposed unsupervised clustering method outperforms EM and its variant, indicating the effectiveness of the joint (a) Actor (b) Director (c) Keyword Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Detailed distribution of searched completion operations for each no- attribute node type on the IMDB dataset using SimpleHGN-AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' optimization of the unsupervised clustering loss and the clas- sification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Figure 4 also shows the convergence of the unsupervised clustering loss LGmoC, which exhibits a stable decreasing trend during the optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Distribution of Searched Completion Operations Figure 5 shows the proportion of attribute completion operations searched by SimpleHGN-AutoAC and MAGNN- AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For different models and datasets, the proportions of searched completion operations are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In SimpleHGN-AutoAC, DBLP tends to select GCN AC, while ACM prefers PPNP AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the same dataset, different GNNs also result in different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Take DBLP as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' MAGNN-AutoAC is more inclined to MEAN AC 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='57% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='68% 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='09%12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='50% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='29% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='29% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='93%GCN_AC PPNP_AC MEAN AC One-hot Ac 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='74% 2:63% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='10%57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45% 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='83% 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='94% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='78%56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='87% 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='03% 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='70% 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='40%GCN AC PPNP AC MEAN AC One-hot_AC 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='90% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='13% 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='76% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='22%GCN_AC PPNP_AC MEAN_AC One-hot AC 80 Percentage(%) 60 40 P 20 0 SimpleHGN_ACM SimpleHGN_IMDB MAGNN_DBLP MAGNN_IMDB SimpleHGN DBLP MAGNN_ACM Model Dataset(a) DBLP (b) ACM (c) IMDB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Performance comparison under different M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' than GCN AC compared to SimpleHGN-AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The results further indicate the necessity of searching for suitable attribute completion operations under different datasets and GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Figure 6 and Figure 7 show the proportion of searched completion operations for each no-attribute node type on ACM and IMDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For ACM, multiple different completion operations are selected even for the same node type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Specifically, more than half of the author and subject nodes choose PPNP AC, while the proportions of other three operations are quite sim- ilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Most term nodes are assigned PPNP AC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='74%), indicating that the term type is more likely to capture the global information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The main reason is that the target node type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', paper) with raw attributes in ACM contains only the paper title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The high-order PPNP AC operations are preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In contrast, GCN AC accounts for the majority of completion operations on IMDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' This is because that the target node type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', movie) has raw attributes and contains rich features, such as length, country, language, likes of movies, and ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, the local completion operation GCN AC is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Next, we analyze the completion operations of concrete actor nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In IMDB, node No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='10797 is the actor Leonardo DiCaprio, who has starred in 22 movies, and the neighbor- hood information is very rich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' As a result, AutoAC chooses GCN AC for him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In contrast, node No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='10799 is the actor Leonie Benesch, who has appeared in only one movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Thus, one-hot AC is automatically selected by AutoAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Hyperparameter Sensitivity 1) Effect of the number of clusters M: Figure 8 shows the performance of AutoAC under different M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Both SimpleHGN- AutoAC and MAGNN-AutoAC can achieve stable perfor- mance, showing that AutoAC has sufficient robustness to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Effect of the loss weighted coefficient λ: We further evaluate the weighted coefficient λ of the auxiliary unsuper- vised clustering loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The available values of λ are set to [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Figure 9 shows the performances of AutoAC under different λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' IMDB is very robust to λ, and the performance change is very insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4 and λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 are suitable for SimpleHGN and MAGNN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For ACM, the choice of λ is slightly sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The effects of the learning rate and the weight decay can be seen in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Impacts of Attribute Missing Rates and Masked Edge Rates 1) Study on the performance of the same dataset with varying attribute missing rates in the node classification task: Table IX shows the performance of SimpleHGN-AutoAC with varying attribute missing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We change attribute miss- ing rates by completing the missing attributes with one-hot (a) DBLP (b) ACM (c) IMDB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Performance comparison under different λ TABLE IX PERFORMANCE OF SIMPLEHGN-AUTOAC WITH VARYING ATTRIBUTE MISSING RATES IN THE NODE CLASSIFICATION TASK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Datasets Attribute Missing Rates Node Types with Missing attributes Macro-F1 Micro-F1 DBLP 0% / 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='19 15% author 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='35±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='17 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='16 30% term, venue 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='13 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='12 45% author, term, venue 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='29 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='26 ACM 0% / 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='85±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='68 17% subject, term 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='10±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='27 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='26 54% author, subject 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='20 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='47±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21 69% author, subject, term 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='18 IMDB 0% / 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='98±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='66 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42 37% keyword 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='57 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='36 67% actor, keyword 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='53 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42 76% director, actor, keyword 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='58 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='41 TABLE X PERFORMANCE OF SIMPLEHGN-AUTOAC WITH VARYING MASKED EDGE RATES IN THE LINK PREDICTION TASK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Datasets Masked Edge Rates Models ROC-AUC MRR DBLP 5% SimpleHGN 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='56 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='44 SimpleHGN-AutoAC 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='36 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='02±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='24 10% SimpleHGN 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='11 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='16 SimpleHGN-AutoAC 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='66 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21 20% SimpleHGN 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='34±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='61 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='41 SimpleHGN-AutoAC 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='72 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='33 30% SimpleHGN 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='66 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='24 SimpleHGN-AutoAC 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='42±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='44 IMDB 5% SimpleHGN 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='89±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='58 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='94 SimpleHGN-AutoAC 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='57±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='36 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='84 10% SimpleHGN 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='92±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='32 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='09±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='40 SimpleHGN-AutoAC 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='73 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='45 20% SimpleHGN 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='39 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='71±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='34 SimpleHGN-AutoAC 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='82 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='32 30% SimpleHGN 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='79 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='57±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='67 SimpleHGN-AutoAC 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='31 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='23±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='21 encoding, which is a commonly used handcrafted attribute completion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' A missing rate of 0% means that all missing attributes are completed manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 45%, 69%, and 76% are inherent attribute missing rates of DBLP, ACM, and IMDB, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=', only one node type has raw attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' From Table IX, we can see that SimpleHGN-AutoAC performs better with higher missing rates, indicating that AutoAC is capable of searching for the suitable completion operation for each no-attribute node and the searched completion operations are superior to the handcrafted completion method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Study on the performance of the same dataset with varying masked edge rates in the link prediction task: Table X shows the performance of SimpleHGN-AutoAC with varying masked edge rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The edges are masked randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We can see that SimpleHGN-AutoAC achieves better performance than SimpleHGN at different masked edge rates, especially on the IMDB dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, the performance of both models decreases as the masked edge rate increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 Peformance(%) SimpeHGN-MacroE1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 SimpleHGN-Micro_F1 MAGNN-Macro F1 MAGNN-Macro F1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 4 8 12 16 M93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 Peformance(%) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 SimpleHGN-Macro F1 SimpleHGN-Micro_F1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 MAGNN-Macro F1 MAGNN-Macro F1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 4 8 12 16 M68 66 Peformance(%) 64 SimpleHGN-MacroE1 SimpleHGN-Micro_F1 MAGNN-Macro F1 62 MAGNN-MacroF1 60 58 4 8 12 16 MSimpleHGN-MacroF1 SimpleHGN-Micro F1 MAGNN-MacroF1 MAGNN-MacroF1★SimpleHGN-Macro F1 SimpleHGN-Micro_F1 MAGNN-Macro_F1 MAGNN-Macro F1SimpleHGN-Macro F1 SimpleHGN-Micro_F1 MAGNN-Macro_F1 MAGNN-Macro_F1VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a differentiable attribute com- pletion framework called AutoAC for automated completion operation search in heterogeneous GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' First, we introduced an expressive completion operation search space and proposed a continuous relaxation scheme to make the search space dif- ferentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Second, we formulated the completion operation search as a bi-level joint optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' To improve search efficiency, we enforced discrete constraints on com- pletion parameters and further proposed a proximal iteration- based search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Moreover, we leveraged an auxiliary unsupervised node clustering task to reduce the dimension of completion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Extensive experimental results reveal that AutoAC is effective to boost the performance of heteroge- neous GNNs and outperforms the SOTA attribute completion method in terms of performance and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' REFERENCES [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Hamilton, Z.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' [47] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Cen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Zou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Zhou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Tang, “Representation learning for attributed multiplex heterogeneous network,” in Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 1358–1368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' [48] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Cantador, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Brusilovsky, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Kuflik, “Second workshop on infor- mation heterogeneity and fusion in recommender systems (hetrec2011),” in Proceedings of the fifth ACM conference on Recommender systems, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 387–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Details of Datasets DBLP3 is a computer science bibliography website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The raw attribute of the paper node is the bag-of-words representation of keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' ACM4 is a citation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The raw attribute of the paper node is also the bag-of-words representation of keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' IMDB5 is a website about movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The attributes of movie nodes are originally present, they are represented by the bag-of-words representation of words extracted for key episodes of movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' LastFM is extracted from last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='fm with timestamps from January 2015 to June 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We use the subset released by [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The target is to predict whether a user likes a certain artist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The raw attribute of the artist node is the one- hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the DBLP dataset, the attributes of the target nodes are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the ACM and IMDB datasets, the target nodes have raw attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Implementations and Configurations of Baselines We use the HGB benchmark to evaluate the performance of all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' In HGB, implementations of baselines are based on their official codes to avoid errors introduced by re-implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Next, we present the configurations of baselines in the node classification and link prediction tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For brevity, we denote the dimension of node embedding as d, the dimension of edge embedding as de, the dimension of attention vector (if exists) as da, the number of GNN layers as L, the number of attention heads as nh, the negative slope of LeakyReLU as s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 1) Node Classification: The baselines in the node classifi- cation task contain HAN, GTN, HetSANN, MAGNN, HGCA, HGT, HetGNN, GCN, GAT, SimpleHGN, and HGNN-AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HAN: We set d = 8, da = 128, nh = 8, and L = 2 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GTN: The adaptive learning rate is employed for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We set d = 64 and the number of GTN channels to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP and ACM, we set L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we set L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HetSANN: For ACM, we set d = 64, L = 3, and nh = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we set d = 32, L = 2, and nh = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP, we set d = 64, L = 2, and nh = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' MAGNN: For DBLP and ACM, we set the batch size to 8, and the number of neighbor samples to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we use the full batch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HGCA: We set d = 64, the temperature parameter τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5, and the loss coefficient λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HGT: We use the layer normalization in each layer, and set d = 64 and nh = 8 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' L is set to 2, 3, 5 for ACM, DBLP and IMDB,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HetGNN: We set d = 128, and the batch size to 200 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For random walk, we set the walk length to 30 and the window size to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 3https://dblp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='uni-trier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='de/ 4http://dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='org/ 5https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='imdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='com (a) DBLP (b) ACM (c) IMDB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Performance comparison under different learning rates (a) DBLP (b) ACM (c) IMDB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Performance comparison under different weight decay values GCN: We set d = 64 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' We set L = 3 for DBLP and ACM, and L = 4 for IMDB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GAT: We set d = 64 and nh = 8 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP and ACM, we set s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='05 and L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we set s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='1 and L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' SimpleHGN: We set d = de = 64, nh = 8, and the edge residual β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='05 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP and ACM, we set L = 3 and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we set L = 6 and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HGNN-AC: We set d = 64, nh = 8, the divided ratio α of N + to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='3, and the loss weighted coefficient λ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 for all datasets, which are consistent with the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' 2) Link prediction: The baselines in the link prediction task contain GATNE, HetGNN, GCN, GAT, and SimpleHGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GATNE: We set d = 200, de = 10, and da = 20 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For the random walk, we set the walk length to 30 and the window size to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For neighbor sampling, we set the number of negative samples for optimization to 5 and the number of neighbor samples for aggregation to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' HetGNN: We set d = 128, and the batch size to 200 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For random walk, we set the walk length to 30 and the window size to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GCN: We set d = 64 and L = 2 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' GAT: For LastFM, We set d = 64, nh = 4, L = 3, and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP, we set d = 64, nh = 8, L = 3, and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we set d = 64, nh = 4, L = 5, and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' SimpleHGN: We set d = 64, de = 32, nh = 2, the edge residual β = 0, and s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='01 for all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For DBLP, we set L = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For LastFM, we set L = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' For IMDB, we set L = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Effects of the learning rate and the weight decay We further evaluate the effect of the learning rate and weight decay when optimizing the completion parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The available learning rates are set to [3e-3, 4e-3, 5e-3, 6e- 3, 7e-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The available weight decay values are set to [5e- 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2 Peformance(%) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 SimpleHGN-MacroE1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='8 SimpleHGN-Micro F1 MAGNN-Macro F1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='6 MAGNN-Macro F1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='4 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='007 The learning rate of94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 Peformance(%) SimpeHGN-MacroE1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 SimpleHGN-Micro F1 MAGNN-Macro F1 MAGNN-Macro F1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='007 The learning rate of68 66 Peformance(%) 64 SimpleHGN-Macro F1 SimpleHGN-Micro_F1 62 MAGNN-Macro F1 MAGNN-Macro F1 60 58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='007 The learning rate of95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='50 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='00 Peformance(%) SimpleHGN-Macro F1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='75 SimpleHGN-Micro_F1 MAGNN-Macro F1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='50 MAGNN-Macro F1 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='25 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='00 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 The weight decay of a 1e-594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 Peformance(%) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 SimpleHGN-Macro F1 SimpleHGN-Micro_F1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 MAGNN-Macro_F1 MAGNN-Macro F1 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 The weight decay of a 1e-568 66 Peformance(%) 64 SimpleHGN-Macro F1 SimpleHGN-Micro F1 62 MAGNN-Macro_F1 MAGNN-Macro F1 60 58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content='0 The weight decay of a 1e-56,1e-5, 2e-5, 3e-5, 4e-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' Figure 10 and Figure 11 show the performances of AutoAC with different learning rates and different weight decay, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' The green and blue lines represent SimpleHGN-AutoAC and MAGNN-AutoAC, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'} +page_content=' From Figure 10 and Figure 11, we can see that AutoAC is very robust to the learning rate and the weight decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE1T4oBgHgl3EQfRQMb/content/2301.03049v1.pdf'}